# Quantum walks on join graphs

Steve Kirkland,1 Hermie Monterde 1

###### Abstract

The join $X\vee Y$ of two graphs $X$ and $Y$ is the graph obtained by joining
each vertex of $X$ to each vertex of $Y$. We explore the behaviour of a
continuous quantum walk on a weighted join graph having the adjacency matrix
or Laplacian matrix as its associated Hamiltonian. We characterize strong
cospectrality, periodicity and perfect state transfer (PST) in a join graph.
We also determine conditions in which strong cospectrality, periodicity and
PST are preserved in the join. Under certain conditions, we show that there
are graphs with no PST that exhibits PST when joined by another graph. This
suggests that the join operation is promising in producing new graphs with
PST. Moreover, for a periodic vertex in $X$ and $X\vee Y$, we give an
expression that relates its minimum periods in $X$ and $X\vee Y$. While the
join operation need not preserve periodicity and PST, we show that
$\big{|}|U_{M}(X\vee
Y,t)_{u,v}|-|U_{M}(X,t)_{u,v}|\big{|}\leq\frac{2}{|V(X)|}$ for all vertices
$u$ and $v$ of $X$, where $U_{M}(X\vee Y,t)$ and $U_{M}(X,t)$ denote the
transition matrices of $X\vee Y$ and $X$ respectively relative to either the
adjacency or Laplacian matrix. We demonstrate that the bound
$\frac{2}{|V(X)|}$ is tight for infinite families of graphs.

Keywords: quantum walk, join graph, perfect state transfer, strong
cospectrality, adjacency matrix, Laplacian matrix

MSC2010 Classification: 05C50; 05C76; 05C22; 15A16; 15A18; 81P45;

11footnotetext: Department of Mathematics, University of Manitoba, Winnipeg,
MB, Canada R3T 2N2

##  1 Introduction

The graphs $K_{2}$, $C_{4}$ $P_{3}$, and $K_{4}\backslash e$ are well-known
examples of small graphs that admit adjacency or Laplacian perfect state
transfer. Kirkland et al. noticed that in these small graphs, the vertices
involved in perfect state transfer share the same neighbours, an observation
that prompted them to examine state transfer between twins [Kirkland2023].
However, it can also be observed that these small graphs are in fact join
graphs. This motivates our investigation on quantum walks on graphs built
using the join operation.

The first infinite family of join graphs revealed to admit Laplacian perfect
state transfer was $K_{n}\backslash e:=O_{2}\vee K_{n-2}$ (the complete graph
on $n$ vertices minus an edge) with $n\equiv 0$ (mod 4) [Bose2008]. Motivated
by this result, Angeles-Canul et al. gave sufficient conditions for adjacency
perfect state transfer to occur between the vertices of
$X\in\\{O_{2},K_{2}\\}$ in the unweighted graph $X\vee Y$, where $Y$ is a
regular graph [Angeles-Canul2010]. Such a graph is called a double cone on
$Y$, and the vertices of $X\in\\{O_{2},K_{2}\\}$ are called the apexes of the
double cone. They also determined sufficient conditions such that adjacency
perfect state transfer in $X$ is preserved under joins of copies of $X$. In a
subsequent paper, Angeles-Canul et al. investigated perfect state transfer in
weighted join graphs, and found that the apexes of a double cone on a
$k$-regular graph admit adjacency perfect state transfer by appropriate choice
of weights of an edge between the apexes and/or loops on the apexes [Angeles-
Canul2009]. More recently, Kirkland et al. fully characterized unweighted
double cones that admit adjacency perfect state transfer between apexes
[Kirkland2023]. For the Laplacian case, Alvir et al. showed that the apexes of
unweighted $O_{2}\vee Y$ admit perfect state transfer if and only if
$|V(Y)|\equiv 2$ (mod 4), while the apexes of unweighted $K_{2}\vee Y$ do not
admit perfect state transfer [Alvir2016]. Joins have also been investigated in
graphs with well-structured eigenbases [Johnston2017, mclaren2023weak], and in
the contexts of fractional revival [Monterde2023a, Chan2021], sedentariness
[Monterde2023] and strong cospectrality [Monterde2022].

Despite its widespread presence in literature, state transfer on join graphs
remains largely unexplored. In this paper, we provide a systematic study of
quantum walks on weighted join graphs having the adjacency and Laplacian
matrices as their associated Hamiltonian. In Section 3, we provide known
results about transition matrices and eigenvalue supports of vertices in join
graphs. In Section 4, we characterize periodic vertices in a join (Theorem 4),
and determine the conditions in which periodicity of a vertex in $X$ and
$X\vee Y$ are equivalent (Corollaries 7 and 9). We also use the join operation
to provide infinite families of graphs that are not integral (resp., Laplacian
integral) whereby each member graph contains periodic vertices (Corollary 12).
We devote Section 5 to finding a relationship between the minimum periods
$\rho_{X}$ and $\rho_{X\vee Y}$ of a vertex that is periodic in both $X$ and
$X\vee Y$, resp. (Theorems 14 and 15). We find that $\rho_{X\vee Y}$ is a
rational multiple of $\rho_{X}$, and if we add the extra hypothesis that $X$
is disconnected, then $\rho_{X\vee Y}$ is an integer multiple of $\rho_{X}$.
Section 6 provides a characterization of strong cospectrality in joins
(Theorem 20). It turns out that strong cospectrality between two vertices of
$X$ in $X\vee Y$ requires strong cospectrality in $X$, except for the case
when $X$ is an empty graph on two vertices. In fact, for the unweighted case,
strong cospectrality in $X$ and $X\vee Y$ are equivalent with a few exceptions
(Corollary 20). In Section 7, we characterize perfect state transfer in joins
(Theorems 23 and 26). Unlike strong cospectrality, perfect state transfer
between two vertices of $X$ in $X\vee Y$ does not require perfect state
transfer between them in $X$. This motivates us in Section 8 to determine
necessary and sufficient conditions such that perfect state transfer in $X$ is
preserved in $X\vee Y$ (Corollaries 28 and 35), and in Section 9 to determine
conditions such that perfect state transfer occurs in $X\vee Y$ given that it
does not occur in $X$ (Theorems 40 and 42, and Corollary 46). In Section 10,
we characterize strong cospectrality and PST in self-joins and iterated join
graphs. The latter result generalizes a previously known characterization of
perfect state transfer in threshold graphs. While the join operation does not
preserve periodicity and PST in general, we show in Section 11 that
$\big{|}|U_{M}(X\vee
Y,t)_{u,v}|-|U_{M}(X,t)_{u,v}|\big{|}\leq\frac{2}{|V(X)|}$ for all $t$ and for
all vertices $u$ and $v$ of $X$ (Corollaries 70 and 72). Consequently, the
behaviour of vertices in $X$ in the quantum walk on $X\vee Y$ mimics the
behaviour of the quantum walk on $X$ as $|V(X)|\rightarrow\infty$ (Corollary
73). We also show that this upper bound is tight for infinite families of
graphs (Examples 75 and 76). Finally, open problems are presented in Section
12.

##  2 Preliminaries

Throughout, we assume that a graph $X$ is weighted and undirected with
possible loops but no multiple edges. We denote the vertex set of $X$ by
$V(X)$, and we allow the edges of $X$ to have positive real weights. We say
that $X$ is simple if $X$ has no loops, and $X$ is unweighted if all edges of
$X$ have weight one. For $u\in V(X)$, we denote the the characteristic vector
of $u$ as $\textbf{e}_{u}$, which is a vector with a $1$ on the entry indexed
by $u$ and zeros elsewhere. The all-ones vector of order $n$, the zero vector
of order $n$, the $m\times n$ all-ones matrix, and the $n\times n$ identity
matrix are denoted by $\textbf{1}_{n}$, $\textbf{0}_{n}$, $\textbf{J}_{m,n}$
and $I_{n}$, respectively. If $m=n$, then we write $\textbf{J}_{m,n}$ as
$\textbf{J}_{n}$, and if the context is clear, then we simply write these
matrices as 1, 0, J and $I$, respectively. We also represent the transpose of
the matrix $H$ by $H^{T}$, and the characteristic polynomial of $H$ in the
variable $t$ by $\phi(H,t)$. Lastly, we denote the simple unweighted empty,
cycle, complete, and path graphs on $n$ vertices as $O_{n}$, $C_{n}$, $K_{n}$,
and $P_{n}$, respectively. We also denote the hypercube on $2^{p}$ vertices by
$Q_{p}$, and the cocktail party graph on $m$ vertices by $CP(m)$, where $m$ is
even.

The adjacency matrix $A(X)$ of $X$ is the matrix defined entrywise as

$A(X)_{u,v}=\begin{cases}\omega_{u,v},&\text{if $u$ and $v$ are adjacent}\\\
0,&\text{otherwise},\end{cases}$

where $\omega_{u,v}$ is the weight of the edge $(u,v)$. The degree matrix
$D(X)$ of $X$ is the diagonal matrix of vertex degrees of $X$, where
$\operatorname{deg}(u)=2\omega_{u,u}+\sum_{j\neq u}\omega_{u,j}$ for each
$u\in V(X)$. The Laplacian matrix of $X$ is the matrix $L(X)=D(X)-A(X)$. We
use $M(X)$ to denote $A(X)$ or $L(X)$. If the context is clear, then we write
$M(X)$, $A(X)$, $L(X)$ and $D(X)$ resp. as $M$, $A$, $L$ and $D$. We say that
$X$ is integral (resp., Laplacian integral) if all eigenvalues of $A(X)$
(resp., $L(X)$) are integers. We say that $X$ is weighted $k$-regular if
$\operatorname{deg}(u)-\omega_{u,u}$ is a constant $k$ for each $u\in V(X)$,
i.e., the row sums of $A(X)$ are a constant $k$. If $X$ is simple, then $X$ is
weighted $k$-regular if and only if $D(X)=kI$.

A (continuous-time) quantum walk on $X$ with respect to $H$ is determined by
the unitary matrix

$U_{H}(X,t)=e^{itH}.$ (1)

The matrix $H$ is called the Hamiltonian of the quantum walk. Typically, $H$
is taken to be $A$ or $L$, but any Hermitian matrix $H$ that respects the
adjacencies of $X$ works in general. Since $M\in\\{A,L\\}$ is real symmetric,
$M$ admits a spectral decomposition

$M=\sum_{j}\lambda_{j}E_{j},$ (2)

where the $\lambda_{j}$’s are the distinct real eigenvalues of $M$ and each
$E_{j}$ is the orthogonal projection matrix onto the eigenspace associated
with $\lambda_{j}$. If the eigenvalues are not indexed, then we also denote by
$E_{\lambda}$ the orthogonal projection matrix corresponding to the eigenvalue
$\lambda$ of $M$. Now, (2) allows us to write (1) as

$U_{M}(X,t)=\sum_{j}e^{it\lambda_{j}}E_{j}.$

We say that perfect state transfer (PST) occurs between two vertices $u$ and
$v$ if $|U_{M}(X,\tau)_{u,v}|=1$ for some time $\tau$. The minimum $\tau>0$
such that $|U_{M}(X,\tau)_{u,v}|=1$ is called the minimum PST time between $u$
and $v$. We say that $u$ is periodic if $|U_{M}(X,\tau)_{u,u}|=1$ for some
time $\tau$. The minimum $\tau>0$ such that $|U_{M}(X,\tau)_{u,u}|=1$ is
called the minimum period of $u$. These properties depend on the matrix $M$,
so we sometimes say adjacency PST (resp., periodicity) when $M=A$; similar
language applies when $M=L$. Further, if $X$ is simple, weighted and
$k$-regular, then
$U_{L}(X,t)=e^{itL}=e^{it(kI-A)}=e^{itk}e^{-itA}=e^{itk}U_{A}(X,-t)$. Hence,
$\left|U_{L}(X,\tau)_{u,v}\right|^{2}=\left|U_{A}(X,\tau)_{u,v}\right|^{2}$
for any $u,v\in V(X)$, so the quantum walks on $A$ and $L$ exhibit the same
state transfer properties.

##  3 Joins

The join $X\vee Y$ of weighted graphs $X$ and $Y$ is the graph obtained by
joining every vertex of $X$ with every vertex of $Y$ with an edge of weight
one. Unless otherwise stated, we make the following assumptions:

  * •

when dealing with $M=L,$ we take $X$ and $Y$ to be simple;

  * •

when dealing with $M=A,$ we take $X$ and $Y$ to be weighted $k$\- and
$\ell$-regular resp. (possibly with loops).

Denote the transition matrices of $X\vee Y$ and $X$ by $U_{M}(X\vee Y,t)$ and
$U_{M}(X,t)$ respectively. The transition matrix of a join with respect to $L$
and $A$ are known (see [Alvir2016, Fact 8] and [Coutinho2021, Lemma 4.3.1],
respectively). Throughout, $m$ and $n$ denote the number of vertices of $X$
and $Y$, respectively.

######  Lemma 1.

If $u,v\in V(X)$ and $w\in V(Y)$, then

$U_{L}(X\vee Y,t)_{u,w}=\frac{1}{m+n}\left(1-e^{it(m+n)}\right)$ (3)

and

$U_{L}(X\vee
Y,t)_{u,v}=e^{itn}U_{L}(X,t)_{u,v}+\frac{1}{m+n}+\frac{ne^{it(m+n)}}{m(m+n)}-\frac{e^{itn}}{m}.$
(4)

######  Lemma 2.

Let $D=(k-\ell)^{2}+4mn$ and
$\lambda^{\pm}=\frac{1}{2}\left(k+\ell\pm\sqrt{D}\right)$. If $u,v\in V(X)$
and $w\in V(Y)$, then

$U_{A}(X\vee
Y,t)_{u,w}=\frac{1}{\sqrt{D}}(e^{it\lambda^{+}}-e^{it\lambda^{-}})$ (5)

and

$U_{A}(X\vee
Y,t)_{u,v}=U_{A}(X,t)_{u,v}+\frac{e^{it\lambda^{+}}(k-\lambda^{-})}{m\sqrt{D}}-\frac{e^{it\lambda^{-}}(k-\lambda^{+})}{m\sqrt{D}}-\frac{e^{itk}}{m}.$
(6)

The eigenvalue support of $u$ with respect to $M$ is the set

$\sigma_{u}(M)=\\{\lambda_{j}:E_{j}\textbf{e}_{u}\neq\textbf{0}\\}.$

The following result determines the elements of the eigenvalue support of a
vertex in a join graph. In what follows, $\sigma_{u}(M(X))$ and
$\sigma_{u}(M)$ denote the eigenvalue supports of vertex $u$ in $X$ and $X\vee
Y$, respectively.

######  Lemma 3.

Let $u\in V(X)$, and define
$\mathcal{S}:=\\{\lambda+n:\lambda\in\sigma_{u}(L(X))\backslash\\{0\\}\\}$.

  1. 1.

We have $\sigma_{u}(L)=\mathcal{S}\cup\mathcal{R}$, where
$\mathcal{R}=\\{0,m+n\\}$ if $X$ is connected and $\mathcal{R}=\\{0,m+n,n\\}$
otherwise.

  2. 2.

We have $\sigma_{u}(A)=\sigma_{u}(A(X))\backslash\\{k\\}\cup\mathcal{R}$,
where $\mathcal{R}=\\{\lambda^{\pm}\\}$ if $X$ is connected and
$\mathcal{R}=\\{\lambda^{\pm},k\\}$ otherwise.

##  4 The ratio condition

A set $S\subseteq\mathbb{R}$ with at least two elements satisfies the ratio
condition if $\frac{\lambda-\theta}{\mu-\eta}\in\mathbb{Q}$ for all
$\lambda,\theta,\mu,\eta\in S$ with $\mu\neq\eta$. A vertex $u$ is periodic if
and only if $\sigma_{u}(M)$ satisfies the ratio condition [Coutinho2021,
Corollary 8.3.1]. Combining this fact with Lemma 3 yields a characterization
of periodicity in joins.

######  Theorem 4.

Let $X$ and $Y$ be weighted graphs and $u\in V(X)$.

  1. 1.

Vertex $u$ is Laplacian periodic in $X\vee Y$ if and only if all eigenvalues
in $\sigma_{u}(L(X))$ are rational. Moreover, if $u$ is Laplacian periodic in
$X\vee Y$, then it is Laplacian periodic in $X$.

  2. 2.

If $X$ is connected (resp., disconnected), then $u$ is adjacency periodic if
and only if $\frac{\lambda^{+}-\lambda}{\sqrt{D}}\in\mathbb{Q}$ for all
$\lambda\in\sigma_{u}(A(X))\backslash\\{k\\}$ (resp.,
$\lambda\in\sigma_{u}(A(X))$). If we assume further that
$k,\ell,\sqrt{D}\in\mathbb{Q}$, then $u$ is adjacency periodic in $X\vee Y$ if
and only if all eigenvalues in $\sigma_{u}(A(X))$ are rational.

The following example illustrates that the converse of the last statement in
Theorem 4(1) does not hold.

######  Example 5.

Consider $X\vee O_{1}$, where $X$ is a weighted $K_{2}$ with positive edge
weight $\eta\notin\mathbb{Q}$. Then $u\in V(X)$ is periodic in $X$ and
$\sigma_{u}(L)=\\{0,3,1+2\eta\\}$. Since
$\frac{1+2\eta-0}{3-0}\notin\mathbb{Q}$, we get that $u$ is not periodic in
$X\vee O_{1}$.

If $\phi(M(X),t)\in\mathbb{Z}[t]$, then we obtain another characterization of
periodicity [Coutinho2021, Corollary 7.6.2].

######  Lemma 6.

Let $S$ be a set of real algebraic integers that is closed under taking
algebraic conjugates. If $S$ satisfies the ratio condition, then either (i)
$S\subseteq\mathbb{Z}$ or (ii) each $\lambda_{j}$ in $S$ is a quadratic
integer of the form $\lambda_{j}=\frac{1}{2}(a+b_{j}\sqrt{\Delta})$, where
$a,b_{j},\Delta\in\mathbb{Z}$ and $\Delta>1$ is square-free. In particular, if
$X$ is a weighted graph with possible loops such that
$\phi(M(X),t)\in\mathbb{Z}[t]$, then vertex $u$ is periodic in $X$ if and only
if either (i) or (ii) holds for $S=\sigma_{u}(M(X))$. In this case,
$\rho=2\pi/g\sqrt{\Delta}$, where each $\lambda_{j}\in\sigma_{u}(M)$,
$g=\operatorname{gcd}\\{(\lambda_{1}-\lambda_{j})/\sqrt{\Delta}\\}$ and
$\Delta=1$ if (i) holds.

We now show that the converse of the last statement in Theorem 4(1) holds when
$\phi(L(X),t)\in\mathbb{Z}[t]$.

######  Corollary 7.

If $\phi(L(X),t)\in\mathbb{Z}[t]$, then the following statements are
equivalent.

  1. 1.

Vertex $u$ is Laplacian periodic in $X$.

  2. 2.

Vertex $u$ is Laplacian periodic in $X\vee Y$ for any simple positively
weighted graph $Y$.

  3. 3.

The eigenvalues in $\sigma_{u}(L(X))$ are all integers.

###### Proof.

Lemma 3(1) and the ratio condition yields (3) implies (2). The second
statement of Theorem 4(1) yields (2) implies (1). Lemma 6(i) and the fact that
$0\in\sigma_{u}(L(X))$ yield (1) implies (3). ∎

Next, we have the following result concerning the adjacency matrix in relation
to Theorem 4(2).

######  Corollary 8.

Let $u\in V(X)$, $\phi(A(X),t)\in\mathbb{Z}[t]$ and $k,\ell\in\mathbb{Z}$.
Consider $D$ in Lemma 2.

  1. 1.

If $X$ is connected, then $u$ is adjacency periodic in $X\vee Y$ if and only
if either:

    1. (a)

The eigenvalues in $\sigma_{u}(A(X))$ are all integers and $D$ is a perfect
square (so that $\lambda^{\pm}\in\mathbb{Z}$). In this case, $u$ is also
adjacency periodic in $X$.

    2. (b)

Each $\lambda_{j}\in\sigma_{u}(A(X))\backslash\\{k\\}$ is a quadratic integer
of the form $\frac{1}{2}(k+\ell+b_{j}\sqrt{\Delta})$, where $\Delta>1$ is
square-free, $\sqrt{D/\Delta}\in\mathbb{Z}$ and $\sqrt{D/\Delta}>b_{j}$ for
each $j$. In this case, $u$ is not periodic in $X$.

  2. 2.

If $X$ is disconnected, then $u$ is adjacency periodic in $X\vee Y$ if and
only if condition (1a) holds.

###### Proof.

To prove (1), suppose $X$ is connected. By Lemma 3(2),
$\sigma_{u}(A)=\sigma_{u}(A(X))\backslash\\{k\\}\cup\\{\lambda^{\pm}\\}$, and
$u$ is periodic in $X\vee Y$ if and only if $\sigma_{u}(A)$ satisfies the
ratio condition. Since $k\in\mathbb{Z}$, the latter condition implies that
$\sigma_{u}(A(X))\backslash\\{k\\}$ satisfies the ratio condition, and as
$\phi(A(X),t)\in\mathbb{Z}[t]$, we get two cases from Lemma 6. First, let
$\sigma_{u}(A(X))\backslash\\{k\\}\subseteq\mathbb{Z}$. Since
$\lambda^{+}=\frac{1}{2}(k+\ell+\sqrt{D})$, we get
$\frac{\lambda^{+}-\lambda}{\sqrt{D}}\in\mathbb{Q}$ if and only if
$\sqrt{D}\in\mathbb{Z}$. Combining this with Theorem 4(2) yields (1a). Next,
suppose each $\lambda_{j}\in\sigma_{u}(A(X))\backslash\\{k\\}$ is of the form
$\frac{1}{2}(a+b_{j}\sqrt{\Delta})$ so that
$\lambda^{+}-\lambda_{j}=\frac{1}{2}\left((k+\ell-a)+(\sqrt{D}-b_{j}\sqrt{\Delta})\right)$.
Then $\frac{\lambda^{+}-\lambda_{j}}{\sqrt{D}}\in\mathbb{Q}$ if and only if
$k+\ell-a=0$ and $\sqrt{\Delta/D}\in\mathbb{Q}$. Now, Perron-Frobenius Theorem
yields $\lambda^{+}=\frac{1}{2}(a+\sqrt{D})$ as the largest eigenvalue of
$A(X\vee Y)$. Thus, $\sqrt{D}>\sqrt{\Delta}$. Since $\sigma_{u}(A)$ satisfies
the ratio condition, we get $\sqrt{D}=b\sqrt{\Delta}$ for some integer $b$,
and so $\sqrt{D/\Delta}=b>b_{j}$ for each $j$. This establishes (1b).

To prove (2), suppose $X$ is disconnected. In this case,
$k\in\sigma_{u}(A(X))$ is an integer, and so we apply Lemma 4(2) to conclude
that $u$ is periodic in $X\vee Y$ if and only if
$\sigma_{u}(A(X))\subseteq\mathbb{Z}$ and $\sqrt{D}\in\mathbb{Q}$. ∎

We now state an analogue of Corollary 7, which is an immediate consequence of
Corollary 8.

######  Corollary 9.

Let $\phi(A(X\vee Y),t)\in\mathbb{Z}[t]$ and $k,\ell\in\mathbb{Z}$. If $u$ is
adjacency periodic in $X$, then $u$ is adjacency periodic in $X\vee Y$ if and
only if $D$ is a perfect square. Moreover, if $D$ is a perfect square, then
the following are equivalent.

  1. 1.

Vertex $u$ is adjacency periodic in $X$.

  2. 2.

Vertex $u$ is adjacency periodic in $X\vee Y$ for any weighted $\ell$-regular
graph $Y$.

  3. 3.

All eigenvalues in $\sigma_{u}(A(X))$ are integers.

######  Remark 10.

For $L$, our assumption on $X$ and $Y$ implies that the join operation
preserves periodicity, and periodicity in the join is inherited by the
underlying graph. However, this does not hold for $A$ given our assumption on
$X$ and $Y$. One can have $\sigma_{u}(A(X))\subseteq\mathbb{Z}$ but
$\sqrt{D}\notin\mathbb{Z}$, and so $u$ is periodic in $X$ but not in $X\vee
Y$. Conversely, periodicity in $X\vee Y$ is not necessarily inherited by $X$
by Corollary 4.

We say that $X$ is periodic if there is a $\tau>0$ such that $U(\tau)=\gamma
I$ for some unit $\gamma\in\mathbb{C}$. Equivalently, each $u\in V(X)$ is
periodic at time $\tau$. We characterize periodic join graphs under mild
conditions.

######  Theorem 11.

Let $X$ and $Y$ be weighted graphs and $\phi(M(X\vee Y),t)\in\mathbb{Z}[t]$.

  1. 1.

If $X$ and $Y$ are simple, then $X\vee Y$ is Laplacian periodic if and only if
$X\vee Y$ is Laplacian integral.

  2. 2.

If $X$ and $Y$ are regular, then $X\vee Y$ is adjacency periodic if and only
if $X\vee Y$ is integral.

###### Proof.

Note that (1) follows from the ratio condition and the fact that $L(X\vee Y)$
is positive semidefinite with $0$ as an eigenvalue. Next, let $X$ and $Y$ be
$k$\- and $\ell$-regular. By way of contradiction, suppose each eigenvalue
$\lambda_{j}$ of $A(X\vee Y)$ is a quadratic integer. Since
$\lambda^{\pm}=\frac{1}{2}(k+\ell\pm\sqrt{D})$ is an eigenvalue of $A(X\vee
Y)$, each $\lambda_{j}$ must be of the form
$\frac{1}{2}(k+\ell+b_{j}\sqrt{\Delta})$. Without loss of generality, let
$k\geq\ell>0$. For each eigenvalue $\lambda_{j}\neq\ell$ of $A(Y)$, we get
$\lambda_{j}=\frac{1}{2}(k+\ell+b_{j}\sqrt{\Delta})\geq\ell+\frac{1}{2}b_{j}\sqrt{\Delta}$
for all each $j$. Similarly,
$\overline{\lambda_{j}}\geq\ell-\frac{1}{2}b_{j}\sqrt{\Delta}$ for all each
$j$. Thus, $\lambda_{j}>\ell$ or $\overline{\lambda_{j}}>\ell$, which cannot
happen because $\ell$ is the largest eigenvalue of $A(Y)$ by the Perron-
Frobenius Theorem. Invoking Lemma 6 establishes (2). ∎

We close this section by providing infinite families of graphs that are not
(Laplacian) integral but contain periodic vertices. The following result is
immediate from Lemma 3 and the ratio condition.

######  Corollary 12.

Let $X$ and $Y$ be weighted graphs, and let $u\in V(X)$.

  1. 1.

If $X$ is Laplacian integral but $Y$ is not, then $X\vee Y$ is not Laplacian
integral, the eigenvalues in $\sigma_{u}(L(X))$ are all integers, and $u$ is
periodic in $X\vee Y$.

  2. 2.

If $X$ is integral but $Y$ is not, then $X\vee Y$ is not integral. Moreover,
if $k,\ell\in\mathbb{Z}$ and $D$ in Lemma 2 is a perfect square, then the
eigenvalues in $\sigma_{u}(A(X))$ are all integers and $u$ is periodic in
$X\vee Y$.

##  5 Minimum periods

Denote the minimum periods of $u$ in $X\vee Y$ and $X$ by $\rho_{X\vee Y}$ and
$\rho_{X}$ respectively. Motivated by Corollaries 7 and 9, we ask: if $u\in
V(X)$ is periodic in $X$ and $X\vee Y$, then how are $\rho_{X\vee Y}$ and
$\rho_{X}$ related? To answer this question, we state a result due to Kirkland
et al. [Kirkland2023, Theorem 4].

######  Theorem 13.

Let $u$ and $v$ be vertices in $X$, and suppose
$\sigma_{u}(M)=\\{\lambda_{1},\ldots,\lambda_{r}\\}$. If $u$ is periodic, then
$\rho=\frac{2\pi q}{\lambda_{1}-\lambda_{2}}$, where
$q=\operatorname{lcm}(q_{2},\ldots,q_{r})$ and each $q_{j}$ is a positive
integer such that
$\frac{\lambda_{1}-\lambda_{j}}{\lambda_{1}-\lambda_{2}}=\frac{p_{j}}{q_{j}}$
for some integer $p_{j}$ satisfying $\text{gcd}(p_{j},q_{j})=1$. In
particular, if $r=2$, then $u$ is periodic with
$\rho=\frac{2\pi}{\lambda_{1}-\lambda_{2}}$.

We now address our above question by finding a constant $c>0$ such that
$\rho_{X\vee Y}=c\rho_{X}$.

######  Theorem 14.

Let $u$ be a non-isolated and periodic vertex in $X$ and $X\vee Y$ with
$\sigma_{u}(L(X))=\\{\lambda_{1},\ldots,\lambda_{r},\lambda_{r+1}\\}$, where
$\lambda_{r+1}=0$. Let $\lambda_{r+2}:=m$ and $\lambda_{r+3}:=-n$.

  1. 1.

Suppose $X$ is connected.

    1. (a)

If $r=1$, then $\rho_{X\vee
Y}=\left(\frac{\lambda_{1}q}{\lambda_{1}+n}\right)\rho_{X}$, where $q$ is a
positive integer such that $\frac{m+n}{\lambda_{1}+n}=\frac{p}{q}$ for some
integer $p$ with $\operatorname{gcd}(p,q)=1$. In particular, if
$\lambda_{1}=m$, then $q=1$.

    2. (b)

If $r=2$ and $\lambda_{2}=m$, then $\rho_{X\vee
Y}=\left(\frac{\lambda_{1}q}{q^{\prime}(\lambda_{1}+n)}\right)\rho_{X}$, where
$q$ is a positive integer defined in (a) and $q^{\prime}$ is a positive
integer such that $m/\lambda_{1}=p^{\prime}/q^{\prime}$ for some integer
$p^{\prime}$ with $\operatorname{gcd}(p^{\prime},q^{\prime})=1$.

    3. (c)

If $r\geq 2$ and $\lambda_{j}\neq m$ for each $j\in\\{1,\ldots,r\\}$, then

$\rho_{X\vee
Y}=\left(\frac{q_{r+2}q_{r+3}\operatorname{gcd}(R_{1},q_{r+1})}{q_{r+1}\operatorname{gcd}(R_{1},q_{r+2})\operatorname{gcd}(R_{2},q_{r+3})}\right)\rho_{X},$
(7)

where $R_{1}=\operatorname{lcm}(q_{3},\ldots,q_{r})$,
$R_{2}=\operatorname{lcm}(q_{2},\ldots,q_{r},q_{r+2})$, the $q_{j}$’s are
positive integers such that
$\frac{\lambda_{1}-\lambda_{j}}{\lambda_{1}-\lambda_{2}}=\frac{p_{j}}{q_{j}}$
for some integers $p_{j}$ with $\operatorname{gcd}(p_{j},q_{j})=1$ and
$\lambda_{1}>\lambda_{2}$.

    4. (d)

If $r\geq 3$ and $\lambda_{r}=m$, then (7) holds with
$R_{1}=\operatorname{lcm}(q_{3},\ldots,q_{r-1})$,
$R_{2}=\operatorname{lcm}(q_{3},\ldots,q_{r-1},q_{r+2})$.

  2. 2.

Suppose $X$ is disconnected.

    1. (a)

If $r=1$, then $\rho_{X\vee Y}=q\rho_{X}$, where
$q=\operatorname{lcm}(q_{3},q_{4})$ and the $q_{j}$’s are positive integers
such that
$\frac{\lambda_{1}-\lambda_{j}}{\lambda_{1}-\lambda_{2}}=\frac{p_{j}}{q_{j}}$
for some $p_{j}$ with $\operatorname{gcd}(p_{j},q_{j})=1$. In particular, if
$\lambda_{1}=m$, then $q=\frac{m}{\operatorname{gcd}(m,n)}$.

    2. (b)

If $r=2$ and $\lambda_{2}=m$, then $\rho_{X\vee Y}=(q/q_{4})\rho_{X}$, where
$q=\operatorname{lcm}(q_{4},q_{5})$ and the $q_{j}$’s are positive integers
such that $\frac{\lambda_{1}-\lambda_{j}}{\lambda_{1}}=\frac{p_{j}}{q_{j}}$
for some integers $p_{j}$ with $\operatorname{gcd}(p_{j},q_{j})=1$.

    3. (c)

If $r\geq 2$ and $\lambda_{j}\neq m$ for each $j\in\\{1,\ldots,r\\}$, then

$\rho_{X\vee
Y}=\left(\frac{q_{r+2}q_{r+3}}{\operatorname{gcd}(q,q_{r+2})\operatorname{gcd}(R,q_{r+3})}\right)\rho_{X},$
(8)

where $R=\operatorname{lcm}(q_{3},\ldots,q_{r+2})$, the $q_{j}$’s are positive
integers such that
$\frac{\lambda_{1}-\lambda_{j}}{\lambda_{1}-\lambda_{2}}=\frac{p_{j}}{q_{j}}$
for some integers $p_{j}$ with $\operatorname{gcd}(p_{j},q_{j})=1$ and
$\lambda_{1}>\lambda_{2}$. Further, the coefficient of $\rho_{X}$ in (8) is an
integer.

    4. (d)

If $r\geq 3$ and $\lambda_{r}=m$, then

$\rho_{X\vee Y}=(Q/q)\rho_{X},$ (9)

where $q=\operatorname{lcm}(q_{3},\ldots,q_{r},q_{r+1})$,
$Q=\operatorname{lcm}(q_{3},\ldots,q_{r-1},q_{r+1},q_{r+2},q_{r+3})$ and the
$q_{j}$’s are positive integers satisfying
$\frac{\lambda_{1}-\lambda_{j}}{\lambda_{1}}=\frac{p_{j}}{q_{j}}$ and
$\operatorname{gcd}(p_{j},q_{j})=1$. In this case, $Q/q$ is an integer.

###### Proof.

Let $X$ be connected. By Lemma 3(1),
$\sigma_{u}(L)=\\{\lambda_{1}+n,\ldots,\lambda_{r}+n\\}\cup\\{\lambda_{r+2}+n,\lambda_{r+3}+n\\}$.
Note that $\lambda_{j}\neq\lambda_{r+3}$ for each $j\in\\{1,\ldots,r\\}$.
However, it is possible that $\lambda_{j}=\lambda_{r+2}=m$ for some
$j\in\\{1,\ldots,r\\}$. In this case, we may instead assume that
$\lambda_{r}=m$. We have the following cases.

  * •

Let $r=1$. Then $\sigma_{u}(L(X))=\\{\lambda_{1},0\\}$ and
$\sigma_{u}(L)=\\{\lambda_{1}+n,m+n,0\\}$, and so Theorem 13 yields
$\rho_{X}=2\pi/\lambda_{1}$ and $\rho_{X\vee Y}=2\pi q/(\lambda_{1}+n)$, where
$q=1$ whenever $\lambda_{1}=m$.

  * •

Let $r=2$ and $\lambda_{r}=m$. Then $\sigma_{u}(L)=\\{\lambda_{1}+n,m+n,0\\}$,
and so $\rho_{X\vee Y}=2\pi q/(\lambda_{1}+n)$ as in the previous case. Since
$\sigma_{u}(L(X))=\\{\lambda_{1},m,0\\}$, Theorem 13 yields $\rho_{X}=2\pi
q^{\prime}/\lambda_{1}$.

  * •

Let $r\geq 2$ and $\lambda_{j}\neq m$ for each $j$. As
$\sigma_{u}(L)=\\{\lambda_{1}+n,\ldots,\lambda_{r}+n,\lambda_{r+2}+n,\lambda_{r+3}+n\\}$
and $u$ is periodic in $X$ and $X\vee Y$, Theorem 13 yields $\rho_{X}=2\pi
q/(\lambda_{1}-\lambda_{2})$ and $\rho_{X\vee Y}=2\pi
Q/(\lambda_{1}-\lambda_{2})$, where
$q=\operatorname{lcm}(q_{3},\ldots,q_{r+1})$,
$Q=\operatorname{lcm}(q_{3},\ldots,q_{r},q_{r+2},q_{r+3})$ and
$\lambda_{1}>\lambda_{2}$. Since
$\operatorname{lcm}(a,b,c)=\operatorname{lcm}(\operatorname{lcm}(a,b),c)$ and
$\operatorname{lcm}(a,b)=\frac{ab}{\operatorname{gcd}(a,b)}$, we may write
$q=\frac{R_{1}q_{r+1}}{\operatorname{gcd}(R_{1},q_{r+1})}$ and

$Q=\frac{R_{1}q_{r+2}q_{r+3}}{\operatorname{gcd}(R_{1},q_{r+2})\operatorname{gcd}(R_{2},q_{r+3})}=qc,$
(10)

where $c$ is the coefficient of $\rho_{X}$ in (7). This implies that
$\rho_{X\vee Y}=c\rho_{X}$.

  * •

Let $r\geq 3$ and $\lambda_{r}=m$. Then
$\sigma_{u}(L)=\\{\lambda_{1}+n,\ldots,\lambda_{r-1}+n,\lambda_{r+2},\lambda_{r+3}\\}$,
and so the same argument in previous subcase yields (10) with
$R_{1}=\operatorname{lcm}(q_{3},\ldots,q_{r-1})$ and
$R_{2}=\operatorname{lcm}(q_{3},\ldots,q_{r-1},q_{r+2})$.

Combining the above cases proves (1).

Next, let $X$ be disconnected. By Lemma 3(1),
$\sigma_{u}(L)=\\{\lambda_{1}+n,\ldots,\lambda_{r}+n\\}\cup\\{\lambda_{r+1}+n,\lambda_{r+2}+n,\lambda_{r+3}+n\\}$.
Moreover, the fact that $u$ is a non-isolated vertex yields $r\geq 1$. We have
the following cases.

  * •

Let $r=1$. Then $\sigma_{u}(L(X))=\\{\lambda_{1},0\\}$ and
$\sigma_{u}(L)=\\{\lambda_{1}+n,\lambda_{2}+n,\lambda_{3}+n,\lambda_{4}+n\\}$.
By Theorem 13, $\rho_{X}=2\pi/\lambda_{1}$ and $\rho_{X\vee Y}=2\pi
q/\lambda_{1}$. In particular, if $\lambda_{1}=m$, then $q_{1}=q_{3}$, and so
$q=q_{4}$.

  * •

Let $r=2$ and $\lambda_{r}=m$. Then
$\sigma_{u}(L)=\\{\lambda_{1}+n,\lambda_{3}+n,\lambda_{4}+n,\lambda_{5}+n\\}$
and $\sigma_{u}(L(X))=\\{\lambda_{1},m,0\\}$. By Theorem 13, we get
$\rho_{X\vee Y}=\frac{2\pi q}{\lambda_{1}-\lambda_{3}}=\frac{2\pi
q}{\lambda_{1}}$ and $\rho_{X}=\frac{2\pi q^{\prime}}{\lambda_{1}}$, where
$p^{\prime},q^{\prime}\in\mathbb{Z}$ such that
$\frac{m}{\lambda_{1}}=\frac{p^{\prime}}{q^{\prime}}\in\mathbb{Q}$ and
$\operatorname{gcd}(p^{\prime},q^{\prime})=1$. Since
$\frac{p_{4}}{q_{4}}=\frac{\lambda_{1}-\lambda_{4}}{\lambda_{1}-\lambda_{3}}=\frac{\lambda_{1}-m}{\lambda_{1}}=1-\frac{p^{\prime}}{q^{\prime}}$,
we get $q_{4}=q^{\prime}$.

  * •

Let $r\geq 2$ and $\lambda_{j}\neq m$ for each $j$. Then
$\sigma_{u}(L)=\\{\lambda_{1}+n,\ldots,\lambda_{r}+n,\lambda_{r+1}+n,\lambda_{r+2}+n,\lambda_{r+3}+n\\}$.
By Theorem 13, $\rho_{X\vee Y}=2\pi Q/(\lambda_{1}-\lambda_{2})$ where
$Q=\operatorname{lcm}(q_{3},\ldots,q_{r+3})$ and $q_{j}$’s are as defined in
(1c). The same argument then yields
$Q=\frac{qq_{r+2}q_{r+3}}{\operatorname{gcd}(q,q_{r+2})\operatorname{gcd}(R,q_{r+3})}=qc,$
where $q=\operatorname{lcm}(q_{3},\ldots,q_{r+1})$.

  * •

Let $r\geq 3$ and $\lambda_{r}=m$. Since
$\sigma_{u}(L)=\\{\lambda_{1}+n,\ldots,\lambda_{r-1}+n,\lambda_{r+1}+n,\lambda_{r+2}+n,\lambda_{r+3}+n\\}$,
we get $\rho_{X}=2\pi q/\lambda_{1}$ and $\rho_{X\vee Y}=2\pi Q/\lambda_{1}$
by Theorem 13. Now, since
$\frac{\lambda_{1}-\lambda_{r}}{\lambda_{1}-\lambda_{r+1}}=\frac{\lambda_{1}-m}{\lambda_{1}}=\frac{p_{r}}{q_{r}}$
and
$\frac{\lambda_{1}-\lambda_{r}}{\lambda_{1}-\lambda_{r+2}}=\frac{\lambda_{1}-m}{\lambda_{1}}=\frac{p_{r+2}}{q_{r+2}}$,
we get $q_{r}=q_{r+2}$, and so
$Q=\operatorname{lcm}(q_{2},\ldots,q_{r+1},q_{r+3})$. Thus,
$Q/q\in\mathbb{Z}$.

Combining the above cases proves (2). ∎

Using the same argument, we get an analogue of Theorem 14 for the adjacency
case.

######  Theorem 15.

Let $u$ be a non-isolated and periodic vertex in $X$ and $X\vee Y$ with
$\sigma_{u}(A(X))=\\{\lambda_{1},\ldots,\lambda_{r},\lambda_{r+1}\\}$, where
$\lambda_{r+1}=k$. Let $\lambda_{r+2}:=\lambda^{-}$ and
$\lambda_{r+3}:=\lambda^{+}$, where $\lambda^{\pm}$ are defined in Lemma 2.

  1. 1.

Suppose $X$ is connected.

    1. (a)

Let $r=1$. Then $\rho_{X\vee
Y}=\left(\frac{q(k-\lambda_{1})}{\sqrt{D}}\right)\rho_{X}$, where $q$ is a
positive integer such that
$\frac{\lambda^{+}-\lambda_{1}}{\sqrt{D}}=\frac{p}{q}$ for some integer $p$
with $\operatorname{gcd}(p,q)=1$. In particular, if $\lambda_{1}=\lambda^{-}$,
then $q=1$.

    2. (b)

If $r=2$ and $\lambda_{2}=\lambda^{-}$, then $\rho_{X\vee
Y}=\left(\frac{(\lambda_{1}-\lambda^{-})q}{q^{\prime}\sqrt{D}}\right)\rho_{X}$,
where $q$ is an integer in (1b) and $q^{\prime}$ is a positive integer such
that
$\frac{\lambda_{1}-k}{\lambda_{1}-\lambda^{-}}=\frac{p^{\prime}}{q^{\prime}}\in\mathbb{Q}$
for some integer $p^{\prime}$ with
$\operatorname{gcd}(p^{\prime},q^{\prime})=1$.

    3. (c)

If $r\geq 2$ and $\lambda_{j}\neq\lambda^{-}$ for each $j\in\\{1,\ldots,r\\}$,
then the conclusion in Theorem 14(1d) holds.

    4. (d)

If $r\geq 3$ and $\lambda_{r}=\lambda^{-}$, then the conclusion in Theorem
14(1e) holds.

  2. 2.

Suppose $X$ is disconnected.

    1. (a)

Let $r=1$. Then $\rho_{X\vee Y}=q\rho_{X}$, where
$q=\operatorname{lcm}(q_{3},q_{4})$ and the $q_{j}$’s are positive integers
such that $\frac{k-\lambda_{j}}{k-\lambda_{1}}=\frac{p_{j}}{q_{j}}$ for some
$p_{j}$ with $\operatorname{gcd}(p_{j},q_{j})=1$. In particular, if
$\lambda_{1}=\lambda^{-}$, then $q=q_{4}$.

    2. (b)

If $r=2$ and $\lambda_{2}=\lambda^{-}$, then $\rho_{X\vee
Y}=\left(q/q_{2}\right)\rho_{X}$, where $q=\operatorname{lcm}(q_{2},q_{5})$,
the $q_{j}$’s are positive integers such that
$\frac{k-\lambda_{j}}{k-\lambda_{1}}=\frac{p_{j}}{q_{j}}$ for some integers
$p_{j}$ with $\operatorname{gcd}(p_{j},q_{j})=1$ and
$\lambda_{1}>\lambda_{2}$.

    3. (c)

If $r\geq 2$ and $\lambda_{j}\neq\lambda^{-}$ for each $j\in\\{1,\ldots,r\\}$,
then the conclusion in Theorem 14(2d) holds.

    4. (d)

If $r\geq 3$ and $\lambda_{r}=\lambda^{-}$, then the conclusion in Theorem
14(2e) holds.

######  Remark 16.

The advantage of Theorems 14-15 is that they apply even if
$\phi(M(X),t)\notin\mathbb{Z}[t]$. We also mention that the constant $c>0$
such that $\rho_{X\vee Y}=c\rho_{X}$ in Theorems 14-15 is always rational. In
fact, $c$ is an integer whenever (i) $X$ is disconnected or (ii) $X$ is
connected and $q_{r+1}=1$ in Theorems 14-15(1c-d).

For the family of complete graphs, Theorem 14(1a) yields $\rho_{X\vee
Y}=\frac{m}{m+n}\rho_{X}$, where $X=K_{m}$ and $Y=K_{n}$. Here,
$c=\frac{m}{m+n}<1$ is rational, and this holds for $M\in\\{A,L\\}$. We
complement this observation by providing an infinite family of connected
graphs $X$ such that $\rho_{X\vee Y}=c\rho_{X}$ for some integer $c\geq 1$.

######  Example 17.

Suppose that $m$ is even and let $X=K_{\frac{m}{2},\frac{m}{2}}$. Since
$\sigma_{u}(L(X))=\\{m/2,m,0\\}$, each $u\in V(X)$ is periodic with
$\rho_{X}=4\pi/m$. Consider $X\vee Y$, where $Y$ is a simple unweighted graph
on $n$ vertices. Applying Theorem 14(1b), we get $\rho_{X\vee Y}=c\rho_{X}$,
where $c=m/2g$ and $g=\operatorname{gcd}(m/2,n)$. Since $g$ divides $m/2$, we
conclude that $c$ is an integer.

######  Example 18.

Let $X=K_{m}$ and $Y$ be a simple unweighted $\ell$-regular graph on $n$
vertices. Each $u\in V(X)$ is periodic with $\rho_{X}=2\pi/m$. By Lemma 3(2),
$\sigma_{u}(A)=\\{\lambda^{\pm},-1\\}$, where
$\lambda^{\pm}=\frac{1}{2}(\ell+m-1\pm\sqrt{D})$. By Corollary 9, $u$ periodic
in $X\vee Y$ if and only if $D=(\ell-m+1)^{2}+4mn$ is a perfect square.
Equivalently, $n=\frac{s(\ell-m+s+1)}{m}$ for some integer $s$ such that $m$
divides $s(\ell+s+1)$. Thus, if $u$ periodic in $X\vee Y$, then
$\frac{\lambda^{+}-(-1)}{\lambda^{+}-\lambda^{-}}=\frac{\ell+s+1}{\ell-m+2s+1}\in\mathbb{Q}$,
from which we get $\rho_{X\vee Y}=2\pi/g$, where
$g=\operatorname{gcd}(\ell+s+1,s-m)$. Hence $\rho_{X\vee Y}=(m/g)\rho_{X}$.
Taking $s=2m$ and $0\leq\ell\leq n-2$ yields $g=\operatorname{gcd}(\ell+1,m)$,
and so $m/g$ is an integer.

##  6 Strong cospectrality

We say that two vertices $u$ and $v$ in a weighted graph $X$ are strongly
cospectral if $E_{j}\textbf{e}_{u}=\pm E_{j}\textbf{e}_{v}$ for each $j$, in
which case we define the sets

$\sigma_{uv}^{+}(M)=\\{\lambda_{j}:E_{j}\textbf{e}_{u}=E_{j}\textbf{e}_{v}\neq\textbf{0}\\}\quad\text{and}\quad\sigma_{uv}^{-}(M)=\\{\lambda_{j}:E_{j}\textbf{e}_{u}=-E_{j}\textbf{e}_{v}\neq\textbf{0}\\}.$
(11)

Equivalently, vertices $u$ and $v$ are strongly cospectral if for each $j$,
either every eigenvector w associated with $\lambda_{j}$ satisfies
$\textbf{w}^{T}\textbf{e}_{u}=\textbf{w}^{T}\textbf{e}_{v}$ or every
eigenvector w associated with $\lambda_{j}$ satisfies
$\textbf{w}^{T}\textbf{e}_{u}=-\textbf{w}^{T}\textbf{e}_{v}$.

If vertices $u$ and $v$ are strongly cospectral, then
$\sigma_{u}(M)=\sigma_{uv}^{+}(M)\cup\sigma_{uv}^{-}(M)$, and $u$ and $v$
belong to the same connected component of $X$ (so neither of them is
isolated). In order to avoid confusion, if $u$ and $v$ are strongly cospectral
in $X$ and $X\vee Y$, then we write the sets in (11) as
$\sigma_{uv}^{\pm}(M(X))$ and $\sigma_{uv}^{\pm}(M)$, respectively.

Strong cospectrality is a necessary condition for PST. Hence, in order to
investigate PST in joins, we first need to characterize strong cospectrality.
In what follows, we let $O_{n}(k)$ denote the empty graph on $n$ vertices with
a loop of weight $k$ on each vertex. If $k=0$, then we simply write $O_{n}(k)$
as $O_{n}$.

######  Theorem 19.

Let $m\geq 2$ and $u,v\in V(X)$ with $u\neq v$.

  1. 1.

Vertices $u$ and $v$ in $X$ are Laplacian strongly cospectral in $X\vee Y$ if
and only if either:

    1. (a)

Vertices $u$ and $v$ are Laplacian strongly cospectral in $X$ and
$m\notin\sigma_{uv}^{-}(L(X))$. In this case,

$\sigma_{uv}^{+}(L)=\\{\lambda+n:0\neq\lambda\in\sigma_{uv}^{+}(L(X))\\}\cup\mathcal{R}\quad\text{and}\quad\sigma_{uv}^{-}(L)=\\{\mu+n:\mu\in\sigma_{uv}^{-}(L(X))\\}$

where $\mathcal{R}=\\{0,m+n\\}$ if $X$ is connected and
$\mathcal{R}=\\{0,m+n,n\\}$ otherwise.

    2. (b)

$X=O_{2}$. In this case, $\sigma_{uv}^{+}(L)=\\{0,n+2\\}$ and
$\sigma_{uv}^{-}(L)=\\{n\\}$.

  2. 2.

Vertices $u$ and $v$ in $X$ are adjacency strongly cospectral in $X\vee Y$ if
and only if either:

    1. (a)

Vertices $u$ and $v$ are adjacency strongly cospectral in $X$ and
$\lambda^{\pm}\notin\sigma_{uv}^{-}(L(X))$. In this case,

$\sigma_{uv}^{+}(A)=\sigma_{uv}^{+}(A(X))\backslash\\{k\\}\cup\mathcal{R}\quad\text{and}\quad\sigma_{uv}^{-}(A)=\sigma_{uv}^{-}(A(X))$

where $\mathcal{R}=\\{\lambda^{\pm}\\}$ if $X$ is connected and
$\mathcal{R}=\\{\lambda^{\pm},k\\}$ otherwise.

    2. (b)

$X=O_{2}(k)$. In this case, $\sigma_{uv}^{+}(L)=\\{\lambda^{\pm}\\}$ and
$\sigma_{uv}^{-}(L)=\\{k\\}$.

  3. 3.

If $w\in V(Y)$, then vertices $u$ and $w$ are not strongly cospectral in
$X\vee Y$.

Moreover, if (1a) or (2a) holds, then vertices $u$ and $v$ belong to the same
connected component in $X$.

###### Proof.

Let $\lambda$ and $\mu$ be nonzero eigenvalues of $L(X)$ and $L(Y)$ with
associated eigenvectors $\textbf{y}_{\lambda}$ and $\textbf{z}_{\mu}$,
respectively. Then $0$, $m+n$, $\lambda+n$ and $\mu+n$ are eigenvalues of $L$
with associated eigenvectors

$\textbf{1}_{m+n},\quad\textbf{u}=\left[\begin{array}[]{ccccc}n\textbf{1}_{m}\\\
-m\textbf{1}_{n}\end{array}\right],\quad\textbf{v}_{\lambda}=\left[\begin{array}[]{cc}\textbf{y}_{\lambda}\\\
\textbf{0}\end{array}\right],\quad\text{and}\quad\textbf{w}_{\mu}=\left[\begin{array}[]{cc}\textbf{0}\\\
\textbf{z}_{\mu}\end{array}\right]$ (12)

respectively. Now, let $u,v\in V(X)$. We have two cases.

  * •

Let $u$ and $v$ be non-isolated in $X$. From the form of
$\textbf{v}_{\lambda}$’s in (12), strong cospectrality in $X$ is required for
strong cospectrality in $X\vee Y$. Hence, we assume $u$ and $v$ are strongly
cospectral in $X$. If $m\in\sigma_{uv}^{-}(L(X))$, then $\textbf{v}_{m}$ is
another eigenvector for $m+n$. In this case,
$\textbf{u}^{T}\textbf{e}_{u}=\textbf{u}^{T}\textbf{e}_{v}$ and
$\textbf{v}_{m}^{T}\textbf{e}_{u}=-\textbf{v}_{m}^{T}\textbf{e}_{v}$, and so
$u$ and $v$ are not strongly cospectral in $X\vee Y$. However, if
$m\notin\sigma_{uv}^{-}(L(X))$, then $u$ and $v$ are strongly cospectral in
$X\vee Y$ with $\sigma_{uv}^{+}(L)$ and $\sigma_{uv}^{-}(L)$ in Theorem 19(1a)
as desired.

  * •

Let $u$ and $v$ be isolated in $X$. Then
$\textbf{v}=\textbf{e}_{u}-\textbf{e}_{v}$ is an eigenvector for $L$
associated to the eigenvalue $n$. If $X=O_{2}$, then [Monterde2022, Corollary
6.9(2)] yields the desired conclusion. However, if $X$ has a connected
component $C$ other than $\\{u\\}$ and $\\{v\\}$, then the vector w which is
constant on each connected component of $X$ and whose sum of all entries is 0
is also an eigenvector for $L$ associated to the eigenvalue $n$. This vector
satisfies $\textbf{w}^{T}\textbf{e}_{u}=\textbf{w}^{T}\textbf{e}_{v}$, and
since $\textbf{v}^{T}\textbf{e}_{u}=-\textbf{v}^{T}\textbf{e}_{v}$, we get
that $u$ and $v$ are not strongly cospectral in $X\vee Y$.

Combining these cases proves (1). Next, let $\lambda\neq k$ and $\mu\neq\ell$
be eigenvalues of $A(X)$ and $L(Y)$ respectively, with associated eigenvectors
$\textbf{y}_{\lambda}$ and $\textbf{z}_{\mu}$. Then $\lambda^{\pm}$, $\lambda$
and $\mu$ are eigenvalues of $A$ with associated eigenvectors

$\textbf{u}=\left[\begin{array}[]{ccccc}(k-\lambda^{\mp})\textbf{1}_{m}\\\
m\textbf{1}_{n}\end{array}\right],\quad\textbf{v}_{\lambda}=\left[\begin{array}[]{cc}\textbf{y}_{\lambda}\\\
\textbf{0}\end{array}\right],\quad\text{and}\quad\textbf{w}_{\mu}=\left[\begin{array}[]{cc}\textbf{0}\\\
\textbf{z}_{\mu}\end{array}\right]$ (13)

respectively. The same argument as the previous case yields the desired
conclusion for (2). Finally, the form of the $\textbf{v}_{\lambda}$’s in
(12-13) yields (3) and the last statement. ∎

From Theorem 19(3), we assume henceforth that strongly cospectral vertices in
$X\vee Y$ belong to $X$.

Note that if $X$ is unweighted. then $m$ is an eigenvalue of $L(X)$ if and
only if $X$ is a join. From (12), we get that $m\in\sigma_{uv}^{-}(L(Z))$ if
and only if $Z=K_{2}$. Combining this with Theorem 19 yields the next result.

######  Corollary 20.

Let $X$ be an unweighted graph with vertices $u$ and $v$.

  1. 1.

Let $X\notin\\{K_{2},O_{2}\\}$. Then $u$ and $v$ in $X$ are Laplacian strongly
cospectral in $X\vee Y$ if and only if they are in $X$.

  2. 2.

Let $X\neq O_{2}$ and suppose $X\vee Y$ is not a complete graph. Then $u$ and
$v$ in $X$ are adjacency strongly cospectral in $X\vee Y$ if and only if they
are in $X$.

###### Proof.

Let $X$ be unweighted. Then $m$ is an eigenvalue of $L(Z)$ if and only if $Z$
is a join. In particular, $m\in\sigma_{uv}^{-}(L(Z))$ if and only if $Z=K_{2}$
by (12). Combining this with Theorem 19 yields (1). To prove (2), note that
$\lambda^{-}=\frac{1}{2}(k+\ell-\sqrt{(k-\ell)^{2}+4mn})=-1$ if and only if
$m=k+1$ and $n=\ell+1$, i.e., $X$ and $Y$ are complete. ∎

######  Example 21.

Consider $X\vee Y$, where $X\in\\{K_{m},O_{m}\\}$. If $m\geq 3$, then the
vertices of $X$ do not admit strongly cospectrality in $X$ and $X\vee Y$ with
respect to $M\in\\{A,L\\}$ [Monterde2022, Corollary 3.10]. If $m=2$, then the
following hold about the apexes $u$ and $v$ of the double cone $X\vee Y$.

  1. 1.

By Theorem 19(1b-2b), $u$ and $v$ are Laplacian and adjacency strongly
cospectral in $O_{2}\vee Y$.

  2. 2.

Since $2\in\sigma_{uv}^{-}(L(X))$, Theorem 19(1a) implies that $u$ and $v$ are
not Laplacian strongly cospectral in $K_{2}\vee Y$ for any graph $Y$. Since
$u$ and $v$ are adjacency strongly cospectral in $K_{2}$, Corollary 20(2)
implies that $u$ and $v$ are strongly cospectral in $K_{2}\vee Y$ if and only
if $Y$ is not a complete graph.

##  7 Perfect state transfer

To determine PST in joins, we make use of a characterization of PST due to
Coutinho. Throughout, we denote the largest power of two that divides an
integer $a$ by $\nu_{2}(a)$. We also denote the minimum PST times between $u$
and $v$ in $X$ and $X\vee Y$ by $\tau_{X}$ and $\tau_{X\vee Y}$, respectively.

######  Theorem 22.

Let $M$ be a real symmetric matrix such that $\phi(M,t)\in\mathbb{Z}[t]$. Then
vertices $u$ and $v$ in $X$ admit perfect state transfer if and only if all of
the following conditions hold.

  1. 1.

Either (i) $\sigma_{u}(M)\subseteq\mathbb{Z}$ or (ii) each
$\lambda_{j}\in\sigma_{u}(M)$ is a quadratic integer of the form
$\frac{1}{2}(a+b_{j}\sqrt{\Delta})$ for some integers $a$, $b_{j}$ and
$\Delta$, where $\Delta>1$ is square-free. Here, we let $\Delta=1$ whenever
(i) holds.

  2. 2.

Vertices $u$ and $v$ are strongly cospectral in $X$.

  3. 3.

For all $\lambda,\eta\in\sigma_{uv}^{+}(M)$ and
$\mu,\theta\in\sigma_{uv}^{-}(M)$, we have
$\nu_{2}\left(\frac{\lambda-\eta}{\sqrt{\Delta}}\right)>\nu_{2}\left(\frac{\lambda-\mu}{\sqrt{\Delta}}\right)=\nu_{2}\left(\frac{\lambda-\theta}{\sqrt{\Delta}}\right)$.

Further, $\tau_{X}=\frac{\pi}{g\sqrt{\Delta}}$, where
$g=\operatorname{gcd}\left(\left\\{\frac{\lambda_{0}-\lambda}{\sqrt{\Delta}}:\lambda\in\sigma_{u}(M)\right\\}\right)$
for some fixed $\lambda_{0}\in\sigma_{uv}^{+}(M)$.

To characterize PST in joins, we now combine Theorem 22 with our results on
periodicity and strong cospectrality in joins from Sections 4 and 6. Note that
it suffices to consider the vertices of $X$ in checking whether PST occurs in
$X\vee Y$ by virtue of Theorem 19(3). We begin with the Laplacian case.

###  7.1 Laplacian case

######  Theorem 23.

Let $m\geq 2$ and $\phi(L(X),t)\in\mathbb{Z}[t]$. Vertices $u$ and $v$ in $X$
admit Laplacian perfect state transfer in $X\vee Y$ if and only if all of the
following conditions hold.

  1. 1.

Either (i) $u$ and $v$ are Laplacian strongly cospectral in $X$ and
$m\notin\sigma_{uv}^{-}(L(X))$ or (ii) $X=O_{2}$.

  2. 2.

The eigenvalues in $\sigma_{u}(L(X))$ are all integers.

  3. 3.

One of the following conditions hold.

    1. (a)

$X$ is connected and one of the following conditions hold for all
$0\neq\lambda\in\sigma_{uv}^{+}(L(X))\cup\\{m\\}$ and
$\mu,\theta\in\sigma_{uv}^{-}(L(X))$.

      1. i.

$\nu_{2}(\lambda)>\nu_{2}(\mu)=\nu_{2}(\theta)$ and $\nu_{2}(n)>\nu_{2}(\mu)$.

      2. ii.

$\nu_{2}(\mu)>\nu_{2}(\lambda)=\nu_{2}(n)$.

      3. iii.

$\nu_{2}(\lambda)=\nu_{2}(\mu)=\nu_{2}(n)$ and
$\nu_{2}\left(\frac{\lambda+n}{2^{\nu_{2}(n)}}\right)>\nu_{2}\left(\frac{\mu+n}{2^{\nu_{2}(n)}}\right)=\nu_{2}\left(\frac{\theta+n}{2^{\nu_{2}(n)}}\right)$.

    2. (b)

$X\neq O_{2}$ is disconnected and (ai) above holds. Here, $u$ and $v$ are in
the same connected component in $X$.

    3. (c)

$X=O_{2}$ and $n\equiv 2$ (mod 4).

Further, $\tau_{X\vee Y}=\frac{\pi}{g}$, where
$g=\operatorname{gcd}\left(\mathcal{T}\right)$,
$\mathcal{T}=\mathcal{S}\cup\mathcal{R}$ and the sets
$\mathcal{S},\mathcal{R}$ are given in Lemma 3.

###### Proof.

By Theorem 19(1) and Corollary 7, conditions (1) and (2) are equivalent resp.
to strong cospectrality and periodicity of $u$ and $v$ in $X\vee Y$. Combining
this with Theorem 22, we get PST between $u$ and $v$ in $X\vee Y$ if and only
if Theorem 22(3) holds. To establish (3a), suppose $X$ is connected. From
Theorem 19(1a), we have
$\sigma_{uv}^{+}(L)=\\{\lambda+n:0\neq\lambda\in\sigma_{uv}^{+}(L(X))\\}\cup\\{0,m+n\\}$
and $\sigma_{uv}^{-}(L)=\sigma_{uv}^{-}(L(X))$. Theorem 22(3) applied to $u$
and $v$ is equivalent to

$\nu_{2}(\lambda+n)>\nu_{2}(\mu+n)=\nu_{2}(\theta+n)$ (14)

for all $0\neq\lambda\in\sigma_{uv}^{+}(L(X))\cup\\{m\\}$ and
$\mu,\theta\in\sigma_{uv}^{-}(L(X))$. We have three cases.

  * •

Let $\nu_{2}(\lambda)>\nu_{2}(\mu)$ for some
$0\neq\lambda\in\sigma_{uv}^{+}(L(X))\cup\\{m\\}$ and
$\mu\in\sigma_{uv}^{-}(L(X))$. Since $\nu_{2}(\lambda+n)>\nu_{2}(\mu+n)$ in
(14), it must be that $\nu_{2}(n)>\nu_{2}(\mu)$ for each
$\mu\in\sigma_{uv}^{-}(L(X))$, in which case, $\nu_{2}(\mu+n)=\nu_{2}(\mu)$.
Now, for the equality in (14) to hold, we need $\nu_{2}(\mu)=\nu_{2}(\theta)$
for all $\mu,\theta\in\sigma_{uv}^{-}(L(X))$. Hence,
$\nu_{2}(\lambda)>\nu_{2}(\mu)$ for all $\mu\in\sigma_{uv}^{-}(L(X))$.
Finally, if $\nu_{2}(\mu)\geq\nu_{2}(\eta)$ for some
$0\neq\eta\in\sigma_{uv}^{+}(L(X))\cup\\{m\\}$, then
$\nu_{2}(n)>\nu_{2}(\eta)$, and so
$\nu_{2}(\eta+n)=\nu_{2}(\eta)\leq\nu_{2}(\mu)=\nu_{2}(\mu+n)$, a
contradiction to (14). Thus, our assumption in this case combined with (14)
gives us $\nu_{2}(\lambda)>\nu_{2}(\mu)=\nu_{2}(\theta)$ and
$\nu_{2}(n)>\nu_{2}(\mu)$ for all
$0\neq\lambda\in\sigma_{uv}^{+}(L(X))\cup\\{m\\}$ and
$\mu,\theta\in\sigma_{uv}^{-}(L(X))$, which proves (3ai).

  * •

Let $\nu_{2}(\mu)>\nu_{2}(\lambda)$ for some
$0\neq\lambda\in\sigma_{uv}^{+}(L(X))\cup\\{m\\}$ and
$\mu\in\sigma_{uv}^{-}(L(X))$. If $\nu_{2}(\lambda)\neq\nu_{2}(n)$, then
$\nu_{2}(\mu+n)\geq\nu_{2}(\lambda+n)$, a contradiction to (14). Hence, we
must have $\nu_{2}(\lambda)=\nu_{2}(n)$ for each
$0\neq\lambda\in\sigma_{uv}^{+}(L(X))\cup\\{m\\}$. Finally, if
$\nu_{2}(n)\geq\nu_{2}(\theta)$ for some $\theta\in\sigma_{uv}^{-}(L(X))$,
then we have $\nu_{2}(\theta+n)\neq\nu_{2}(n)=\nu_{2}(\mu+n)$, which again
contradicts (14). Thus, $\nu_{2}(\mu)>\nu_{2}(n)=\nu_{2}(\lambda)$ for all for
some $0\neq\lambda\in\sigma_{uv}^{+}(L(X))\cup\\{m\\}$ and
$\theta\in\sigma_{uv}^{-}(L(X))$. This proves (3aii).

  * •

Let $\nu_{2}(\mu)=\nu_{2}(\lambda)$ for some
$0\neq\lambda\in\sigma_{uv}^{+}(L(X))\cup\\{m\\}$ and
$\mu\in\sigma_{uv}^{-}(L(X))$. For the strict inequality in (14) to hold, it
is required that $\nu_{2}(\mu)=\nu_{2}(\lambda)=\nu_{2}(n)$ for all
$0\neq\lambda\in\sigma_{uv}^{+}(L(X))\cup\\{m\\}$ and
$\mu\in\sigma_{uv}^{-}(L(X))$. From this, the conditions in (3aiii) are
straightforward.

Combining these cases proves (3a). Now, if $X$ is disconnected, then
$n\in\sigma_{uv}^{+}(L(X))$ by Theorem 19(1b), and so Theorem 22(3) holds if
and only if (14) holds and $\nu_{2}(n)>\nu_{2}(\mu+n)$ for all
$\mu\in\sigma_{uv}^{-}(L(X))$. Equivalently, (3ai) holds. This proves (3b). If
$X=O_{2}$, then Theorem 22(3) holds if and only if $\nu_{2}(n+2)>\nu_{2}(n)$,
i.e., $n\equiv 2$ (mod 4). This proves (3c). The minimum PST time follows from
Theorem 22. ∎

######  Remark 24.

If $X\neq K_{2}$ is unweighted, then we may drop the condition
$m\in\sigma_{uv}^{-}(L(X))$ in Theorem 23(1i).

The following result is immediate from Theorem 23(3).

######  Corollary 25.

Let $m\geq 2$ and $\phi(L(X),t)\in\mathbb{Z}[t]$. If $m+n$ is odd, then $X\vee
Y$ has no Laplacian perfect state transfer. Moreover, if $m$ or $n$ is odd and
$X$ is either disconnected or admits Laplacian perfect state transfer, then
$X$ does not admit Laplacian perfect state transfer in $X\vee Y$.

###  7.2 Adjacency case

######  Theorem 26.

Let $m\geq 2$, $k,\ell\in\mathbb{Z}$ and $\phi(A(X),t)\in\mathbb{Z}[t]$.
Vertices $u$ and $v$ in $X$ admit adjacency perfect state transfer in $X\vee
Y$ if and only if all of the following conditions hold.

  1. 1.

Either (i) $u$ and $v$ are adjacency strongly cospectral in $X$ and
$\lambda^{-}\notin\sigma_{uv}^{-}(L(X))$ or (ii) $X=O_{2}(k)$.

  2. 2.

One of the following conditions hold.

    1. (a)

The eigenvalues in $\sigma_{u}(A(X))$ are all integers and $D$ is a perfect
square.

    2. (b)

$X$ is connected, each $\lambda_{j}\in\sigma_{u}(A(X))\backslash\\{k\\}$ is of
the form $\frac{1}{2}(k+\ell+b_{j}\sqrt{\Delta})$ and
$\lambda^{\pm}=\frac{1}{2}(k+\ell\pm b\sqrt{\Delta})$, where $b_{j},b,\Delta$
are integers with $b>b_{j}$ for each $j$ and $\Delta>1$ is square-free.

  3. 3.

For all $\lambda,\eta\in\sigma_{uv}^{+}(A(X))\backslash\\{k\\}\cup\mathcal{R}$
and $\mu,\theta\in\sigma_{uv}^{-}(A(X))$,
$\nu_{2}\left(\frac{\lambda-\eta}{\sqrt{\Delta}}\right)>\nu_{2}\left(\frac{\lambda-\mu}{\sqrt{\Delta}}\right)=\nu_{2}\left(\frac{\lambda-\theta}{\sqrt{\Delta}}\right)$,
where $\Delta=1$ whenever (2a) holds, $\mathcal{R}$ is given in Theorem 19(2).

Further, $\tau_{X\vee Y}=\frac{\pi}{g\sqrt{\Delta}}$, where
$g=\operatorname{gcd}\left(\mathcal{T}\right)$,
$\mathcal{T}=\left\\{\frac{\lambda_{0}-\lambda}{\sqrt{\Delta}}:\lambda\in\sigma_{u}(A)\backslash\\{k\\}\cup\mathcal{R}\right\\}$
and $\lambda_{0}\in\sigma_{uv}^{+}(A)\backslash\\{k\\}\cup\mathcal{R}$ is
fixed.

###### Proof.

This is a direct consequence of Theorems 8(2), 19(2) and 22. ∎

The next result can be viewed as an analogue of Corollary 25.

######  Corollary 27.

If $k+\ell$ is odd in Theorem 26, then $X\vee Y$ has no adjacency perfect
state transfer.

###### Proof.

If $k+\ell=\lambda^{+}+\lambda^{+}$ is odd, then so is
$\lambda^{+}-\lambda^{-}$, a contradiction to Theorem 26(3). ∎

##  8 PST in $X$ and $X\vee Y$

We now utilize our results from the previous section to determine when PST is
preserved in the join.

###  8.1 Laplacian case

######  Corollary 28.

Let $m\geq 2$ and $\phi(L(X),t)\in\mathbb{Z}[t]$. Suppose Laplacian perfect
state transfer occurs between vertices $u$ and $v$ in $X$ and
$\tau_{X}=\frac{\pi}{h}$. Then it occurs between $u$ and $v$ in $X\vee Y$ if
and only if $m\notin\sigma_{uv}^{-}(L(X))$ and $m,n\equiv 0$ (mod
$2^{\alpha}$), where $\alpha>\nu_{2}(h)$. In this case, $\tau_{X\vee
Y}=\frac{\pi}{g}$, where $g=\operatorname{gcd}(\mathcal{T})$, $\mathcal{T}$ is
given in Theorem 23 and $\nu_{2}(g)=\nu_{2}(h)$. Moreover, if $X$ is
connected, then $\frac{h}{g}$ is rational. Otherwise,
$\frac{h}{g}=\operatorname{lcm}(h/h_{1},h/h_{2})$ is an odd integer where
$h_{1}=\operatorname{gcd}(m,h)$ and $h_{2}=\operatorname{gcd}(n,h)$.

###### Proof.

If PST occurs in $X$, then $u$ and $v$ are strongly cospectral in $X$ and
Theorem 23(2) holds. Hence, PST occurs in $X\vee Y$ if and only if (1i) and
(3ai) of Theorem 23 holds. Equivalently, $m\notin\sigma_{uv}^{-}(L(X))$,
$\nu_{2}(m)>\nu_{2}(\mu)$ and $\nu_{2}(n)>\nu_{2}(\mu)$ for all
$\mu\in\sigma_{uv}^{-}(L(X))$. As $\nu_{2}(h)=\nu_{2}(\mu)=\nu_{2}(\mu+n)$ for
all $\mu\in\sigma_{uv}^{-}(L(X))$, these conditions are equivalent to
$m,n\equiv 0$ (mod $2^{\alpha}$), where $\alpha>\nu_{2}(h)$. The minimum PST
time follows from Theorem 23. ∎

######  Corollary 29.

Let $X$ be an unweighted graph on $m=2^{p}$ vertices. Then conditions
(3aii-3aiii) of Theorem 23 do not hold. If we add that Laplacian perfect state
transfer occurs between vertices $u$ and $v$ in $X$, then:

  1. 1.

If $p=1$, then Laplacian perfect state transfer does not occur between $u$ and
$v$ in $K_{2}\vee Y$ for any $Y$.

  2. 2.

If $p\geq 2$, then Laplacian perfect state transfer occurs between $u$ and $v$
in $X\vee Y$ if and only if $n\equiv 0$ (mod $2^{\alpha}$), where
$\alpha>\nu_{2}(g)$ and $\frac{\pi}{h}$ is the minimum PST time between $u$
and $v$ in $X$.

###### Proof.

The assumption implies that each eigenvalue $\lambda$ of $L(X)$ is at most
$m$. Thus, $\nu_{2}(\lambda)<\nu_{2}(m)$, and so conditions (3aii-iii) of
Theorem 22 do not hold. Now, from Corollary 28, PST occurs between $u$ and $v$
in $X\vee Y$ if and only if $m\notin\sigma_{uv}^{-}(L(X))$ and $n\equiv 0$
(mod $2^{\alpha}$), where $\alpha>\nu_{2}(h)$. Thus, if $p=1$, then Example
21(1) implies that $u$ and $v$ are not strongly cospectral in $K_{2}\vee Y$,
and so (1) holds. However, if $p\geq 2$, then $m\notin\sigma_{uv}^{-}(L(X))$,
and so (2) holds. ∎

In Corollaries 28 and 29(2), the $Y$ with the least number of vertices and
edges that works is $Y=O_{n}$, where $n=2^{\nu_{2}(g)+1}$. Next, combining
Corollary 28 and Theorem 23(3ai) also yields the following result.

######  Corollary 30.

Let $u$ and $v$ be strongly cospectral vertices in $X\vee Y$ and
$\sigma_{u}(L(X))\subseteq\mathbb{Z}$. If condition (3ai) of Theorem 23 holds,
then Laplacian perfect state transfer occurs between $u$ and $v$ in $X$ and
$X\vee Y$.

From Theorem 23(3a), it is evident that PST in $X\vee Y$ does not necessarily
yield PST in $X$. Thus, the above result can be viewed as a characterization
of the equivalence of PST in a graph and its join.

Using Corollary 28, we provide an infinite family of graphs such that $X$ and
$X\vee Y$ both have PST. Recall that a graph $X$ is Hadamard diagonalizable if
$L(X)$ is diagonalizable by a Hadamard matrix [Barik2011].

######  Example 31.

Let $X$ be a Hadamard diagonalizable graph on $m\geq 4$ vertices with PST
between $u$ and $v$. From [Johnston2017], $\sigma_{uv}^{+}(L(X))$ and
$\sigma_{uv}^{-}(L(X))$ respectively consist of integers $\lambda\equiv 0$ and
$\mu\equiv 2$ (mod 4). Since $m\equiv 0$ (mod 4), Corollary 28 yields PST
between $u$ and $v$ in $X\vee Y$ if and only if $n\equiv 0$ (mod 4).

Our next result shows that for a graph $X$ with PST between $u$ and $v$, it is
possible for PST to fail between $u$ and $v$ in $X\vee Y$ for any graph $Y$,
yet occur in $(X\cup Z)\vee Y$ for some graph $Z$.

######  Corollary 32.

Let $m\geq 2$ and $\phi(L(X),t)\in\mathbb{Z}[t]$. If Laplacian perfect state
transfer occurs between vertices $u$ and $v$ in $X$ with
$m\in\sigma_{uv}^{-}(L(X))$, then the following hold.

  1. 1.

For any graph $Y$, Laplacian perfect state transfer does not occur between $u$
and $v$ in $X\vee Y$.

  2. 2.

Let $Z$ be a graph on $r$ vertices such that $m+r\notin\sigma_{uv}^{-}(L(X))$.
Then Laplacian perfect state transfer occurs between $u$ and $v$ in $(X\cup
Z)\vee Y$ if and only if $\nu_{2}(n)>\nu_{2}(m)=\nu_{2}(r)$.

###### Proof.

Since $m\in\sigma_{uv}^{-}(L(X))$, $u$ and $v$ are not strongly cospectral in
$X\vee Y$ by Theorem 19(1). We now prove (2). Since
$m+r\notin\sigma_{uv}^{-}(L(X))$, Corollary 28 implies that PST occurs between
$u$ and $v$ in $X\vee Y$ if and only if $\nu_{2}(m+r)>\nu_{2}(m)$ and
$\nu_{2}(n)>\nu_{2}(m)$ for all $\mu\in\sigma_{uv}^{-}(L(X))$. Equivalently,
$\nu_{2}(n)>\nu_{2}(m)=\nu_{2}(r)$. ∎

In Corollary 32(2), the $Z$ and $Y$ with the least number of vertices and
edges that work are $Z=O_{r}$ and $Y=O_{n}$ respectively, where
$r=2^{\nu_{2}(m)}$ and $r=2^{\nu_{2}(m)+1}$.

######  Example 33.

Let $X=K_{2}$ with vertices $u$ and $v$. From Corollary 29(1), PST between $u$
and $v$ fails in $K_{2}\vee Y$ for any graph $Y$. Since
$r+2\notin\sigma_{uv}^{-}(L(K_{2}))$ for all $r\geq 1$, Corollary 32 yields
PST between $u$ and $v$ in $(K_{2}\cup Z)\vee Y$ if and only if
$\nu_{2}(n)>\nu_{2}(r)=1$. In particular, we may take $Z=O_{2}$ and $Y=O_{4}$.

The following remark can be used to generate infinite families of weighted
graphs $X$ whereby PST occurs between $u$ and $v$ in $X$ and $(X\cup Z)\vee Y$
for some graph $Z$, but not in $X\vee Y$ for any graph $Y$.

######  Remark 34.

Let $m\geq 2$ and $\phi(L(X),t)\in\mathbb{Z}[t]$. Suppose vertices $u$ and $v$
admit Laplacian PST in $X$, and suppose $\mu$ divides $m$ for some
$\mu\in\sigma_{uv}^{-}(L(X))$. Let $X^{\prime}$ be the graph obtained by
scaling all edge weights of $X$ by a factor of $\frac{m}{\mu}$. Then
$m\in\sigma_{uv}^{-}(L(X^{\prime}))$, and so PST does not occur between $u$
and $v$ in $X^{\prime}\vee Y$ for any graph $Y$ by Corollary 32(1). But since
$\frac{m}{\mu}$ is an integer, we get $\phi(L(X^{\prime}),t)\in\mathbb{Z}[t]$,
and so we may apply Corollary 32(2) to obtain a graph $Z$ such that PST occurs
between $u$ and $v$ in $(X^{\prime}\cup Z)\vee Y$.

###  8.2 Adjacency case

In contrast with the Laplacian case in Corollary 32(2), it turns out that if
$X$ is regular and has adjacency PST between two vertices, then we can always
choose a regular graph $Y$ such that adjacency PST is preserved in $X\vee Y$.

######  Corollary 35.

Let $m\geq 2$, $k,\ell\in\mathbb{Z}$ and $\phi(A(X),t)\in\mathbb{Z}[t]$.
Suppose adjacency perfect state transfer occurs between vertices $u$ and $v$
in $X$ and $\tau_{X}=\frac{\pi}{h}$. Then it occurs between them in $X\vee Y$
if and only if

  1. 1.

$n=\frac{s(k-\ell+s)}{m}$ for some integer $s$ such that $m$ divides
$s(k-\ell+s)$ and $\ell-s\notin\sigma_{uv}^{-}(A(X))$, and

  2. 2.

for all $\mu\in\sigma_{uv}^{-}(A(X))$, $\nu_{2}(s)>\nu_{2}(k-\mu)$ and
$\nu_{2}(\ell-k)>\nu_{2}(k-\mu)$.

In this case, $\tau_{X\vee Y}=\frac{\pi}{g}$, where
$g=\operatorname{gcd}(\mathcal{T})$, $\mathcal{T}$ is given in Theorem 26,
$\lambda_{0}\in\sigma_{uv}^{+}(A)\backslash\\{k\\}\cup\mathcal{R}$ is fixed
and $\nu_{2}(g)=\nu_{2}(h)$. Moreover, if $X$ is connected, $\frac{h}{g}$ is
rational. Otherwise, $\frac{h}{g}=\operatorname{lcm}(h/h_{1},h/h_{2})$ is an
odd integer and $h_{1}=\operatorname{gcd}(\lambda_{0}-\lambda^{+},h)$ and
$h_{2}=\operatorname{gcd}(\lambda_{0}-\lambda^{-},h)$.

###### Proof.

From Theorem 26, it follows that PST occurs between $u$ and $v$ in $X\vee Y$
if and only if

  * (a)

$D$ is a perfect square and $\lambda^{-}\notin\sigma_{uv}^{-}(A(X))$, and

  * (b)

$\nu_{2}(\lambda^{\pm}-\eta)>\nu_{2}(\lambda^{\pm}-\mu)=\nu_{2}(\lambda^{\pm}-\theta)$
for all $\eta\in\sigma_{uv}^{+}(A(X))\backslash\\{k\\}\cup\mathcal{S}$ and
$\mu,\theta\in\sigma_{uv}^{-}(A(X))$.

By the choice of $s$ in (1), $D=(k-\ell)^{2}+4mn=(k-\ell+2s)^{2}$. Since
$\lambda^{\pm}=\frac{1}{2}(k+\ell\pm\sqrt{D})$, we get $\lambda^{+}=k+s$ and
$\lambda^{-}=\ell-s$. Thus, (1) ensures that condition (a) above holds. Using
the fact that $X$ has PST between $u$ and $v$, Theorem 22(3) yields

$\nu_{2}(k-\eta)>\nu_{2}(k-\mu)=\nu_{2}(k-\theta)$ (15)

for all $\eta\in\sigma_{uv}^{+}(A(X))\backslash\\{k\\}$ and
$\mu,\theta\in\sigma_{uv}^{-}(A(X))$. Observe that for each
$\lambda\in\sigma_{u}(A(X))\backslash\\{k\\}$, we can write
$\lambda^{+}-\lambda=(\lambda^{+}-k)+(k-\lambda)=s+(k-\lambda)$. Similarly,
$\lambda^{-}-\lambda=(\ell-s-k)+(k-\lambda)$. Thus,

$\nu_{2}(\lambda^{+}-\lambda)\geq\min\\{\nu_{2}(s),\nu_{2}(k-\lambda)\\}\quad\text{and}\quad\nu_{2}(\lambda^{-}-\lambda)\geq\min\\{\nu_{2}(\ell-
s-k),\nu_{2}(k-\lambda)\\}.$ (16)

We have the following cases.

  * •

Suppose $\nu_{2}(s)>\nu_{2}(k-\mu)$ for some $\mu\in\sigma_{uv}^{-}(A(X))$.
Then (15) implies that $\nu_{2}(s)>\nu_{2}(k-\mu)$ for all
$\mu\in\sigma_{uv}^{-}(A(X))$, and combining this with (16) yields
$\nu_{2}(\lambda^{+}-\mu)=\nu_{2}(\lambda^{+}-\theta)=\nu_{2}(k-\mu)$.
Moreover, (15) and (16) give us $\nu_{2}(\lambda^{+}-\eta)>\nu_{2}(k-\mu)$ for
all $\mu\in\sigma_{uv}^{-}(A(X))$, from which we get
$\nu_{2}(\lambda^{+}-\eta)>\nu_{2}(\lambda^{+}-\mu)=\nu_{2}(\lambda^{+}-\theta)$
for all $\eta\in\sigma_{uv}^{+}(A(X))\backslash\\{k\\}$ and
$\mu,\theta\in\sigma_{uv}^{-}(A(X))$.

  * •

Suppose $\nu_{2}(s)\leq\nu_{2}(k-\mu)$ for some $\mu\in\sigma_{uv}^{-}(A(X))$.
Then one can check using (15) and (16) that
$\nu_{2}(\lambda^{+}-\eta)=\nu_{2}(s)\leq\nu_{2}(\lambda^{+}-\mu)$, a
violation of condition (b).

Consequently,
$\nu_{2}(\lambda^{+}-\eta)>\nu_{2}(\lambda^{+}-\mu)=\nu_{2}(\lambda^{+}-\theta)$
for all $\eta\in\sigma_{uv}^{+}(A(X))\backslash\\{k\\}$ and
$\mu,\theta\in\sigma_{uv}^{-}(A(X))$ if and only if
$\nu_{2}(s)>\nu_{2}(k-\mu)$ for all $\mu\in\sigma_{uv}^{-}(A(X))$. Using this
fact and again arguing similarly as in the above two cases using (15) and (16)
yields
$\nu_{2}(\lambda^{+}-\eta)>\nu_{2}(\lambda^{+}-\mu)=\nu_{2}(\lambda^{+}-\theta)$
for all $\eta\in\sigma_{uv}^{+}(A(X))\backslash\\{k\\}$ and
$\mu,\theta\in\sigma_{uv}^{-}(A(X))$ if and only if
$\nu_{2}(\ell-k)>\nu_{2}(k-\mu)$ for all $\mu\in\sigma_{uv}^{-}(A(X))$. Thus,
the assumption in (2) is equivalent to condition (b) above, which yields the
desired conclusion. ∎

The following example illustrates Corollary 35.

######  Example 36.

Let $X$ be a simple integer-weighted Hadamard diagonalizable graph on $m\geq
4$ vertices with PST between vertices $u$ and $v$. Then $m\equiv 0$ (mod 4)
and $X$ is $p$-regular for some integer $p$. From Example 31,
$\sigma_{uv}^{+}(A(X))$ consists of integers $p-\lambda$, where $\lambda\equiv
0$ (mod 4), while $\sigma_{uv}^{-}(A(X))$ consists of integers $p-\mu$ such
that $\mu\equiv 2$ (mod 4). Invoking Corollary 35(2), we get PST between $u$
and $v$ in $X\vee Y$ if and only if (i) $n=\frac{s(p-\ell+s)}{m}$, where $s$
is an integer such that $m$ divides $s(p-\ell+s)$ and
$\ell-s\notin\sigma_{uv}^{-}(A(X))$ and (ii) $\nu_{2}(s)\geq 2$ and
$\nu_{2}(\ell-p)\geq 2$. In particular, if $X=Q_{p}$, then we may take
$\ell=p$ and $s=2^{(p+1)/2}a$ (i.e., $n=2a$) if $p$ is odd and $s=2^{p/2}a$
(i.e., $n=a$) otherwise, to get PST in both $Q_{p}$ and $Q_{p}\vee Y$.

######  Remark 37.

If $X=K_{2}$, then $k=-\mu=1$, and so the condition
$\ell-s\notin\sigma_{uv}^{-}(A(X))$ in Corollary 35(1) is equivalent to
$\ell\neq n-1$, while the conditions $\nu_{2}(s)>1$ and $\nu_{2}(\ell-1)>1$ in
Corollary 35(2) are equivalent to $\nu_{2}(s-2)=1$ and $\nu_{2}(\ell+3)>1$.
Thus, taking $X=K_{2}$ in Corollary 35 recovers a known characterization of
PST in connected double cones [Kirkland2023, Theorem 12(1)].

Next, we have an analogue of Corollary 30, which determines when PST in $X\vee
Y$ is inherited by $X$.

######  Corollary 38.

Let $m\geq 2$, $k,\ell\in\mathbb{Z}$ and $\phi(A(X),t)\in\mathbb{Z}[t]$.

  1. 1.

Let $X$ be disconnected. If perfect state transfer occurs between vertices $u$
and $v$ in $X\vee Y$, then it also occurs between them in $X$ if and only if
$X\neq O_{2}(k)$ for any $k\in\mathbb{Z}$.

  2. 2.

Suppose $X$ is connected. If perfect state transfer occurs between vertices
$u$ and $v$ in $X\vee Y$, then perfect state transfer occurs between $u$ and
$v$ in $X$ if and only if both conditions below hold.

    1. (a)

The eigenvalues in $\sigma_{u}(A(X))$ are all integers and $D$ is a perfect
square.

    2. (b)

$\nu_{2}(k-\eta)>\nu_{2}(k-\mu)=\nu_{2}(k-\theta)$ for any
$\eta\in\sigma_{uv}^{+}(A(X))\backslash\\{k\\}$ and
$\mu,\theta\in\sigma_{uv}^{-}(A(X))$.

##  9 PST in $X\vee Y$ but not in $X$

In Corollary 28, we characterized graphs with PST that also admit PST in the
join. In this section, our goal is to show that PST can be induced in the join
despite the lack of PST in the underlying graph.

###  9.1 Laplacian case

The following is immediate from Theorem 23.

######  Corollary 39.

Let $m\geq 2$ and $\phi(L(X),t)\in\mathbb{Z}[t]$. Suppose $u$ and $v$ are
strongly cospectral vertices in $X$ with
$\sigma_{u}(L(X))\subseteq\mathbb{Z}$. Then Laplacian perfect state transfer
occurs between $u$ and $v$ in $X\vee Y$ but not in $X$ if and only if
$m\notin\sigma_{uv}^{-}(L(X))$, $\nu_{2}(m)=\nu_{2}(n)$ and either (i) Theorem
23(3aii-3aiii) holds or (ii) $X=O_{2}$.

For strongly cospectral vertices in $X$ whose elements in
$\sigma_{uv}^{-}(L(X))$ consists of integers whose largest powers of two in
their factorizations are larger than those in $\sigma_{uv}^{+}(L(X))$, Theorem
22 yields no PST between them. Nonetheless, we show that Laplacian PST can be
induced in $X\vee Y$ by appropriate choice of $n$. The following result
follows directly from Corollary 39 and Theorem 23(3aii).

######  Theorem 40.

Let $m\geq 2$ and $\phi(L(X),t)\in\mathbb{Z}[t]$. Suppose $u$ and $v$ are
strongly cospectral vertices in $X$ with $\sigma_{u}(L)\subseteq\mathbb{Z}$
such that the $\nu_{2}(\mu)>\nu_{2}(\lambda)$ for all
$0\neq\lambda\in\sigma_{uv}^{+}(L(X))$ and $\mu\in\sigma_{uv}^{-}(L(X))$. Then
Laplacian perfect state transfer occurs between $u$ and $v$ in $X\vee Y$ if
and only if $m\notin\sigma_{uv}^{-}(L(X))$, $X$ is connected and
$\nu_{2}(\lambda)=\nu_{2}(m)=\nu_{2}(n)$ for all
$0\neq\lambda\in\sigma_{uv}^{+}(L(X))$.

######  Example 41.

Let $X=CP(m)$, where $m\equiv 2$ (mod 4). Then $X$ is connected,
$(m-2)$-regular and any two non-adjacent vertices $u$ and $v$ in $X$ are
strongly cospectral with $\sigma_{uv}^{+}(L(X))=\\{0,m\\}$ and
$\sigma_{uv}^{-}(L(X))=\\{m-2\\}$. As $\nu_{2}(m-2)>\nu_{2}(m)$, Laplacian PST
does not occur between $u$ and $v$ in $X$ by Theorem 22(3). Applying Theorem
40, Laplacian PST occurs between $u$ and $v$ in $X\vee Y$ if and only if
$n\equiv 2$ (mod 4). In particular, if we take $n=2$, then $X\vee Y=CP(m+2)$
admits PST between any two non-adjacent vertices.

For strongly cospectral vertices in $X$ whose eigenvalue supports consist of
integers with equal largest powers of two in their factorizations, Theorem 22
again yields no PST between them. Despite this fact, we show that under
certain assumptions, Laplacian PST can be induced in $X\vee Y$ by appropriate
choice of $n$.

######  Theorem 42.

Let $m\geq 2$ and $\phi(L(X),t)\in\mathbb{Z}[t]$. Suppose $u$ and $v$ are
strongly cospectral vertices in $X$ with $\sigma_{u}(L)\subseteq\mathbb{Z}$
such that the $\nu_{2}(\lambda)$’s are equal for all
$\lambda\in\sigma_{u}(L)$, say to $\alpha$. Write each
$0\neq\lambda_{r}\in\sigma_{uv}^{+}(L(X))$ as
$\lambda_{r}=2^{\alpha}(2p_{r}-1)$ and each $\mu_{s}\in\sigma_{uv}^{-}(L(X))$
as $\mu_{s}=2^{\alpha}(2q_{s}-1)$ for some $p_{r},q_{s}\in\mathbb{Z}$. Then
Laplacian perfect state transfer occurs between $u$ and $v$ in $X\vee Y$ if
and only if $X$ is connected, $m\notin\sigma_{uv}^{-}(L(X))$,
$\nu_{2}(m)=\nu_{2}(n)=\alpha$ (so we may write $m=2^{\alpha}(2y-1)$ and
$n=2^{\alpha}(2z+1)$ for some $y,z\in\mathbb{Z}$), and one of the following
conditions below hold.

  1. 1.

All $\nu_{2}(q_{s})$’s are equal, $\nu_{2}(p_{r})>\nu_{2}(q_{s})$, and
$\nu_{2}(y)\geq\nu_{2}(z)\geq\nu_{2}(q_{s})+1$ for all $r,s$.

  2. 2.

All $\nu_{2}(p_{r})$’s are equal and
$\nu_{2}(p_{r})=\nu_{2}(y)=\nu_{2}(z)<\nu_{2}(q_{s})$ for all $s$.

  3. 3.

For any $r,s,t$, we have
$\nu_{2}(p_{r})=\nu_{2}(q_{s})=\nu_{2}(y)=\nu_{2}(z)$,
$\nu_{2}\left(\frac{p_{r}+z}{2^{\nu_{2}(z)}}\right)>\nu_{2}\left(\frac{q_{s}+z}{2^{\nu_{2}(z)}}\right)=\nu_{2}\left(\frac{q_{t}+z}{2^{\nu_{2}(z)}}\right)$
and
$\nu_{2}\left(\frac{y+z}{2^{\nu_{2}(z)}}\right)>\nu_{2}\left(\frac{q_{s}+z}{2^{\nu_{2}(z)}}\right)$.

###### Proof.

Corollary 39 implies that PST occurs between $u$ and $v$ in $X\vee Y$ if and
only if $\nu_{2}(m)=\nu_{2}(n)=\alpha$ and Theorem 23(3aii) holds. Let
$m=2^{\alpha}(2y-1)$ and $n=2^{\alpha}(2z+1)$ for some integers $y,z$. Note
that $m+n=2^{\alpha+1}(y+z)$, $\lambda_{r}+n=2^{\alpha+1}(p_{r}+z)$ and
$\mu_{s}+n=2^{\alpha+1}(q_{s}+z)$. If $X$ is disconnected, then
$n\in\sigma_{uv}^{+}(L(X))$. Now, Theorem 23(3aii) holds if and only if $X$ is
connected and

$\nu_{2}(p_{r}+z)>\nu_{2}(q_{s}+z)=\nu_{2}(q_{t}+z)\quad\text{and}\quad\nu_{2}(y+z)>\nu_{2}(q_{s}+z)$
(17)

for any $r,s,t$. One can then proceed by cases similar to the proof of Theorem
23(2a). ∎

######  Remark 43.

Let $X$ be a simple unweighted graph with strongly cospectral vertices $u$ and
$v$ such that $\sigma_{u}(L)\subseteq\mathbb{Z}$. If $X$ has an odd number of
vertices and an odd number of spanning trees, then the matrix-tree theorem
implies that all nonzero eigenvalues in $\sigma_{u}(L)$ are odd. Thus, Theorem
42 applies with $\alpha=0$.

We illustrate Theorem 42 and Remark 43 using a simple example.

######  Example 44.

Let $X=P_{3}$ with end vertices $u$ and $v$. Then
$\sigma_{uv}^{+}(L(X))=\\{0,3\\}$, $\sigma_{uv}^{-}(L(X))=\\{1\\}$, and so by
Theorem 22(2), $u$ and $v$ does not admit Laplacian PST in $X$. Applying
Theorem 42(1), we get Laplacian PST between $u$ and $v$ in $X\vee Y$ if and
only if $n\equiv 1$ (mod 4). In particular, taking $Y=O_{n}$ yields $X\vee
Y=K_{d}\backslash e$ with $d\equiv 0$ (mod 4), a graph known to have PST
between non-adjacent vertices.

We close this subsection with a result that determines all graphs with
isolated vertices that exhibit Laplacian PST in the join. It is immediate from
Theorems 19(1a) and 23(3c).

######  Theorem 45.

If $u$ and $v$ are two isolated vertices in $X$, then Laplacian perfect state
transfer occurs between $u$ and $v$ in $X\vee Y$ if and only if $X=O_{2}$ and
$n\equiv 2$ (mod 4).

From Theorem 45, $O_{2}$ is the only graph with isolated vertices that
exhibits PST in the join. This coincides with a known characterization of
Laplacian PST in double cones [Alvir2016, Corollary 4].

###  9.2 Adjacency case

For the adjacency case, we give one scenario where PST can be induced in
$X\vee Y$. The following is immediate from Theorems 22(3) and 26(3).

######  Corollary 46.

Let $m\geq 2$, $k,\ell\in\mathbb{Z}$ and $\phi(A(X),t)\in\mathbb{Z}[t]$.
Suppose $X$ is connected and vertices $u$ and $v$ are strongly cospectral in
$X$ such that $\sigma_{u}(A(X))\subseteq\mathbb{Z}$ and

$\nu_{2}\left(\lambda-\eta\right)>\nu_{2}\left(\lambda-\mu\right)=\nu_{2}\left(\lambda-\theta\right),$
(18)

for all $\lambda,\eta\in\sigma_{uv}^{+}(A(X))\backslash\\{k\\}$ and
$\mu,\theta\in\sigma_{uv}^{-}(A(X))$. If (18) does not hold whenever
$\lambda=k$, then

  1. 1.

adjacency perfect state transfer does not occur between $u$ and $v$ in $X$;
and

  2. 2.

adjacency perfect state transfer occurs between $u$ and $v$ in $X\vee Y$ if
and only if $\lambda^{-}\notin\sigma_{uv}^{-}(A(X))$, $D$ is a perfect square
and (18) also holds whenever $\lambda\in\\{\lambda^{\pm}\\}$.

Using Corollary 39, we generate an infinite family of graphs with no PST but
which admit PST in the join.

######  Example 47.

Let $m\geq 6$ and $X=CP(m)$, where $m\equiv 2$ (mod 4). From Example 41, $X$
is connected, $(m-2)$-regular and PST does not occur between $u$ and $v$ in
$X$ by Corollary 46(1). Applying Corollary 46(2), we get adjacency PST between
$u$ and $v$ in $X\vee Y$ if and only if $n=\frac{z(m-\ell-2+z)}{m}$ for some
integer $z$ such that $m$ divides $z(m-\ell-2+z)$,
$\nu_{2}(\ell)>\nu_{2}(z)=1$ and $\ell-z\neq 0$. The latter three conditions
are equivalent to $D$ being a perfect square,
$\nu_{2}(\lambda^{+}-\lambda^{-})=\nu_{2}(\lambda^{-}-0)=2$ and
$\lambda^{-}\neq 0$, respectively. In particular, we may take $\ell=0$ and
$z=2$ to get $n=2$. In this case, $X\vee Y=CP(m+2)$ which admits PST between
any two non-adjacent vertices.

We finish this section with an analogue of Theorem 45. It is immediate from
Theorem 26.

######  Theorem 48.

If $u$ and $v$ are two isolated vertices in $X$, then adjacency perfect state
transfer occurs between $u$ and $v$ in $X\vee Y$ if and only if $X=O_{2}(k)$,
$D$ is a perfect square and $\nu_{2}(\lambda^{+}-k)=\nu_{2}(\lambda^{-}-k)$.

##  10 Families of joins

We now determine quantum state transfer in self-joins and iterated join
graphs.

###  10.1 Self-joins

Denote the $r$-fold join of $X$ with itself by $X^{r}:=\bigvee_{j=1}^{r}X$.
The following characterizes periodicity in $X^{r}$. It follows from
Corollaries 7 and 9, and Theorem 11.

######  Theorem 49.

Let $M\in\\{A,L\\}$ and $\phi(M(X),t)\in\mathbb{Z}[t]$. Vertex $u$ is periodic
in $X$ if and only if it is periodic in $X^{r}$ for all $r$, if and only if
$\sigma_{u}(M(X))\subseteq\mathbb{Z}$. Moreover, $X$ is periodic if and only
if $X^{r}$ is periodic for all $r$, if and only if $X$ is (Laplacian)
integral.

Next, we characterize strong cospectrality in $X^{r}$.

######  Theorem 50.

Suppose vertices $u$ and $v$ are strongly cospectral vertices in $X$ with
respect to $M$.

  1. 1.

If $m\notin\sigma_{uv}^{-}(L(X))$, then $u$ and $v$ are Laplacian strongly
cospectral in $X^{r}$ for all $r\geq 1$ with

$\sigma_{uv}^{+}(L)=\\{\lambda+(r-1)m:\lambda\in\sigma_{uv}^{+}(L(X))\backslash\\{0\\}\\}\cup\mathcal{R}\quad\text{and}\quad\sigma_{uv}^{-}(L)=\\{\mu+(r-1)m:\mu\in\sigma_{uv}^{-}(L(X))\\},$

where $\mathcal{R}=\\{0,rm\\}$ if $X$ is connected and
$\mathcal{R}=\\{0,rm,(r-1)m\\}$ otherwise. If we add that $X\neq K_{2}$ is
unweighted, then $u$ and $v$ are strongly cospectral in $X^{r}$ for all $r\geq
1$.

  2. 2.

If $k-m\notin\sigma_{uv}^{-}(A(X))$, then $u$ and $v$ are adjacency strongly
cospectral in $X^{r}$ for all $r\geq 1$ with

$\sigma_{uv}^{+}(A)=\sigma_{uv}^{+}(A(X))\backslash\\{k\\}\cup\mathcal{R}\quad\text{and}\quad\sigma_{uv}^{-}(A)=\sigma_{uv}^{-}(A(X)),$

where $\mathcal{R}=\\{k-m,k+(r-1)m\\}$ if $X$ is connected and
$\mathcal{R}=\\{k-m,k+(r-1)m,k\\}$ otherwise.

###### Proof.

Note that (1) follows from Theorem 19(1) and Remark 20. To prove (2), note
that $X^{r}$ is $(k+(r-1)m)$-regular with $rm$ vertices. Taking $Y=X^{r-1}$,
we get $X^{r}=X\vee Y$ and $D=((k+(r-2)m)-k)^{2}+4(r-1)m^{2}=r^{2}m^{2}$, and
so $\lambda^{-}=k-m$ and $\lambda^{+}=k+(r-1)m$. Invoking Theorem 19(2) yields
(2). ∎

Taking $Y=X^{r-1}$ in Theorem 23 yields a characterization of Laplacian PST in
$r$-fold joins.

######  Theorem 51.

Let $r,m\geq 2$ and $\phi(L(X),t)\in\mathbb{Z}[t]$. Vertices $u$ and $v$ in
$X$ admit Laplacian perfect state transfer in $X^{r}$ if and only if either:

  1. 1.

$X=O_{2}$, in which case $X^{r}=CP(2r)$, where $r$ is even.

  2. 2.

Vertices $u$ and $v$ belong to the same connected component of $X$ and all of
the conditions below hold.

    1. (a)

Vertices $u$ and $v$ are Laplacian strongly cospectral in $X$ and
$m\notin\sigma_{uv}^{-}(L(X))$.

    2. (b)

The eigenvalues in $\sigma_{u}(L(X))$ are all integers.

    3. (c)

One of the following conditions hold.

      1. i.

$X$ is connected and one of the following conditions hold for all
$0\neq\lambda\in\sigma_{uv}^{+}(L(X))\cup\\{m\\}$ and
$\mu,\theta\in\sigma_{uv}^{-}(L(X))$.

        1. A.

$\nu_{2}(\lambda)>\nu_{2}(\mu)=\nu_{2}(\theta)$ and $r$ is any integer.

        2. B.

$\nu_{2}(\mu)>\nu_{2}(\lambda)$ and $r$ is even.

        3. C.

$\nu_{2}(\lambda)=\nu_{2}(\mu)$,
$\nu_{2}\left(\frac{\lambda+(r-1)m}{2^{\nu_{2}(m)}}\right)>\nu_{2}\left(\frac{\mu+(r-1)m}{2^{\nu_{2}(m)}}\right)=\nu_{2}\left(\frac{\theta+(r-1)m}{2^{\nu_{2}(m)}}\right)$
and $r$ is even.

      2. ii.

$X\neq O_{2}$ is disconnected and (iA) above holds.

If these conditions hold, then $\tau_{X^{r}}=\frac{\pi}{g}$, where
$g=\operatorname{gcd}\left(\mathcal{T}\right)$,
$\mathcal{T}=\\{rm\\}\cup\\{\lambda-m:\lambda\in\sigma_{u}(L(X))\backslash\\{0\\}\\}$
if $X$ is connected, and
$\mathcal{T}=\\{m\\}\cup\sigma_{u}(L(X))\backslash\\{0\\}$ otherwise.

######  Example 52.

Let $X=P_{3}$ with end vertices $u$ and $v$, and consider $X^{r}$ for any
integer $r$. From Example 44, $\sigma_{uv}^{+}(L(X))=\\{0,3\\}$,
$\sigma_{uv}^{-}(L(X))=\\{1\\}$ and Laplacian PST does not occur in $X$. Since
$\lambda=m=3$ and $\mu=1$ are odd, we have
$\nu_{2}(\lambda+(r-1)m)=\nu_{2}(r)$ and $\nu_{2}(\mu+(r-1)m)=1$. By Theorem
51(2ci), PST occurs between $u$ and $v$ in $X^{r}$ if and only if $r\equiv 0$
(mod 4). In this case, $P_{3}^{r}=(O_{2}\vee O_{1})^{r}=CP(2r)\vee K_{r}$.
Indeed, Theorem 51(1) yields PST in $CP(2r)$ between non-adjacent vertices
with minimum time $\frac{\pi}{2}$, and as $r\equiv 0$ (mod 4), Corollary 28
implies that this PST is preserved in $CP(2r)\vee K_{r}$ at the same time.

Using Corollary 28, we characterize when Laplacian PST in preserved in self-
joins.

######  Corollary 53.

Let $m\geq 2$ and $\phi(L(X),t)\in\mathbb{Z}[t]$. Suppose Laplacian perfect
state transfer occurs between vertices $u$ and $v$ in $X$ and
$\tau_{X}=\frac{\pi}{h}$. Then it occurs between them in $X^{r}$ if and only
if $m\notin\sigma_{uv}^{-}(L(X))$ and $m\equiv 0$ (mod $2^{\alpha}$), where
$\alpha>\nu_{2}(h)$. In this case, $\tau_{X^{r}}=\frac{\pi}{g}$, where
$g=\operatorname{gcd}(\mathcal{T})$, $\mathcal{T}$ is given in Theorem 51 and
$\nu_{2}(g)=\nu_{2}(h)$. Moreover, if $X$ is connected, then $\frac{h}{g}$ is
rational. Otherwise, $\frac{h}{g}=\frac{h}{\operatorname{gcd}(h,m)}$ is odd.

In Theorem 51(2a) and Corollary 53, we may drop the condition
$m\notin\sigma_{uv}^{-}(L(X))$ if $X\neq K_{2}$ is unweighted. Taking
$Y=X^{r-1}$ in Theorem 26 yields a characterization of adjacency PST in
$r$-fold joins.

######  Theorem 54.

Let $r,m\geq 2$, $k\in\mathbb{Z}$ and $\phi(A(X),t)\in\mathbb{Z}[t]$. If $X$
is simple, then vertices $u$ and $v$ in $X$ admit adjacency perfect state
transfer in $X^{r}$ if and only if either:

  1. 1.

$X=O_{2}$, in which case $X^{r}=CP(2r)$, where $r$ is even.

  2. 2.

Vertices $u$ and $v$ belong to the same connected component of $X$ and all of
the conditions below hold.

    1. (a)

Vertices $u$ and $v$ are adjacency strongly cospectral in $X$ and
$k-m\notin\sigma_{uv}^{-}(L(X))$.

    2. (b)

The eigenvalues in $\sigma_{u}(A(X))$ are all integers.

    3. (c)

One of the following conditions hold.

      1. i.

$X$ is connected and one of the following conditions hold for all
$0\neq\lambda\in\sigma_{uv}^{+}(A(X))\backslash\\{k\\}$ and
$\mu,\theta\in\sigma_{uv}^{-}(A(X))$.

        1. A.

$\nu_{2}(m)>\nu_{2}(k-\mu)$, $\nu_{2}(k-\lambda)>\nu_{2}(k-\mu)$ and $r$ is
any integer.

        2. B.

$\nu_{2}(k-\mu)>\nu_{2}(m)=\nu_{2}(k-\lambda)$ and $r$ is even

        3. C.

$\nu_{2}(m)=\nu_{2}(k-\eta)=\nu_{2}(k-\mu)$,
$\nu_{2}\left(\frac{k-m-\lambda}{2^{\nu_{2}(m)}}\right)>\nu_{2}\left(\frac{k-m-\mu}{2^{\nu_{2}(m)}}\right)=\nu_{2}\left(\frac{k-m-\theta}{2^{\nu_{2}(m)}}\right)$
and $r$ is an integer such that
$\nu_{2}(r)>\nu_{2}\left(\frac{k-m-\mu}{2^{\nu_{2}(m)}}\right)$.

      2. ii.

$X\neq O_{2}$ is disconnected and (iA) above holds.

If these conditions hold, then $\tau_{X^{r}}=\frac{\pi}{g}$, where
$g=\operatorname{gcd}\left(\mathcal{T}\right)$,
$\mathcal{T}=\\{rm\\}\cup\\{k-m-\lambda:\lambda\in\sigma_{u}(A(X))\backslash\\{k\\}\\}$
if $X$ is connected and
$\mathcal{T}=\\{m\\}\cup\\{k-\lambda:\lambda\in\sigma_{u}(A(X))\backslash\\{k\\}\\}$
otherwise.

######  Example 55.

Let $X=CP(m)$, where $m\equiv 2$ (mod 4). From Example 41,
$\sigma_{uv}^{+}(A(X))=\\{m-2,-2\\}$, $\sigma_{uv}^{-}(A(X))=\\{0\\}$ and
adjacency PST does not occur between non-adjacent vertices $u$ and $v$. If we
let $k=m-2$, $\lambda=-2$ and $\mu=0$, then
$\nu_{2}(k-\mu)>\nu_{2}(m)=\nu_{2}(k-\lambda)$. By Theorem 54(2ciB), adjacency
PST occurs between $u$ and $v$ in $X^{r}$ if and only if $r$ is even. In this
case, $X^{r}=CP(m)^{r}=CP(rm)$. Since $rm=2r(m/2)$, we may also use Theorem
54(1) to conclude that $X^{r}$ has PST for all even $r$.

We also characterize when adjacency PST in preserved in self-joins using
Corollary 35.

######  Corollary 56.

Let $r,m\geq 2$, $k\in\mathbb{Z}$ and $\phi(A(X),t)\in\mathbb{Z}[t]$. Suppose
$X$ is simple and adjacency perfect state transfer occurs between vertices $u$
and $v$ in $X$ and $\tau_{X}=\frac{\pi}{h}$. Then it occurs between them in
$X^{r}$ if and only if $k-m\notin\sigma_{uv}^{-}(A(X))$ and
$\nu_{2}(m)>\nu_{2}(k-\mu)$ for all $\mu\in\sigma_{uv}^{-}(A(X))$. Here,
$\tau_{X^{r}}=\frac{\pi}{g}$, where $g=\operatorname{gcd}(\mathcal{T})$,
$\mathcal{T}$ is given in Theorem 54 and $\nu_{2}(g)=\nu_{2}(h)$. Further, if
$X$ is connected, $\frac{h}{g}$ is rational. Otherwise,
$\frac{h}{g}=\frac{h}{\operatorname{gcd}(h,m)}$ is odd.

###  10.2 Iterated join graphs

In this subsection, we characterize Laplacian strong cospectrality and PST in
iterated join graphs.

An iterated join graph is a weighted graph that is either of the form:

  * (i)

$((((X_{m_{1}}\vee X_{m_{2}})\cup X_{m_{3}})\vee X_{m_{4}})\ldots)\vee
X_{m_{2k}}:=\Gamma(X_{m_{1}},\ldots,X_{m_{2k}})$

  * (ii)

$((((X_{m_{1}}\cup X_{m_{2}})\vee X_{m_{3}})\cup X_{m_{4}})\ldots)\vee
X_{m_{2k+1}}:=\Gamma(X_{m_{1}},\ldots,X_{m_{2k+1}})$,

where $X_{m_{1}},\ldots,X_{m_{2r}}$ are weighted graphs on
$m_{1},\ldots,m_{2r}\geq 1$ vertices, respectively.

If we take $X_{j}=O_{m_{j}}$ for odd $j$ and $X_{j}=K_{m_{j}}$ otherwise, then
$\Gamma(O_{m_{1}},K_{m_{2}}\ldots,K_{m_{2k}})$ is a connected threshold graph.
$\Gamma(K_{m_{1}},O_{m_{2}},\ldots,K_{m_{2k+1}})$ is also a threshold graph,
where $X_{j}=O_{m_{j}}$ for even $j$ and $X_{j}=K_{m_{j}}$ otherwise. In fact,
graphs of either form completely describe the class of connected threshold
graphs [Severini]. Thus, iterated join graphs are generalizations of connected
theshold graphs.

Lemma 3 and the fact that the union preserves eigenvalue supports yield the
following result which determines the eigenvalue support of each vertex in an
iterated join graph.

######  Proposition 57.

The following hold.

  1. 1.

Let $X=\Gamma(X_{m_{1}},\ldots,X_{m_{2k}})$ and suppose $u\in V(X_{m_{j}})$
for some $j\in\\{1,\ldots,2k\\}$. Define
$\alpha_{h}=\sum_{\ell=1}^{h}m_{\ell}$,
$\beta_{h}=\sum_{\ell=1}^{(2k-h)/2}m_{h+2\ell}$ when $h\neq 2k$ and
$\beta_{2k}=0$. The following hold.

    1. (a)

If $j$ is odd, then
$\sigma_{u}(L(X))=\\{\lambda+\beta_{j-1}:\lambda\in\sigma_{u}(L(X_{m_{j}}))\backslash\\{0\\}\\}\cup\\{\alpha_{h}+\beta_{h}:j\leq
h\leq 2k\ \text{is even}\\}\cup\mathcal{R}$, where $\mathcal{R}=\\{0\\}$
whenever $j=1$ and $X_{m_{1}}$ is connected, and
$\mathcal{R}=\\{0,\beta_{j-1}\\}$ otherwise.

    2. (b)

If $j$ is even, then
$\sigma_{u}(L(X))=\\{\lambda+\alpha_{j-1}+\beta_{j}:\lambda\in\sigma_{u}(L(X_{m_{j}}))\backslash\\{0\\}\\}\cup\\{\alpha_{h}+\beta_{h}:j\leq
h\leq 2k\ \text{is even}\\}\cup\mathcal{R}$, where $\mathcal{R}=\\{0\\}$
whenever $X_{j}$ is connected and $\mathcal{R}=\\{0,\alpha_{j-1}+\beta_{j}\\}$
otherwise.

  2. 2.

Let $X=\Gamma(X_{m_{1}},\ldots,X_{m_{2k+1}})$ and suppose $u\in V(X_{m_{j}})$
for some $j\in\\{1,\ldots,2k+1\\}$. Define
$\alpha_{h}=\sum_{\ell=1}^{h}m_{\ell}$,
$\beta_{h}=\sum_{\ell=1}^{(2k+1-h)/2}m_{h+2\ell}$ when $h\neq 2k+1$ and
$\beta_{2k+1}=0$. The following hold.

    1. (a)

If $j=1$, then
$\sigma_{u}(L(X))=\\{\lambda+\beta_{1}:\lambda\in\sigma_{u}(L(X_{m_{j}}))\backslash\\{0\\}\\}\cup\\{\alpha_{h+1}+\beta_{h}:1\leq
h\leq 2k+1\ \text{is odd}\\}\cup\mathcal{R}$, where
$\mathcal{R}=\\{0,\beta_{1}\\}$.

    2. (b)

If $j\geq 3$ is odd, then
$\sigma_{u}(L(X))=\\{\lambda+\alpha_{j-1}+\beta_{j}:\lambda\in\sigma_{u}(L(X_{m_{j}}))\backslash\\{0\\}\\}\cup\\{\alpha_{h}+\beta_{h}:j\leq
h\leq 2k+1\ \text{is odd}\\}\cup\mathcal{R}$, where $\mathcal{R}=\\{0\\}$ if
$X$ is connected and $\mathcal{R}=\\{0,\alpha_{j-1}+\beta_{j}\\}$ otherwise.

    3. (c)

If $j$ is even, then
$\sigma_{u}(L(X))=\\{\lambda+\beta_{j-1}:\lambda\in\sigma_{u}(L(X_{m_{j}}))\backslash\\{0\\}\\}\cup\\{\alpha_{h}+\beta_{h}:j+1\leq
h\leq 2k+1\ \text{is odd}\\}\cup\mathcal{R}$, where
$\mathcal{R}=\\{0,\beta_{j-1}\\}$.

Using Proposition 57 and the same argument in the proof of Theorem 19 yield
the following characterization of Laplacing strong cospectrality in iterated
join graphs.

######  Theorem 58.

Let $X=\Gamma(X_{m_{1}},\ldots,X_{m_{2k}})$ and consider $\alpha_{h}$ and
$\beta_{h}$ in Proposition 57(1). Two vertices $u$ and $v$ of $X_{m_{j}}$ are
Laplacian strongly cospectral in $X$ if and only if one of the following
conditions hold.

  1. 1.

$j$ is odd and either:

    1. (a)

Vertices $u$ and $v$ are Laplacian strongly cospectral in $X_{m_{j}}$ and
$\alpha_{j}\notin\sigma_{uv}^{-}(L(X_{m_{j}}))$. In this case,
$\sigma_{uv}^{+}(L)=\\{\lambda+\beta_{j-1}:\lambda\in\sigma_{uv}^{+}(L(X_{m_{j}}))\backslash\\{0\\}\\}\cup\\{\alpha_{h}+\beta_{h}:j\leq
h\leq 2k\ \text{is even}\\}\cup\mathcal{R}$ and
$\sigma_{uv}^{-}(L)=\\{\mu+\beta_{j-1}:\mu\in\sigma_{uv}^{-}(L(X_{m_{j}}))\\}$,
where $\mathcal{R}$ is given in Proposition 57(1a).

    2. (b)

$j=1$ and $X_{m_{1}}=O_{2}$ with
$\sigma_{uv}^{+}(L)=\\{\alpha_{h}+\beta_{h}:1\leq h\leq 2k\ \text{is
even}\\}\cup\\{0\\}$ and $\sigma_{uv}^{-}(L)=\\{\beta_{0}\\}$.

  2. 2.

$j$ is even and either:

    1. (a)

Vertices $u$ and $v$ are Laplacian strongly cospectral in $X_{m_{j}}$ and
$m_{j}\notin\sigma_{uv}^{-}(L(X_{m_{j}}))$. In this case,
$\sigma_{uv}^{+}(L)=\\{\lambda+\alpha_{j-1}+\beta_{j}:\lambda\in\sigma_{uv}^{+}(L(X_{m_{j}}))\backslash\\{0\\}\\}\cup\\{\alpha_{h}+\beta_{h}:j\leq
h\leq 2k\ \text{is even}\\}\cup\mathcal{R}$ and
$\sigma_{uv}^{-}(L)=\\{\mu+\alpha_{j-1}+\beta_{j}:\mu\in\sigma_{uv}^{-}(L(X_{m_{j}}))\\}$,
where $\mathcal{R}$ is given in Proposition 57(1b).

    2. (b)

$X_{m_{j}}=O_{2}$ with $\sigma_{uv}^{+}(L)=\\{\alpha_{h}+\beta_{h}:j\leq h\leq
2k\ \text{is even}\\}\cup\\{0\\}$ and
$\sigma_{uv}^{-}(L)=\\{\alpha_{j-1}+\beta_{j}\\}$.

######  Theorem 59.

Let $X=\Gamma(X_{m_{1}},\ldots,X_{m_{2k+1}})$ and consider $\alpha_{h}$,
$\beta_{h}$ and $\mathcal{R}$ in Proposition 57(2). Two vertices $u$ and $v$
of $X_{m_{j}}$ are Laplacian strongly cospectral in $X$ if and only if one of
the following conditions hold.

  1. 1.

$j=1$, vertices $u$ and $v$ are Laplacian strongly cospectral in $X_{m_{1}}$
and $m_{1}+m_{2}\notin\sigma_{uv}^{-}(L(X_{m_{1}}))$. In this case,
$\sigma_{uv}^{+}(L)=\\{\lambda+\beta_{1}:\lambda\in\sigma_{uv}^{+}(L(X_{m_{1}}))\backslash\\{0\\}\\}\cup\\{\alpha_{h+1}+\beta_{h}:1\leq
h\leq 2k+1\ \text{is odd}\\}\cup\mathcal{R}$ and
$\sigma_{uv}^{-}(L)=\\{\mu+\beta_{1}:\mu\in\sigma_{uv}^{-}(L(X_{m_{1}}))\\}$,
where $\mathcal{R}$ is given in Proposition 57(2a).

  2. 2.

$j\geq 3$ is odd and either:

    1. (a)

Vertices $u$ and $v$ are Laplacian strongly cospectral in $X_{m_{j}}$ and
$m_{j}\notin\sigma_{uv}^{-}(L(X_{m_{j}}))$. In this case,
$\sigma_{uv}^{+}(L)=\\{\lambda+\alpha_{j-1}+\beta_{j}:\lambda\in\sigma_{uv}^{+}(L(X_{m_{j}}))\backslash\\{0\\}\\}\cup\\{\alpha_{h}+\beta_{h}:j\leq
h\leq 2k+1\ \text{is odd}\\}\cup\mathcal{R}$ and
$\sigma_{uv}^{-}(L)=\\{\mu+\alpha_{j-1}+\beta_{j}:\mu\in\sigma_{uv}^{-}(L(X_{m_{j}}))\\}$,
where $\mathcal{R}$ is given in Proposition 57(2b).

    2. (b)

$X_{m_{j}}=O_{2}$ with $\sigma_{uv}^{+}(L)=\\{\alpha_{h}+\beta_{h}:j\leq h\leq
2k\ \text{is odd}\\}\cup\\{0\\}$ and
$\sigma_{uv}^{-}(L)=\\{\alpha_{j-1}+\beta_{j}\\}$.

  3. 3.

$j$ is even, vertices $u$ and $v$ are Laplacian strongly cospectral in
$X_{m_{j}}$, and $\alpha_{j}\notin\sigma_{uv}^{-}(L(X_{m_{j}}))$. In this
case,
$\sigma_{uv}^{+}(L)=\\{\lambda+\beta_{j-1}:\lambda\in\sigma_{uv}^{+}(L(X_{m_{j}}))\backslash\\{0\\}\\}\cup\\{\alpha_{h}+\beta_{h}:j\leq
h\leq 2k+1\ \text{is odd}\\}\cup\mathcal{R}$ and
$\sigma_{uv}^{-}(L)=\\{\mu+\beta_{j-1}:\mu\in\sigma_{uv}^{-}(L(X_{m_{j}}))\\}$,
where $\mathcal{R}$ is given in Proposition 57(2c).

######  Remark 60.

From Theorems 58 and 59, a pair of strongly cospectral vertices in an iterated
join graph belong to the same $X_{m_{j}}$. Consequently, if $u$ and $v$ belong
to $X_{m_{j}}$ and $X_{m_{\ell}}$ respectively, then PST cannot occur between
them.

Our next endeavour is to characterize PST in iterated join graphs, starting
with those of the form $\Gamma(X_{m_{1}},\ldots,X_{m_{2k}})$. We separate the
cases when $j$ is odd and $j$ is even. The following result follows from
combining Corollary 7, Theorem 58, the same argument in the proof of Theorem
23, and the fact that

$\alpha_{h}+\beta_{h}=\alpha_{j}+\gamma_{h}+\beta_{j-1}$ (19)

for a fixed odd $j$ and for all even $j\leq h\leq 2k$ in Theorem 58(1) with
$\gamma_{h}=0$ whenever $h=j+1$ and
$\gamma_{h}=m_{j+2}+m_{j+4}++m_{j+6}\ldots+m_{h-1}$ otherwise, and

$\alpha_{h}+\beta_{h}=\alpha_{j-1}+\delta_{h}+\beta_{j}$ (20)

for a fixed even $j$ and for all even $j\leq h\leq 2k$ in Theorem 58(2) with
$\delta_{h}=m_{j}+m_{j+1}+m_{j+3}+\ldots+m_{h-1}$.

######  Theorem 61.

Let $X=\Gamma(X_{m_{1}},\ldots,X_{m_{2k}})$, $\phi(L(X),t)\in\mathbb{Z}[t]$
and $m_{j}\geq 2$ for some odd $j$. Consider $\alpha_{h}$ and $\beta_{h}$ in
Proposition 57(1) and $\gamma_{h}$ in (19). Two vertices $u$ and $v$ in
$X_{m_{j}}$ admit Laplacian perfect state transfer in $X$ if and only if all
of the following conditions hold.

  1. 1.

Either (i) $u$ and $v$ are Laplacian strongly cospectral in $X_{m_{j}}$ and
$\alpha_{j}\notin\sigma_{uv}^{-}(L(X_{m_{j}}))$ or (ii) $j=1$ and
$X_{m_{1}}=O_{2}$.

  2. 2.

The eigenvalues in $\sigma_{u}(L(X_{m_{j}}))$ are all integers.

  3. 3.

One of the following conditions hold.

    1. (a)

$X_{m_{j}}$ is connected and one of the following conditions hold for all
$\lambda\in\sigma_{uv}^{+}(L(X_{m_{j}}))\backslash\\{0\\}$,
$\mu,\theta\in\sigma_{uv}^{-}(L(X_{m_{j}}))$ and even $j\leq h\leq 2k$.

      1. i.

$\nu_{2}(\beta_{j-1})>\nu_{2}(\mu)=\nu_{2}(\theta)$,
$\nu_{2}(\lambda)>\nu_{2}(\mu)$ and
$\nu_{2}(\alpha_{j}+\gamma_{h})>\nu_{2}(\mu)$.

      2. ii.

$\nu_{2}(\mu)>\nu_{2}(\beta_{j-1})=\nu_{2}(\lambda)=\nu_{2}(\alpha_{j}+\gamma_{h})$.

      3. iii.

$\nu_{2}(\beta_{j-1})=\nu_{2}(\mu)=\nu_{2}(\lambda)=\nu_{2}(\alpha_{j}+\gamma_{h})$,
$\nu_{2}\left(\frac{\lambda+\beta_{j-1}}{2^{\nu_{2}(\beta_{j-1})}}\right)>\nu_{2}\left(\frac{\mu+\beta_{j-1}}{2^{\nu_{2}(\beta_{j-1})}}\right)=\nu_{2}\left(\frac{\theta+\beta_{j-1}}{2^{\nu_{2}(\beta_{j-1})}}\right)$
and
$\nu_{2}\left(\frac{\alpha_{h}+\beta_{h}}{2^{\nu_{2}(\beta_{j-1})}}\right)>\nu_{2}\left(\frac{\mu+\beta_{j-1}}{2^{\nu_{2}(\beta_{j-1})}}\right).$

    2. (b)

$X_{m_{j}}$ is disconnected and (ai) above holds.

    3. (c)

$j=1$, $X_{m_{1}}=O_{2}$ and
$\nu_{2}(2+\gamma_{h}+\beta_{0})>\nu_{2}(\beta_{0})$ for each even $1\leq
h\leq 2k$.

Further, $\tau_{X\vee Y}=\frac{\pi}{g}$, where
$g=\operatorname{gcd}\left(\mathcal{T}\right)$, where
$\mathcal{T}=\sigma_{u}(L(X))$ is given in Proposition 57(1).

######  Theorem 62.

Let $X=\Gamma(X_{m_{1}},\ldots,X_{m_{2k}})$, $\phi(L(X),t)\in\mathbb{Z}[t]$
and $m_{j}\geq 2$ for some even $j$. Consider $\alpha_{h}$ and $\beta_{h}$ in
Proposition 57(1) and $\delta_{h}$ in (20) Two vertices $u$ and $v$ in
$X_{m_{j}}$ admit Laplacian perfect state transfer in $X$ if and only if all
of the following conditions hold.

  1. 1.

Either (i) $u$ and $v$ are Laplacian strongly cospectral in $X_{m_{j}}$ and
$m_{j}\notin\sigma_{uv}^{-}(L(X_{m_{j}}))$ or (ii) $X_{m_{j}}=O_{2}$.

  2. 2.

The eigenvalues in $\sigma_{u}(L(X_{m_{j}}))$ are all integers.

  3. 3.

One of the following conditions hold.

    1. (a)

$X_{m_{j}}$ is connected and one of the following conditions hold for all
$\lambda\in\sigma_{uv}^{+}(L(X_{m_{j}}))\backslash\\{0\\}$,
$\mu,\theta\in\sigma_{uv}^{-}(L(X_{m_{j}}))$ and even $j\leq h\leq 2k$.

      1. i.

$\nu_{2}(\alpha_{j-1}+\beta_{j})>\nu_{2}(\mu)=\nu_{2}(\theta)$,
$\nu_{2}(\lambda)>\nu_{2}(\mu)$ and $\nu_{2}(\delta_{h})>\nu_{2}(\mu)$.

      2. ii.

$\nu_{2}(\mu)>\nu_{2}(\alpha_{j-1}+\beta_{j})=\nu_{2}(\lambda)=\nu_{2}(\delta_{h})$.

      3. iii.

$\nu_{2}(\alpha_{j-1}+\beta_{j})=\nu_{2}(\mu)=\nu_{2}(\lambda)=\nu_{2}(\delta_{h})$,
$\nu_{2}\left(\frac{\lambda+\alpha_{j-1}+\beta_{j}}{2^{\nu_{2}(\alpha_{j-1}+\beta_{j})}}\right)>\nu_{2}\left(\frac{\mu+\alpha_{j-1}+\beta_{j}}{2^{\nu_{2}(\alpha_{j-1}+\beta_{j})}}\right)=\nu_{2}\left(\frac{\theta+\alpha_{j-1}+\beta_{j}}{2^{\nu_{2}(\alpha_{j-1}+\beta_{j})}}\right)$
and
$\nu_{2}\left(\frac{\alpha_{h}+\beta_{h}}{2^{\nu_{2}(\alpha_{j-1}+\beta_{j})}}\right)>\nu_{2}\left(\frac{\mu+\alpha_{j-1}+\beta_{j}}{2^{\nu_{2}(\alpha_{j-1}+\beta_{j})}}\right).$

    2. (b)

$X_{m_{j}}$ is disconnected and (ai) above holds.

    3. (c)

$X_{m_{j}}=O_{2}$ and
$\nu_{2}(\alpha_{j-1}+\delta_{h}+\beta_{j})>\nu_{2}(\alpha_{j-1}+\beta_{j})$
for each even $j\leq h\leq 2k$.

Further, $\tau_{X\vee Y}=\frac{\pi}{g}$, where
$g=\operatorname{gcd}\left(\mathcal{T}\right)$, where
$\mathcal{T}=\sigma_{u}(L(X))$ is given in Proposition 57(1).

######  Remark 63.

We have the following.

  1. 1.

We obtain a characterization for PST between vertices in $X_{m_{j}}$ in the
graph $X=\Gamma(X_{m_{1}},\ldots,X_{m_{2k+1}})$ whenever $m_{j}\geq 2$ for
some even $j\geq 2$ by replacing ‘even $j\leq h\leq 2k$’ with ‘odd $j\leq
h\leq 2k+1$’ in Theorem 61.

  2. 2.

We obtain a characterization for PST between vertices in $X_{m_{j}}$ in the
graph $X=\Gamma(X_{m_{1}},\ldots,X_{m_{2k+1}})$ whenever $m_{j}\geq 2$ for
some odd $j\geq 3$ by replacing ‘even $j\leq h\leq 2k$’ with ‘odd $j\leq h\leq
2k+1$’ in Theorem 62.

In order to complete our characterization of PST in iterated join graphs, it
suffices to determine PST between vertices in $X_{m_{1}}$ in the graph
$X=\Gamma(X_{m_{1}},\ldots,X_{m_{2k+1}})$ by virtue of Remark 63. To do this,
we combine Corollary 7, Theorem 58, the same argument in the proof of Theorem
23, and the fact that

$\alpha_{h+1}+\beta_{h}=\phi_{h}+\beta_{1}$ (21)

for all odd $1\leq h\leq 2k+1$ in Theorem 59(1) for a fixed odd $j$ with
$\phi_{h}=m_{1}+m_{2}+m_{4}+m_{6}+\ldots+m_{h-1}$.

######  Theorem 64.

Let $X=\Gamma(X_{m_{1}},\ldots,X_{m_{2k+1}})$, $m_{1}\geq 2$ and
$\phi(L(X),t)\in\mathbb{Z}[t]$. Consider $\alpha_{h}$ and $\beta_{h}$ in
Proposition 57(2) and $\phi_{h}$ in (21). Two vertices $u$ and $v$ in
$X_{m_{1}}$ admit Laplacian perfect state transfer in $X$ if and only if all
of the following conditions hold.

  1. 1.

Vertices $u$ and $v$ are Laplacian strongly cospectral in $X_{m_{1}}$ and
$m_{1}+m_{2}\notin\sigma_{uv}^{-}(L(X_{m_{1}}))$.

  2. 2.

The eigenvalues in $\sigma_{u}(L(X_{m_{1}}))$ are all integers.

  3. 3.

One of the following conditions hold.

    1. (a)

$X_{m_{1}}$ is connected and one of the following conditions hold for all
$\lambda\in\sigma_{uv}^{+}(L(X_{m_{1}}))\backslash\\{0\\}$,
$\mu,\theta\in\sigma_{uv}^{-}(L(X_{m_{1}}))$ and odd $1\leq h\leq 2k+1$.

      1. i.

$\nu_{2}(\beta_{1})>\nu_{2}(\mu)=\nu_{2}(\theta)$,
$\nu_{2}(\lambda)>\nu_{2}(\mu)$ and $\nu_{2}(\phi_{h})>\nu_{2}(\mu)$.

      2. ii.

$\nu_{2}(\mu)>\nu_{2}(\beta_{1})=\nu_{2}(\lambda)=\nu_{2}(\phi_{h})$.

      3. iii.

$\nu_{2}(\beta_{1})=\nu_{2}(\mu)=\nu_{2}(\lambda)=\nu_{2}(\phi_{h})$,
$\nu_{2}\left(\frac{\lambda+\beta_{1}}{2^{\nu_{2}(\beta_{1})}}\right)>\nu_{2}\left(\frac{\mu+\beta_{1}}{2^{\nu_{2}(\beta_{1})}}\right)=\nu_{2}\left(\frac{\theta+\beta_{1}}{2^{\nu_{2}(\beta_{1})}}\right)$
and
$\nu_{2}\left(\frac{\alpha_{h}+\beta_{h}}{2^{\nu_{2}(\beta_{1})}}\right)>\nu_{2}\left(\frac{\mu+\beta_{1}}{2^{\nu_{2}(\beta_{1})}}\right).$

    2. (b)

$X_{m_{1}}$ is disconnected and (ai) above holds.

In both cases, $u$ and $v$ belong to the same component in $X_{m_{1}}$.

Further, $\tau_{X\vee Y}=\frac{\pi}{g}$, where
$g=\operatorname{gcd}\left(\mathcal{T}\right)$, where
$\mathcal{T}=\sigma_{u}(L(X))$ is given in Proposition 57(2).

######  Remark 65.

If $X_{m_{j}}\neq K_{2}$ is unweighted, then we may drop the condition
$m_{1}+m_{2}\in\sigma_{uv}^{-}(L(X_{m_{1}}))$ in Theorem 59(1), the condition
$\alpha_{j}\notin\sigma_{uv}^{-}(L(X_{m_{j}}))$ in Theorems 58(1a) and 59(3),
and the condition $m_{j}\notin\sigma_{uv}^{-}(L(X_{m_{j}}))$ in Theorems
58(2a) and 59(2a). The same applies to Theorems 61, 62 and 64.

######  Remark 66.

The conditions in Theorem 61(3c) are equivalent to $m_{1}=2$ and

$\nu_{2}(2+\beta_{0})>\nu_{2}(\beta_{0}),\quad\nu_{2}(2+m_{3}+\beta_{0})>\nu_{2}(\beta_{0}),\quad\ldots\quad\nu_{2}(2+m_{3}+m_{5}+\ldots+m_{2k-1}+\beta_{0})>\nu_{2}(\beta_{0}).$

The first inequality above is equivalent to $\beta_{0}\equiv 2$ (mod 4), and
the latter ones yield $m_{j}\equiv 0$ (mod 4) for all odd $j\geq 3$. Since
$X=Z_{1}\vee X_{m_{2k}}$ for some disconnected graph $Z_{1}$, applying Theorem
23(3ab) yields $\nu_{2}(m_{2k})>\nu_{2}(\beta_{0})$ and
$\nu_{2}(m_{1}+m_{2}+m_{4}+\ldots+m_{2k-2})>\nu_{2}(\beta_{0})$. Now,
Corollary 30 implies that PST also occurs between $u$ and $v$ in $Z_{2}$,
where $Z_{2}$ is the graph such that $Z_{1}=Z_{2}\cup X_{m_{2k-1}}$. The same
argument applied $k-2$ times yields $\nu_{2}(m_{j})>\nu_{2}(\beta_{0})$ for
all even $j\geq 3$ and PST between $u$ and $v$ in $Z_{m_{k-2}}=O_{2}\vee
X_{m_{2}}$. As $\beta_{0}\equiv 2$ (mod 4), the former condition is equivalent
to $m_{j}\equiv 0$ (mod 4) for all even $j\geq 3$, while the latter is
equivalent to $m_{2}\equiv 2$ (mod 4) by virtue of Theorem 23(3c). These
considerations imply that Theorem 61(3c) holds if and only if $m_{1}=2$,
$m_{2}\equiv 2$ (mod 4) and $m_{j}\equiv 0$ (mod 4) for all $j\geq 3$. One can
also check that Theorem 64 holds for $X_{m_{1}}=K_{2}$ if and only if
$m_{1}=2$, $m_{2}\equiv 2$ (mod 4) and $m_{j}\equiv 0$ (mod 4) for all $j\geq
3$. In both cases, the minimum PST time is $\frac{\pi}{2}$.

######  Corollary 67.

Let $X$ be a connected threshold graph. Then Laplacian perfect state transfer
occurs in $X$ if and only if $m_{1}=2$, $m_{2}\equiv 2$ (mod 4) and
$m_{j}\equiv 2$ (mod 4) for all $j\geq 3$. In this case, Laplacian perfect
state transfer occurs between vertices $u$ and $v$ in
$X_{m_{1}}\in\\{O_{2},K_{2}\\}$ with minimum PST time is $\frac{\pi}{2}$.

###### Proof.

By assumption,
$X\in\\{\Gamma(O_{m_{1}},K_{m_{1}},\ldots,K_{m_{2k}}),\Gamma(K_{m_{1}},O_{m_{1}},\ldots,K_{m_{2k+1}})\\}$.
First, suppose $X=\Gamma(O_{m_{1}},K_{m_{1}},\ldots,K_{m_{2k}})$. For two
vertices $u$ and $v$ to be strongly cospectral in $X$, Theorem 19(3) implies
that they both belong to $O_{m_{j}}$ if $j$ is odd or $K_{m_{j}}$ if $j$ is
even. Hence, we have three cases.

  * •

Let $j=1$. Then vertices $u$ and $v$ of $O_{m_{1}}$ are strongly cospectral in
$X$ if and only if $m_{1}=2$.

  * •

Suppose $j\geq 3$ is odd so that vertices $u$ and $v$ both belong to
$O_{m_{j}}$. Then we can write $X=(((Y\cup O_{m_{j}})\vee
K_{m_{j+1}})\cup\ldots)\vee K_{m_{2k}}$ for some graph $Y$. Now, Theorem 19(1)
and an inductive argument implies that either (i) $u$ and $v$ are strongly
cospectral in $(Y\cup X_{m_{j}})\vee X_{m_{j+1}}$ and
$m_{j}\notin\sigma_{uv}^{-}(L(X_{j}))$ or (ii) $Y\cup X_{m_{j}}=O_{2}$. Since
$Y$ has at least one vertex, the latter case does not hold. Hence, it must be
that $u$ and $v$ are strongly cospectral in $(Y\cup O_{m_{j}})\vee
K_{m_{j+1}}$. But this implies that $u$ and $v$ are strongly cospectral in
$O_{m_{j}}$, which cannot happen since by Theorem 19, the only graph with
isolated vertices that admit strong cospectrality in the join is $O_{2}$,
which is not equal to $Y\cup O_{m_{j}}$.

  * •

Suppose $j\geq 3$ is even so that $u$ and $v$ belong to $K_{m_{j}}$. Then the
same argument as the previous case implies that $u$ and $v$ are strongly
cospectral in $Y\vee K_{m_{j}}$ and $m_{j}\notin\sigma_{uv}^{-}(L(X_{j}))$.
But this implies that $u$ and $v$ are strongly cospectral in $K_{m_{j}}$,
i.e., $m_{j}=2\in\sigma_{uv}^{-}(L(X_{j}))$, which is a contradiction.

Thus, in $X=\Gamma(O_{m_{1}},K_{m_{1}},\ldots,K_{m_{2k}})$, only the vertices
of $O_{m_{1}}$ admit strong cospectrality in $X$, in which case $m_{1}=2$. The
same holds for $X=\Gamma(O_{m_{1}},K_{m_{1}},\ldots,K_{m_{2k}})$. Cmbining
this with Remark 66 completes the proof. ∎

##  11 Bounds

From the previous sections, PST between $u$ and $w$ in $X$ need not guarantee
PST between $u$ and $v$ in $X\vee Y$. The same can be said about periodicity
of a vertex. Thus, we ask, if $u,v\in V(X)$ then how far can $|U_{M}(X\vee
Y,t)_{u,v}|$ be from $|U_{M}(X,t)_{u,v}|$ as $t$ ranges over $\mathbb{R}$? We
answer this question by providing an upper bound for the absolute value of
$|U_{M}(X\vee Y,t)_{u,v}|-|U_{M}(X,t)_{u,v}|$. To do this, we define

$\alpha_{M}(t)=\begin{cases}\dfrac{me^{-itn}+ne^{itm}-(m+n)}{m(m+n)},&\text{if
}M=L\\\
\dfrac{e^{it\lambda^{+}}(k-\lambda^{-})}{m\sqrt{D}}-\dfrac{e^{it\lambda^{-}}(k-\lambda^{+})}{m\sqrt{D}}-\dfrac{e^{itk}}{m},&\text{if
}M=A.\end{cases}$ (22)

and

$T_{M}=\begin{cases}\\{2j\pi/g:j\in\mathbb{Z}\\},&\text{if }M=L\\\
\\{t\in\mathbb{R}:e^{itk}=e^{it\lambda^{+}}=e^{it\lambda^{-}}\\},&\text{if
}M=A,\end{cases}$ (23)

where $g=\operatorname{gcd}(m,n)$ and $\lambda^{\pm}$ are given in Lemma 2. If
the context is clear, then we simply write as $\alpha_{M}(t)$ and $T_{M}$ as
$\alpha(t)$ and $T$, respectively. If we add that $k,\ell\in\mathbb{Z}$ and
$\Delta$ in Lemma 2 is a perfect square, then we may write
$T_{A}=\\{2j\pi/h:j\in\mathbb{Z}\\}$, where
$h=\operatorname{gcd}(\lambda^{+}-k,\lambda^{-}-k)$.

Combining (4) and (6) with (22) yields the following lemma.

######  Lemma 68.

We have $\alpha_{M}(t)=0$ if and only if $t\in T_{M}$. Moreover, for any
$u,v\in V(X)$ and for all $t$,

$U_{L}(X\vee
Y,t)_{u,v}-e^{itn}U_{L}(X,t)_{u,v}=e^{itn}\alpha_{L}(t)\quad\text{and}\quad
U_{A}(X\vee Y,t)_{u,v}-U_{A}(X,t)_{u,v}=\alpha_{A}(t).$

###  11.1 Laplacian case

######  Theorem 69.

For all $u,v\in V(X)$ and for all $t$

$\big{|}\ U_{L}(X\vee Y,t)_{u,v}-e^{itn}U_{L}(X,t)_{u,v}\ \big{|}\leq 2/m$
(24)

with equality if and only if $\nu_{2}(m)=\nu_{2}(n)$, in which case equality
holds in (24) at time $\tau=j\pi/g$, where $j$ is any odd integer and
$g=\operatorname{gcd}(m,n)$. Moreover, if equality holds in (24), then

$|\ U_{L}(X\vee Y,\tau)_{u,v}-e^{i\tau n}U_{L}(X,\tau)_{u,v}\ |=U_{L}(X\vee
Y,\tau)_{u,v}+U_{L}(X,\tau)_{u,v}=2/m.$ (25)

###### Proof.

Lemma 68 yields (24) with equality if and only if $e^{i\tau m}=e^{i\tau n}=-1$
for some $\tau>0$. Equivalently, $\nu_{2}(m)=\nu_{2}(n)$, in which case
$\tau=j\pi/g$ for some odd $j$. From this, (25) is immediate. ∎

######  Corollary 70.

For all $u,v\in V(X)$ and for all $t$,

$\bigg{|}\ |U_{L}(X\vee Y,t)_{u,v}|-|U_{L}(X,t)_{u,v}|\ \bigg{|}\leq 2/m$ (26)

with equality if and only if $\nu_{2}(m)=\nu_{2}(n)$,
$U_{L}(X,\tau)_{u,v}\in\mathbb{R}$ and either $U_{L}(X,\tau)_{u,v}\leq 0$ or
$U_{L}(X,\tau)_{u,v}\geq 2/m$, in which case equality holds in (26) at time
$\tau$ in Theorem 69.

###### Proof.

Applying the triangle inequality to (24) yields (26). Using (25), we get
equality in (26) if and only if $\big{|}\ |U_{L}(X\vee
Y,\tau)_{u,v}|-|U_{L}(X,\tau)_{u,v}|\ \big{|}=\big{|}\
\big{|}\frac{2}{m}-U_{L}(X,\tau)_{u,v}\big{|}-|U_{L}(X,\tau)_{u,v}|\
\big{|}=\frac{2}{m}$. Result follows. ∎

###  11.2 Adjacency case

The following is an analogue of Theorem 69 for the adjacency case.

######  Theorem 71.

For all $u,v\in V(X)$ and for all $t$,

$\big{|}\ U_{A}(X\vee Y,t)_{u,v}-U_{A}(X,t)_{u,v}\ \big{|}\leq 2/m$ (27)

with equality if and only if there is a time $\tau>0$ such that
$e^{i\tau\lambda^{+}}=e^{i\tau\lambda^{-}}=-e^{i\tau k}$, in which case
$\alpha_{A}(t)=-2e^{i\tau k}/m$. If we add that $k,\ell\in\mathbb{Z}$ and
$\Delta$ in Lemma 2 is a perfect square, then the latter condition yields
$\\{k,\lambda^{\pm}\\}\subseteq\mathbb{Z}$ and
$\nu_{2}(\lambda^{+}-k)=\nu_{2}(\lambda^{-}-k)$, in which case $\tau=j\pi/h$,
where $h=\operatorname{gcd}(\lambda^{+}-k,\lambda^{-}-k)$ and $j$ is any odd.

###### Proof.

Since $\lambda^{\pm}=\frac{1}{2}(k+\ell\pm\sqrt{D})$, one checks that

$\begin{split}\frac{e^{it\lambda^{+}}(k-\lambda^{-})}{m\sqrt{D}}-\frac{e^{it\lambda^{-}}(k-\lambda^{+})}{m\sqrt{D}}&=\frac{e^{it\lambda^{+}}(k-\ell+\sqrt{D})}{2m\sqrt{D}}-\frac{e^{it\lambda^{-}}(k-\ell-\sqrt{D})}{2m\sqrt{D}}\\\
&=\frac{1}{m\sqrt{D}}\left[\sqrt{D}\cos(t\sqrt{D}/2)+i(k-\ell)\sin(t\sqrt{D}/2)\right].\end{split}$
(28)

Consequently, the following equation holds for all $t$

$\begin{split}\left|\frac{e^{it\lambda^{+}}(k-\lambda^{-})}{m\sqrt{D}}-\frac{e^{it\lambda^{-}}(k-\lambda^{+})}{m\sqrt{D}}\right|&=\frac{1}{m\sqrt{D}}\sqrt{D+((k-\ell)^{2}-D)\sin^{2}(t\sqrt{D}/2)}.\end{split}$
(29)

Since $mn>0$ and $D=(k-\ell)^{2}+4mn$, we get that $(k-\ell)^{2}-D=-4mn<0$.
Hence, (29) gives us
$\left|\frac{e^{it\lambda^{+}}(k-\lambda^{-})}{m\sqrt{D}}-\frac{e^{it\lambda^{-}}(k-\lambda^{+})}{m\sqrt{D}}\right|\leq\frac{1}{m}$.
Combining this with (6) yields

$\big{|}U_{A}(X\vee
Y,t)_{u,v}-U_{A}(X,t)_{u,v}\big{|}\leq\left|\frac{e^{it\lambda^{+}}(k-\lambda^{-})}{m\sqrt{D}}-\frac{e^{it\lambda^{-}}(k-\lambda^{+})}{m\sqrt{D}}\right|+\frac{1}{m}\leq\frac{2}{m}$

which establishes (27). Using the first equality in (28), we can write (6) as

$U_{A}(X\vee
Y,t)_{u,v}-U_{A}(X,t)_{u,v}=\frac{e^{it\lambda^{+}}(k-\ell+\sqrt{D})}{2m\sqrt{D}}-\frac{e^{it\lambda^{-}}(k-\ell-\sqrt{D})}{2m\sqrt{D}}-\frac{e^{itk}}{m}$.

Thus, equality holds in (27) if and only if there is a time $\tau$ such that

$e^{i\tau\lambda^{+}}=e^{i\tau\lambda^{-}}=-e^{i\tau k},$ (30)

in which case, $U_{A}(X\vee Y,t)_{u,v}-U_{A}(X,t)_{u,v}=-\frac{2e^{itk}}{m}$.
This implies that $\\{k,\lambda^{\pm}\\}$ satisfies the ratio condition. Thus,
if $k,\ell\in\mathbb{Z}$ and $D$ is a perfect square, then Lemma 6 yields
$\\{k,\lambda^{\pm}\\}\subseteq\mathbb{Z}$. Hence, (30) holds if and only if
$\nu_{2}(\lambda^{+}-k)=\nu_{2}(\lambda^{-}-k)$, in which case $\tau=j\pi/g$
for any odd integer $j$. ∎

######  Corollary 72.

For all $u,v\in V(X)$ and for all $t$,

$\begin{split}\bigg{|}\ |U_{A}(X\vee Y,t)_{u,v}|-|U_{A}(X,t)_{u,v}|\
\bigg{|}&\leq 2/m\end{split}$ (31)

with equality if and only if the bound in (27) holds at time $\tau$ in Theorem
71, $U_{A}(X,\tau)_{u,v}=|U_{A}(X,\tau)_{u,v}|e^{i\tau k}$ and
$|U_{A}(X,\tau)_{u,v}|\geq 2/m$.

###### Proof.

Again applying triangle inequality to (27) yields (31). Making use of Theorem
71, we get equality in (31) if and only if $\big{|}\
|U_{A}(X,\tau)_{u,v}-2e^{i\tau k}/m\ |-|U_{A}(X,\tau)_{u,v}|\ \big{|}=2/m$.
Thus, our result is immediate. ∎

Combining Corollary 70 and 72 yields the following result.

######  Corollary 73.

Let $M\in\\{A,L\\}$. For all $u,v\in V(X)$ and for all $t$,

$\bigg{|}\ |U_{M}(X\vee Y,t)_{u,v}|-|U_{M}(X,t)_{u,v}|\ \bigg{|}\rightarrow
0\quad\text{as $m\rightarrow\infty$}.$

From Corollary 73, we find that for vertices in $X$, the state transfer
properties of the quantum walk on $X\vee Y$ tend to mimic those of the quantum
walk on $X,$ as we increase the number of vertices of $X$.

###  11.3 Tightness

Define $F(t)_{u,v}:=|U_{M}(X\vee Y,t)_{u,v}|-|U_{M}(X,t)_{u,v}|$. From Lemma
68,
$F(t)_{u,v}=\big{|}U_{M}(X,t)_{u,v}+\alpha_{M}(t)\big{|}-|U_{M}(X,t)_{u,v}|$.
for any $u,v\in V(X)$. We now illustrate that the values of $F(t)_{u,v}$ can
be zero, positive or negative.

######  Lemma 74.

The following are immediate from (23).

  1. 1.

If $\tau\in T_{M}$ then $\alpha_{M}(\tau)=0$, and so $F(\tau)_{u,v}=0$.

  2. 2.

If $U_{M}(X,\tau)_{u,v}=0$ for some $\tau\in\mathbb{R}\backslash T_{M}$, then
$F(\tau)_{u,v}=|\alpha_{M}(\tau)|>0$. If $X$ is disconnected and $u$ and $v$
are in different connected components, then $U_{M}(X,t)_{u,v}=0$ for all
$t\in\mathbb{R}$, and so $F(t)_{u,v}>0$ for all $t\in\mathbb{R}\backslash
T_{M}$.

  3. 3.

Let $|U_{M}(X,\tau)_{u,v}|=1$ for some $\tau\in\mathbb{R}\backslash T_{M}$.
Since $\alpha_{M}(t)\neq 0$ and $|U_{M}(X\vee Y,t)_{u,v}|\leq 1$, we get
$F(\tau)_{u,v}<0$. If $u$ is isolated in $X$, then $U_{M}(X,t)_{u,u}=1$ for
all $t$, and so $F(t)_{u,u}<0$ for all $t\in\mathbb{R}\backslash T_{M}$.

We now use Lemma 74 to show that under mild conditions on $m$ and $n$, if $X$
has PST between $u$ and $v$, then there are choices of $t$ for which
$|F(t)_{u,v}|=2/m$. Thus, the bounds in Corollaries 70 and 72 are tight.

######  Example 75.

Suppose $X$ exhibits Laplacian PST between $u$ and $v$ at $\pi/2$. Let
$\tau=r\pi$ for any odd $r$ so that $U_{L}(X,\tau/2)_{u,v}=1$ and
$U_{L}(X,\tau/2)_{u,u}=0$. If $\nu_{2}(m)=\nu_{2}(n)=1$, then we can write
$T_{L}=\\{j\pi/g^{\prime}:j\in\mathbb{Z}\\}$, where $g^{\prime}=g/2$ is odd.
Hence, $\tau/2\in\mathbb{R}\backslash T_{L}$. In this case,
$\alpha_{L}(\tau/2)=-\frac{2}{m}$, and so
$F(\tau/2)_{u,u}=-F(\tau/2)_{u,v}=\frac{2}{m}$.

######  Example 76.

Suppose $X$ admits adjacency PST between $u$ and $v$ at $\pi/2$. Let
$k,\ell\in\mathbb{Z}$ and $\Delta$ be a perfect square so that,
$k,\lambda^{+}$ and $\lambda^{-}$ are integers. Let $\tau=r\pi$ for any odd
$r$ so that $U_{A}(X,\tau/2)_{u,v}=e^{i\tau k/2}$ and
$U_{A}(X,\tau/2)_{u,u}=0$. If $\nu_{2}(\lambda^{+}-k)=1$ and
$\nu_{2}(\lambda^{-}-k)=1$, then we can write
$T_{A}=\\{j\pi/h^{\prime}:j\in\mathbb{Z}\\}$ where $h^{\prime}=h/2$ is odd.
Thus, $\tau/2\in\mathbb{R}\backslash T_{A}$, and so
$\alpha_{A}(\tau/2)=-\frac{2e^{i\tau k/2}}{m}$, which yields
$F(\tau/2)_{u,u}=-F(\tau/2)_{u,v}=\frac{2}{m}$.

One checks that disconnected double cones with PST at $\frac{\pi}{2}$ satisfy
the conditions in Examples 75 and 76, and so they provide infinite families of
graphs whereby the upper bound in Corollaries 70 and 72 is met.

##  12 Future work

In this paper, we investigated strong cospectrality, periodicity and perfect
state transfer in join graphs, and characterized the conditions in which these
properties are preserved or induced in a join. In order to inspire further
systematic study of quantum walks on join graphs, we present some open
questions.

If $X$ is a weighted join graph on an odd number of vertices and
$\phi(L(X),t)\in\mathbb{Z}[t]$, then Corollary 25 implies that Laplacian PST
does not occur in $X$. This is also known to be true for unweighted graphs on
$n\in\\{3,5\\}$ vertices, which are not necessarily joins. This leads us to
conjecture the following.

######  Conjecture 77.

If $X$ is a simple connected unweighted graph on an odd number of vertices,
then Laplacian perfect state transfer does not occur in $X$.

We note however that if $X$ is a weighted join graph with an odd number of
vertices, then it is possible to get PST in $X$ if we drop the condition that
$\phi(L(X),t)\in\mathbb{Z}[t]$. Indeed, if $X$ is a weighted $K_{2}$ with
vertices $u$ and $v$ joined by an edge of weight $\frac{a}{b}>0$ where $a>0$
is odd and $b\equiv 0$ (mod 4), then $\phi(L(X),t)\notin\mathbb{Z}[t]$, but
one checks that $X\vee O_{1}$ is a weighted $K_{3}$ with PST between $u$ and
$v$ at time $\frac{2\pi}{3}$.

Next, observe that $X$ is an induced subgraph of $X\vee Y$. In Corollaries 70
and 72, we showed for $M\in\\{A,L\\}$ that $\big{|}|U_{M}(X\vee
Y,t)_{u,v}|-|U_{M}(X,t)_{u,v}|\big{|}\leq f(|V(X)|)$ for all $t$, where
$f(|V(X)|)=2/|V(X)|$, and this inequality is tight for infinite families. In
general, if $X$ is a subgraph of $Z$ induced by $W\subseteq V(Z)$ with
$|W|\geq 2$, then for $u,v\in V(X)$, we ask: what is an upper bound for
$\big{|}|U_{M}(Z,t)_{u,v}|-|U_{M}(X,t)_{u,v}|\big{|}$ in terms of $|W|$? In
particular, if $X$ is either complete or empty, what is
$\sup_{t>0}\big{|}|U_{M}(Z,t)_{u,v}|-|U_{M}(X,t)_{u,v}|\big{|}$? In line with
Corollary 73, we further ask: for which induced subgraphs $X$ of $Z$ does
$\big{|}|U_{M}(Z,t)_{u,v}|-|U_{M}(X,t)_{u,v}|\big{|}\rightarrow 0$ for all $t$
as $|V(X)|\rightarrow\infty$?

If $X=K_{m}$ and $Y$ is a graph on $m$ vertices, then we have proof for
$M\in\\{A,L\\}$ that $|U_{M}(X\vee Y,t)_{u,v}|-|U_{M}(X,t)_{u,v}|=0$ if and
only if $t$ belongs to some countable set. Hence, we pose the question, are
there families of graphs such that for some set $T$ with positive measure,
$|U_{M}(X\vee Y,t)_{u,v}|-|U_{M}(X,t)_{u,v}|=0$ for all $t\in T$?

Finally, one of the surprising results in this paper is that the join
operation can induce PST in the resulting graph under mild conditions. Owing
to the fact that adjacency PST is rare in unweighted graphs [Godsil2012a,
Corollary 10.2], it would be interesting to determine other graph operations
that induce PST in the resulting graph. In [bhattacharjya2023quantum, Theorem
12], the authors show that the blow-up operation exhibits this property for
some families of graphs (such as the blow-up of two copies of $K_{1,n}$).
However, under Cartesian products, PST between vertices $(u,v)$ and $(w,x)$
with $(u,v)\neq(w,x)$ requires that either (i) $u$ and $w$, and $v$ and $x$
admit PST in the underlying graphs (ii) $u$ and $w$ admit PST and $v=x$ is
periodic in the underlying graphs or (iii) $u=w$ is periodic and $v$ and $w$
admit PST in the underlying graphs. Hence, the Cartesian product cannot induce
PST in the resulting graph.

## Acknowledgements

H. Monterde is supported by the University of Manitoba Faculty of Science and
Faculty of Graduate Studies. S. Kirkland is supported by NSERC Discovery Grant
RGPIN-2019-05408.

## References