# Sedentariness in quantum walks

Hermie Monterde,1

###### Abstract

We formalize the notion of a sedentary vertex and present a relaxation of the
concept of a sedentary family of graphs introduced by Godsil [Linear Algebra
Appl. 614:356-375, 2021]. We provide sufficient conditions for a given vertex
in a graph to exhibit sedentariness. We also show that a vertex with at least
two twins (vertices that share the same neighbours) is sedentary. We prove
that there are infinitely many graphs containing strongly cospectral vertices
that are sedentary, which reveals that, even though strong cospectrality is a
necessary condition for pretty good state transfer, there are strongly
cospectral vertices which resist high probability state transfer to other
vertices. Moreover, we derive results about sedentariness in products of
graphs which allow us to construct new sedentary families, such as Cartesian
powers of complete graphs and stars.

Keywords: quantum walks, sedentary walks, twin vertices, strongly cospectral
vertices, adjacency matrix, Laplacian matrix

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

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

###### Contents

  1. 1 Introduction
  2. 2 Sedentariness
  3. 3 Products
  4. 4 A sufficient condition
  5. 5 Twin vertices
  6. 6 Cones
  7. 7 Strongly cospectral vertices
  8. 8 Trees
  9. 9 Other types of state transfer

##  1 Introduction

A (continuous-time) quantum walk (CTQW) on $X$ describes the propagation of
quantum states across a quantum spin network modelled by the graph $X$, where
the qubits in the spin network and the interactions between them are
represented by the vertices and edges of $X$, respectively. If $H$ is a real
symmetric matrix that encodes the adjacencies in $X$, i.e., $H_{u,v}=0$ if and
only if $[u,v]$ is not an edge in $X$, then the CTQW on $X$ with respect to
$H$ is determined by the complex symmetric unitary matrix

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

We call $U_{H}(t)$ and $H$ resp. the transition matrix and the Hamiltonian of
the quantum walk. If $H$ in (1) is clear from the context, then we write
$U_{H}(t)$ as $U(t)$. As $U(t)$ is unitary, $\sum_{j\in V(X)}\lvert
U(t)_{u,j}\rvert^{2}=1$ for all $t$, and so $\lvert U(t)_{u,v}\rvert^{2}$ is
interpreted as the probability of quantum state transfer between $u$ and $v$
at time $t$.

The concept of a sedentary family of graphs was introduced by Godsil [SedQW],
which was mainly motivated by the behaviour of quantum walks on complete
graphs. Godsil defined a family $\mathscr{F}$ of graphs to be sedentary if
there is a constant $a>0$ such that for each $X\in\mathscr{F}$ and each vertex
$u$ of $X$, we have $\lvert U(t)_{u,u}\rvert\geq 1-\frac{a}{\lvert
V(X)\rvert}$ for all $t$. Consequently, $\lvert U(t)_{u,u}\rvert$ tends to 1
for each $u\in V(X)$ as $\lvert V(X)\rvert$ increases. Godsil showed that
large classes of strongly regular graphs are sedentary at any vertex. While
the main focus in [SedQW] was to study sedentary families of graphs, Godsil
also investigated sedentariness of a single vertex in a graph by showing that
cones over $k$-regular graphs exhibit varying degrees of sedentariness at
their apexes with respect to the adjacency matrix depending on the value $k$.
Frigerio and Paris showed that cones are also sedentary at the apex with
respect to the Laplacian matrix [chiral]. To the best of our knowledge, no
other families of finite graphs are known to exhibit sedentariness.

In order to better understand the notion of single vertex sedentariness and
obtain more sedentary families of graphs, we formalize the definition of a
sedentary vertex and propose a relaxation of the notion of a sedentary family
of graphs. Let $0<C\leq 1$ be a constant. We say that a vertex $u$ of $X$ is
$C$-sedentary if $\lvert U(t)_{u,u}\rvert\geq C$ for all $t$. We say that a
family $\mathscr{F}$ of graphs is $C$-sedentary if there exists a real-valued
function $f$ satisfying $0<f(s)\leq 1$ for all $s>0$ such that (i) for each
$X\in\mathscr{F}$, some vertex $u$ of $X$ is $f(\lvert V(X)\rvert)$-sedentary
and (ii) $f(s)\rightarrow C$ as $s$ increases. If $f(s)=1-\frac{a}{s}$ for
some $a>0$, then $C=1$, and if we add that each vertex of each
$X\in\mathscr{F}$ is $f(\lvert V(X)\rvert)$-sedentary, then the concept of a
$C$-sedentary family coincides with Godsil’s notion of a sedentary family. If
$C=0$ and $\inf_{t>0}\lvert U(t)_{u,u}\rvert=f(\lvert V(X)\rvert)$ for each
$X\in\mathscr{F}$, then the family is said to be quasi-sedentary, a concept
first introduced in this paper. We emphasize that these properties depend on
the matrix $H$, which we later choose to be a generalized adjacency matrix or
a generalized normalized adjacency matrix of $X$.

The main goal of this paper is to provide sufficient conditions for
$C$-sedentariness of a vertex and construct families of graphs that are
$C$-sedentary. We prove our main result, which states that by an appropriate
choice of a subset $S$ of the eigenvalue support of a vertex $u$, one may be
able to show that $u$ is sedentary. We then use this result to establish that
for any vertex $u$ in a set of twins $T$, $\lvert U(t)_{u,u}\rvert\geq
1-\frac{2}{\lvert T\rvert}$ for all $t$. Consequently, vertices with at least
two twins are sedentary, which allows us to construct new families of graphs
that are $C$-sedentary. This includes graphs built from joins, graphs with
tails and blow-ups of graphs. We also show that sedentariness is preserved
under Cartesian products, which provides another way to construct
$C$-sedentary families. Another result, which is rather unexpected, is that
there are infinitely many graphs containing strongly cospectral vertices that
are sedentary. This reveals that some strongly cospectral vertices resist high
probability transfer to other vertices. We also discuss the connection of
sedentariness to other types of quantum state transfer. Even though sedentary
vertices do not exhibit pretty good state transfer, we show that there are
$C$-sedentary and quasi-sedentary families whose each member graph exhibits
proper fractional revival at the sedentary vertices. For local uniform mixing,
we show that this is only possible for quasi-sedentary families.

Throughout this paper, we assume that $X$ is a connected weighted undirected
graph with possible loops but no multiple edges. We denote the vertex and edge
sets of $X$ resp. by $V(X)$ and $E(X)$, and we allow the edges of $X$ to have
nonzero real weights (i.e., an edge can have either positive or negative
weight). We denote an edge between vertices $u$ and $v$ by $[u,v]$. 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 set of neighbours of $u$
in $X$ as $N_{X}(u)$, and the characteristic vector of $u$ as
$\textbf{e}_{u}$, which is a vector with a $1$ on the entry indexed by $u$ and
$0$’s 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 resp. by $\textbf{1}_{n}$, $\textbf{0}_{n}$, $\textbf{J}_{m,n}$ and
$I_{n}$. 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 resp. as 1, 0, J
and $I$. We also represent the transpose of $M$ by $M^{T}$. We denote the
simple unweighted empty, cycle, complete, and path graphs on $n$ vertices
resp. as $O_{n}$, $C_{n}$, $K_{n}$, and $P_{n}$. We also denote the simple
unweighted complete bipartite graph with partite sets of sizes
$n_{1},\ldots,n_{k}$ as $K_{n_{1},\ldots,n_{k}}$.

For two graphs $X$ and $Y$, the join $X\vee Y$ is the resulting graph after
adding all edges $[u,v]$ of weight one, where $u\in V(X)$ and $v\in V(Y)$,
while the union $X\cup Y$ is the resulting graph with $V(X\cup Y)=V(X)\cup
V(X)$ and $E(X\cup Y)=E(X)\cup E(Y)$. The Cartesian product $X\square Y$ is a
graph with vertex set $V(X)\times V(Y)$ where $(u,x)$ and $(v,y)$ are adjacent
in $X\square Y$ if either $u=v$ and $[x,y]$ is an edge in $Y$ or $x=y$ and
$[u,v]$ is an edge in $X$. The weight of the edge between $(u,x)$ and $(v,y)$
is equal to the weight of $[u,v]$ if $x=y$ and $[x,y]$ if $u=v$. The direct
product $X\times Y$ is the graph with vertex set $V(X)\times V(Y)$ where
$(u,x)$ and $(v,y)$ are adjacent in $X\times Y$ if $[u,v]$ and $[x,y]$ are
edges resp. in $X$ and $Y$. The weight of the edge between $(u,x)$ and $(v,y)$
is equal to the product of the weights of the edges $[u,v]$ and $[x,y]$.

We define the adjacency matrix $A(X)$ of $X$ entrywise as

$A(X)_{u,v}=\begin{cases}\omega_{u,v},&\text{if $u$ and $v$ are adjacent}\\\
0,&\text{otherwise},\end{cases}$ (2)

where $0\neq\omega_{u,v}\in\mathbb{R}$ is the weight of $[u,v]$. 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
$u\in V(X)$. We say that $X$ is weighted-regular if $D(X)$ is a scalar
multiple of the identity matrix. As $X$ is weighted, it is possible that
$\operatorname{deg}(u)=0$ without $u$ being isolated. Assuming that
$\operatorname{deg}(u)\geq 0$ for all $u\in V(X)$, we define
$D(X)^{-\frac{1}{2}}$ as the diagonal matrix whose $(u,u)$ entry is
$1/\sqrt{\operatorname{deg}(u)}$ if $\operatorname{deg}(u)>0$ and $0$
otherwise.

Let $a,b,c\in\mathbb{R}$ with $a\neq 0$. A matrix of the form
$\mathbf{A}(X)=cI+bD(X)+aA(X)$ is called a generalized adjacency matrix
$\mathbf{A}(X)$ of $X$, and a matrix of the form
$\mathcal{A}(X)=bI+aD(X)^{-\frac{1}{2}}A(X)D(X)^{-\frac{1}{2}}$ is called a
generalized normalized adjacency matrix $\mathcal{A}(X)$ of $X$. These two
matrices were first studied in [MonterdeELA] in the context of quantum state
transfer (in particular, in relation to the concept of strong cospectrality).
We consider these two types of matrices, which are generalizations of well-
known matrices associated to graphs. Indeed, if $c=0$, $b=1$ and $a=-1$, then
$\mathbf{A}$ becomes the Laplacian matrix $L(X)$ of $X$, while
$\mathcal{A}(X)$ becomes the normalized Laplacian matrix $\mathcal{L}(X)$ of
$X$. Since the quantum walks determined by $bI+aH$ and $H$ are equivalent, we
simplify the discussion by considering the matrices

$\mathbf{A}(X)=\alpha
D(X)+A(X)\quad\text{and}\quad\mathcal{A}(X)=D(X)^{-\frac{1}{2}}A(X)D(X)^{-\frac{1}{2}},$
(3)

and note that the quantum walks determined by A are equivalent for all
$\alpha\in\mathbb{R}$ whenever the graph is weighted-regular. We use $M(X)$ to
denote $\textbf{A}(X)$ or $\mathcal{A}(X)$, and use $H=M(X)$ in (1). If the
context is clear, then we write $M(X)$, $A(X)$, $L(X)$, $\mathcal{L}(X)$ and
$D(X)$ resp. as $M$, $A$, $L$, $\mathcal{L}$ and $D$. Finally, if $U_{X\square
Y}(t)$ is the transition matrix of $X\square Y$ with respect to $\mathbf{A}$,
then it is known that

$\begin{split}U_{X\square Y}(t)=U_{X}(t)\otimes U_{Y}(t),\end{split}$ (4)

while if $U_{X\times Y}(t)$ is the transition matrix of $X\times Y$ with
respect to $\mathcal{A}$, then $U_{X\times Y}(t)=U_{X}(t)\otimes U_{Y}(t)$
whenever $X$ and $Y$ are simple. Here, $A\otimes B$ denotes the Kronecker
product of matrices $A$ and $B$.

##  2 Sedentariness

We begin with the definition of a sedentary vertex.

######  Definition 1.

We say that vertex $u$ of $X$ is $C$-sedentary if for some constant $0<C\leq
1$,

$\inf_{t>0}\lvert U_{M}(t)_{u,u}\rvert\geq C.$ (5)

If equality holds in (5), then we say that $u$ is sharply $C$-sedentary, while
if the infimum in (5) is attained for some $t>0$, then we say that $u$ is
tightly $C$-sedentary.

We also say that $u$ is not sedentary if $\inf_{t>0}\lvert
U_{M}(t)_{u,u}\rvert=0$. Note that for a sharply $C$-sedentary vertex, $C$ is
the best lower bound one can get for $\lvert U_{M}(t)_{u,u}\rvert$ for all
$t$. It is also clear that a tightly sedentary vertex is sharply sedentary,
but the converse is not true. If $C$ is not important, then we resp. say
sedentary, sharply sedentary, and tightly sedentary. Sedentariness of $X$ at
$u$ implies that $\lvert U_{M}(t)_{u,u}\rvert$ is bounded away from $0$, and
as a result, the quantum state initially at vertex $u$ tends to stay at $u$.

As $M$ is real symmetric, it admits a spectral decomposition
$M=\sum_{j}\lambda_{j}E_{j}$, and so we can write (1) as

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

where the $\lambda_{j}$’s are the distinct eigenvalues of $M$ and $E_{j}$ is
the orthogonal projection matrix associated with $\lambda_{j}$. The eigenvalue
support of vertex $u$ with respect to $M$ is the set
$\sigma_{u}(M)=\\{\lambda_{j}:E_{j}\textbf{e}_{u}\neq\textbf{0}\\}$. We say
that two vertices $u$ and $v$ are cospectral if $(E_{j})_{u,u}=(E_{j})_{v,v}$
for each $j$. It is immediate that if $X$ has an automorphism mapping $u$ to
$v$, then they are cospectral.

Let $S_{1}$ and $S_{2}$ be two non-empty disjoint proper subsets of $V(X)$.
Order the vertices of $X$ in a way that $S_{1}$ comes first, followed by
$S_{2}$ and then $V(X)\backslash(S_{1}\cup S_{2})$. We say that there is
pretty good state transfer (PGST) from $S_{1}$ and $S_{2}$ if for each
$\epsilon>0$, there exists a time $t_{\epsilon}$ such that
$U_{M}(t_{\epsilon})$ has the block form

$U_{M}(t_{\epsilon})=\left[\begin{array}[]{cccc}*&U_{\epsilon}&*\\\
U^{T}_{\epsilon}&*&*\\\ &*&*\end{array}\right],$

where $U_{\epsilon}$ is an $\lvert S_{1}\rvert$-by-$\lvert S_{2}\rvert$ matrix
satisfying $\|U_{\epsilon}\|>1-\epsilon$. Clearly, if $\lvert
S_{1}\rvert=\lvert S_{2}\rvert=1$, then we get PGST between two vertices. If
$\|U_{\epsilon}\|=1$, then we say that perfect state transfer (PST) occurs
from $S_{1}$ and $S_{2}$, a notion that is equivalent to group state transfer
(GST) introduced by Brown et al. [Brown2021]. As $U_{M}(t_{1})$ is non-
singular, if PST occurs from $S_{1}$ and $S_{2}$, then $\lvert
S_{1}\rvert\leq\lvert S_{2}\rvert$, and the case $\lvert S_{1}\rvert=\lvert
S_{2}\rvert=1$ yields PST between two vertices. PST, PGST and GST (and later
on, uniform mixing and fractional revival) fall under the general notion of
quantum state transfer, which is an important physical concept.

The following basic properties of $C$-sedentary vertices are immediate from
the fact that $U(t)$ is unitary.

######  Proposition 2.

Let $X$ be a graph with vertex $u$.

  1. 1.

If $X$ is (sharply or tightly) $C_{1}$-sedentary at $u$, where $0<C_{1}\leq
1$, then the following hold.

    1. (a)

$X$ is also $C_{2}$-sedentary at $u$ whenever $0<C_{2}\leq C_{1}$.

    2. (b)

If $u$ and $v$ are cospectral vertices, then $X$ is also (sharply or tightly)
$C_{1}$-sedentary at $v$.

    3. (c)

Any subset $S$ of $V(X)$ containing $u$ cannot be involved in pretty good
state transfer in $X$.

    4. (d)

For any vertex $v\neq u$, $\sup_{t>0}\lvert
U_{M}(t)_{u,v}\rvert\leq\sqrt{1-C_{1}^{2}}$.

  2. 2.

If for each vertex $v\neq u$ of $X$, there is a constant $C_{v}<1$ such that
$\sup_{t>0}\lvert U_{M}(t)_{u,v}\rvert=C_{v}$, then $u$ is sharply
$C$-sedentary if and only if $1-\sum_{v\neq u}C_{v}^{2}>0$, in which case
$C=1-\sqrt{\sum_{v\neq u}C_{v}^{2}}$.

By Proposition 2(1a), it is desirable to find the least $C<1$ such that a
vertex is $C$-sedentary, i.e., the $C$ such that $u$ is sharply $C$-sedentary.
Proposition 2(1c) implies that PGST and sedentariness are mutually exclusive.
Thus, our investigation of sedentariness is motivated in the same way as the
study of PGST, in a sense that identifying sedentary vertices rules out the
existence of PGST. Proposition 2(1d) tells us that a necessary condition for
sedentariness of $u$ is that $\lvert U(t)_{u,v}\rvert$ is bounded away from 1
for any $v\neq u$, while Proposition 2(2) provides a sufficient condition for
sedentariness. But since not much is known about pairs of vertices such that
$\lvert U(t)_{u,v}\rvert$ bounded away from 1, Proposition 2(2) will not be
very useful to us. Instead, we present a sufficient condition for
sedentariness in Section 4 that only depends on the diagonal entries of the
$E_{j}$’s. Next, we define what it means for a family of graphs to be
$C$-sedentary.

######  Definition 3.

Let $0\leq C\leq 1$ and $\mathscr{F}$ be a (countable) family of graphs. We
call $\mathscr{F}$ is $C$-sedentary if there is a function
$f:\mathbb{R}^{+}\rightarrow(0,1]$ such that (i) for each $X\in\mathscr{F}$
and some $u\in V(X)$, the graph $X$ is $f(\lvert V(X)\rvert)$-sedentary at $u$
and (ii) $f(s)\rightarrow C$ as $s\rightarrow\infty$. Further, if $C=1$, then
we call $\mathscr{F}$ is sedentary; if each $X\in\mathscr{F}$ is sharply
(resp., tightly) $f(\lvert V(X)\rvert)$-sedentary at $u$, then call
$\mathscr{F}$ is sharply (resp., tightly) $C$-sedentary; and if $C=0$ and
$\mathscr{F}$ is sharply $C$-sedentary, then call $\mathscr{F}$ is quasi-
sedentary.

Note that if $C=1$, then the above notion coincides with Godsil’s definition
of sedentary quantum walks. For example, if $\mathscr{K}$ is the family of
complete graphs on $n\geq 3$ vertices, then for all $t$,

$\lvert U(t)_{u,u}\rvert=\frac{\lvert n-1+e^{itn}\rvert}{n}\geq 1-\frac{2}{n}$
(6)

with equality if and only if $t=\frac{j\pi}{n}$ for odd $j$. Thus,
$\mathscr{K}$ is a sedentary family. The case $0\leq C<1$ is a more relaxed
version of $C$-sedentariness than the case $C=1$. In [SedQW], Godsil showed
that cones on $d$-regular graphs on $n$ vertices are $C$-sedentary at the apex
with respect to $A$, where $C=\frac{d^{2}}{d^{2}+4n}$. Thus, the concept of
$C$-sedentariness for $0<C\leq 1$ is not entirely new, although this concept
is first formalized in this paper. Quasi-sedentariness, on the other hand, is
a new concept introduced in this paper, and may be regarded as the weakest
form of sedentariness for families of graphs.

##  3 Products

Consider A and $\mathcal{A}$ in (3). In this section, we derive results about
sedentariness in products of graphs.

######  Theorem 4.

Let $X_{1},\ldots,X_{n}$ be weighted graphs with possible loops, and
$Z=\square_{j=1}^{n}X_{j}$.

  1. 1.

If each $X_{j}$ is $C_{j}$-sedentary at $u_{j}$, then $Z$ is
$\prod_{j=1}^{n}C_{j}$-sedentary at $(u_{1},\ldots,u_{n})$. In particular, if
each $X_{j}$ is sharply $C_{j}$-sedentary at $u_{j}$, then $Z$ is sharply
$C^{\prime}$-sedentary at $(u_{1},\ldots,u_{n})$ with
$C^{\prime}\geq\prod_{j=1}^{n}C_{j}$.

  2. 2.

If $Z$ is $C$-sedentary at $(u_{1},\ldots,u_{n})$, then each $X_{j}$ is
sharply $C_{j}$-sedentary at $u_{j}$ for some $0<C_{j}\leq 1$.

  3. 3.

If each $X_{j}$ is tightly $C_{j}$-sedentary at $u_{j}$ and there exists a
time $t_{1}$ such that $\lvert U_{X_{j}}(t_{1})_{u_{j},u_{j}}\rvert=C_{j}$ for
each $j$, then $Z$ is tightly $C$-sedentary at $(u_{1},\ldots,u_{n})$ with
$\lvert
U_{Z}(t_{1})_{(u_{1},\ldots,u_{n}),(u_{1},\ldots,u_{n})}\rvert=\prod_{j=1}^{n}C_{j}$.

###### Proof.

From (4), we have $\lvert U_{X_{1}\square
X_{2}}(t)_{(u_{1},u_{2}),(u_{1},u_{2})}\rvert=\lvert
U_{X_{1}}(t)_{u_{1},u_{1}}\rvert\cdot\lvert U_{X_{2}}(t)_{u_{2},u_{2}}\rvert$.
Using the fact that $\inf_{t>0}f(t)g(t)\geq\inf_{t>0}f(t)\inf_{t>0}g(t)$ for
all nonnegative functions $f$ and $g$ yields (1-3). ∎

By Theorem 4, Cartesian products of graphs with sedentary vertices also
contain sedentary vertices. Consequently, Cartesian products of sedentary
families also yield a sedentary family.

######  Corollary 5.

Let $\mathscr{F}_{1},\ldots,\mathscr{F}_{n}$ be families of weighted graphs
with possible loops. If each $\mathscr{F}_{j}$ is $C_{j}$-sedentary, then
$\mathscr{F}=\left\\{\square_{j=1}^{n}X_{j}:X_{j}\in\mathscr{F}_{j}\right\\}$
is $\prod_{j=1}^{n}C_{j}$-sedentary.

If the graphs involved are simple, then Theorem 4 and Corollary 5 also hold
for the direct product. We also note that if $X$ and $Y$ are simple and
weighted-regular, then so are $X\square Y$ and $X\times Y$, and so the quantum
walks determined by A and $\mathcal{A}$ are equivalent. In this case, Theorem
4 and Corollary 5 apply to $\mathcal{A}$, and their analogs for the direct
product also apply to $\mathbf{A}$. However, if $X$ is not weighted-regular,
then it is not clear how to obtain simple expressions for
$e^{it\mathcal{A}(X\square Y)}$ and $e^{it\mathbf{A}(X\times Y)}$.

Next, we examine Cartesian products of complete graphs. Since these are
regular, our results apply to A and $\mathcal{A}$. We use $\nu_{2}(b)$ to
denote the largest power of two that divides an integer $b$.

######  Theorem 6.

Let $n_{1},\ldots,n_{m}\geq 2$ and $X=\square_{j=1}^{m}K_{n_{j}}$. The
following hold.

  1. 1.

If $n_{j}=2$ for some $j$, then $X$ is not sedentary at any vertex.

  2. 2.

If each $n_{j}\geq 3$, then $X$ is $C$-sedentary at any vertex, where
$C=\prod_{j=1}^{m}(1-\frac{2}{n_{j}})$. In particular, if the
$\nu_{2}(n_{j})$’s are all equal, then $\lvert U_{X}(t_{1})_{w,w}\rvert\geq C$
for any vertex $w$ with equality at $t_{1}=\pi/2^{\nu_{2}(n_{1})}$.

###### Proof.

Let $n_{j}=2$. If $X$ is sedentary at some vertex $(u_{1},\ldots,u_{m})$,
where $u_{j}\in V(K_{2})$, then Theorem 4(2) implies that $K_{2}$ is sedentary
at $u_{j}$, which is a contradiction because $K_{2}$ exhibits PST. This proves
(1). Now, if each $n_{j}\geq 3$, then (6) and Theorem 4 imply that $X$ is
$C$-sedentary. If we add that the $\nu_{2}(n_{j})$’s are all equal, then each
$n_{j}^{\prime}=n_{j}/2^{\nu_{2}(n_{1})}$ is odd, and so (6) implies that each
$K_{n_{j}}$ is tightly $(1-\frac{2}{n_{j}})$-sedentary at any vertex at time
$t_{1}=\pi/2^{\nu_{2}(n_{1})}$. Invoking Theorem 4(3) completes the proof of
(2). ∎

The following corollary is immediate from Theorem 6.

######  Corollary 7.

Fix $k$ and let $\mathscr{F}$ be a family of graphs of the form
$\square_{j=1}^{k}K_{n_{j}}$, where each $n_{j}\geq 3$.

  1. 1.

If $n_{j}$ is fixed for some $j$, then $\mathscr{F}$ is
$(1-\frac{2}{n_{j}})$-sedentary at any vertex.

  2. 2.

If each $n_{j}$ increases as $\prod_{j=1}^{k}n_{j}\rightarrow\infty$, then
$\mathscr{F}$ is sedentary at any vertex.

If we add that the $\nu_{2}(n_{j})$’s are equal for all
$\square_{j=1}^{k}K_{n_{j}}\in\mathscr{F}$, then the sedentariness in (1) and
(2) is tight.

The Hamming graph $H(k,n)$ is obtained by taking the Cartesian product of
$k\geq 1$ copies of $K_{n}$. Combining Theorem 6(2) and Corollary 7(2) yields
the following result about Hamming graphs.

######  Corollary 8.

Let $u$ be a vertex of $H(k,n)$ and $\mathscr{F}$ be a family of Hamming
graphs $H(k,n)$. If $n\geq 3$, then $u$ is tightly sedentary in $H(k,n)$ and
$\mathscr{F}$ is a sedentary family of graphs.

##  4 A sufficient condition

We say that vertex $u$ is periodic in $X$ with respect to $M$ if $\lvert
U_{M}(t_{1})_{u,u}\rvert=1$ for some time $t_{1}$, and the minimum such
$t_{1}>0$ is called the minimum period of $u$, denote by $\rho$. If $u$ is
periodic, then $\lvert U_{M}(t)_{u,u}\rvert$ is a periodic function because
$\lvert U_{M}(t+\rho)_{u,u}\rvert=\lvert(U_{M}(t)U(\rho))_{u,u}\rvert=\lvert
U_{M}(t)_{u,u}\rvert\cdot\lvert U_{M}(\rho)_{u,u}\rvert=\lvert
U_{M}(t)_{u,u}\rvert$. In this case, $\inf_{t>0}\lvert
U_{M}(t)_{u,u}\rvert=\min_{t\in[0,\rho]}\lvert U_{M}(t)_{u,u}\rvert$, and so
the following is immediate.

######  Lemma 9.

If $u$ is periodic, then $u$ is tightly sedentary if and only if
$U_{M}(t)_{u,u}\neq 0$ for all $t\in[0,\rho]$.

From Lemma 9, a periodic sedentary vertex is tightly sedentary. Since a rook
graph has all integer eigenvalues, it is periodic. By Theorem 6(2), it follows
that each vertex in a rook graph is tightly sedentary.

######  Example 10.

By Theorem 6(2), the rook graphs $X=K_{3}\square K_{4}$ and $Y=K_{3}\square
K_{5}$ resp. are $\frac{1}{6}$\- and $\frac{1}{5}$-sedentary at any vertex.
Since both are periodic, Lemma 9 implies that both are tightly sedentary.
Invoking Theorem 6(2), we get $\min_{t>0}\lvert
U_{Y}(t)_{w,w}\rvert=\frac{1}{5}$ is attained at $t_{1}=\pi$, and so $Y$ is
tightly $\frac{1}{5}$-sedentary at any vertex. But since
$\nu_{2}(3)\neq\nu_{2}(4)$, we cannot say that $X$ is tightly
$\frac{1}{6}$-sedentary. Indeed, by computing $U_{K_{3}}(t)$ and
$U_{K_{4}}(t)$, and using the fact that $\lvert U_{X}(t)_{w,w}\rvert=\lvert
U_{K_{3}}(t)_{u,u}\rvert\cdot\lvert U_{K_{4}}(t)_{v,v}\rvert$, where
$w=(u,v)$, one checks that $\min_{t>0}\lvert U_{X}(t)_{w,w}\rvert\approx
0.2064$ is attained at $t_{1}\approx 0.9556$. Thus, $X$ is tightly
$C$-sedentary at any vertex, where $C\approx 0.7936$. Moreover, since
$\min_{t>0}\lvert U_{K_{3}}(t)_{u,u}\rvert=\frac{1}{3}$ and $\min_{t>0}\lvert
U_{K_{4}}(t)_{v,v}\rvert=\frac{1}{2}$, which are attained at
$t_{1}=\frac{\pi}{3}$ and $t_{1}=\frac{\pi}{4}$ resp., we conclude that the
converse of Theorem 4(3) does not hold.

We now prove the main result in this section which could be used to prove that
a vertex is sedentary.

######  Theorem 11.

Let $u$ be a vertex of $X$ with
$\sigma_{u}(M)=\\{\lambda_{1},\ldots,\lambda_{r}\\}$, where $E_{j}$ is the
orthogonal projection matrix corresponding to $\lambda_{j}$. If $S$ is a non-
empty proper subset of $\sigma_{u}(M)$, say
$S=\\{\lambda_{1},\ldots,\lambda_{s}\\}$, such that

$\sum_{j=1}^{s}(E_{j})_{u,u}=a$ (7)

for some $\frac{1}{2}\leq a<1$, then

$\lvert
U_{M}(t)_{u,u}\rvert\geq\left\lvert\sum_{j=1}^{s}e^{it\lambda_{j}}(E_{j})_{u,u}\right\rvert-(1-a)\quad\text{for
all}\ t.$ (8)

If there exists a time $t_{1}>0$ such that

$\left\lvert\sum_{j=1}^{s}e^{it_{1}\lambda_{j}}(E_{j})_{u,u}\right\rvert\geq
1-a,$ (9)

and for all $j\in\\{1,\ldots,s\\}$ and $k\in\\{s+1,\ldots,r\\}$,

$e^{it_{1}(\lambda_{1}-\lambda_{j})}=1\quad\text{and}\quad
e^{it_{1}(\lambda_{1}-\lambda_{k})}=-1,$ (10)

then equality holds in (8), in which case $\lvert
U_{M}(t_{1})_{u,u}\rvert=2a-1$ and $u$ is periodic at time $2t_{1}$.

###### Proof.

For brevity, let $\alpha_{j}=(E_{j})_{u,u}$ for each $j=1,\ldots,r$. We know
that $U_{M}(t)_{u,u}=\sum_{j=1}^{r}\alpha_{j}e^{it\lambda_{j}}$. Suppose (7)
holds, where $1\leq s<r$ and $\frac{1}{2}\leq a<1$. Then
$\sum_{j=s+1}^{r}\alpha_{j}=1-a$, and because $\alpha_{j}>0$ for each $j$, we
obtain
$\left\lvert\sum_{k=s+1}^{r}\alpha_{k}e^{it\lambda_{k}}\right\rvert\leq\sum_{k=s+1}^{r}\alpha_{k}=1-a$
by triangle inequality. Hence, for all $t$, we have

$\lvert
U_{M}(t)_{u,u}\rvert\stackrel{{\scriptstyle(*)}}{{\geq}}\left\lvert\sum_{j=1}^{s}\alpha_{j}e^{it\lambda_{j}}\right\rvert-\left\lvert\sum_{k=s+1}^{r}\alpha_{k}e^{it\lambda_{k}}\right\rvert\stackrel{{\scriptstyle(**)}}{{\geq}}\left\lvert\sum_{j=1}^{s}\alpha_{j}e^{it\lambda_{j}}\right\rvert-(1-a)$.

This proves (8). Equality holds in $(**)$ if and only if for some $t_{1}$ and
$\gamma\in\mathbb{C}$, $e^{it_{1}\lambda_{k}}=-\gamma$ for each
$k\in\\{s+1,\ldots,r\\}$. This reduces $(*)$ to
$\left\lvert\sum_{j=1}^{s}\alpha_{j}e^{it_{1}(\lambda_{j}-\lambda_{s+1}-\pi)}-(1-a)\right\rvert\geq\
\left\lvert\sum_{j=1}^{s}\alpha_{j}e^{it_{1}\lambda_{j}}\right\rvert-(1-a)$,
which is an equality if and only if
$\left\lvert\sum_{j=1}^{s}\alpha_{j}e^{it_{1}\lambda_{j}}\right\rvert\geq 1-a$
and $e^{it_{1}(\lambda_{j}-\lambda_{s+1})}=-1$ for each $j$. The latter yields
$e^{it_{1}\lambda_{j}}=\gamma$ for $j=1,\ldots,r$. This proves (9) and (10).
If equality holds in $(*)$ and $(**)$, then
$\left\lvert\sum_{j=1}^{s}\alpha_{j}e^{it_{1}\lambda_{j}}\right\rvert=a$, and
so $\lvert U_{M}(t_{1})_{u,u}\rvert=2a-1$. The statement about periodicity is
straightforward. ∎

The following lemma helps us identify sharply sedentary vertices which are not
tightly sedentary.

######  Lemma 12.

Suppose the premise of Theorem 11 holds. If $\ell_{j}$ and $m_{j}$ are
integers such that

$\sum_{j=1}^{s}m_{j}\lambda_{j}+\sum_{j=s+1}^{r}\ell_{j}\lambda_{j}=0\quad\text{and}\quad\sum_{j=1}^{s}m_{j}+\sum_{j=s+1}^{r}\ell_{j}=0$

implies that $\sum_{j=1}^{s}m_{j}$ is even, then there exists a sequence
$\\{t_{k}\\}$ such that $\lim_{k\rightarrow\infty}\lvert
U_{M}(t_{k})_{u,u}\rvert=2a-1$.

The proof of Lemma 12 is similar to the proof of a characterization of PGST
between two vertices [Kempton2017a, Lemma 2.2], except that we replace the
sets $\sigma_{uv}^{+}(M)$ and $\sigma_{uv}^{-}(M)$ resp. by
$\sigma_{u}(M)\backslash S$ and $S$.

Using Theorem 11 and Lemma 12, we obtain the following sufficient conditions
for sedentariness.

######  Corollary 13.

Let $u$ be a vertex of $X$ and suppose $\varnothing\neq
S\subseteq\sigma_{u}(M)$.

  1. 1.

Let $S=\\{\lambda_{1}\\}$. If $(E_{1})_{u,u}=a$, then $\lvert
U(t)_{u,u}\rvert\geq 2a-1$ for all $t$. The following also hold.

    1. (a)

If $a>\frac{1}{2}$, then $u$ is $(2a-1)$-sedentary. This is tight (resp.,
sharp) whenever (10) (resp., Lemma 12) holds. Moreover, if $u$ is periodic,
then $u$ is tightly $C$-sedentary for some $C\geq 2a-1$.

    2. (b)

Suppose (9) and (10) hold, or Lemma 12 holds. If $a=\frac{1}{2}$, then $u$ is
not sedentary.

  2. 2.

Let $\lvert S\rvert\geq 2$, $b>0$ and
$F(t)=\left\lvert\sum_{j=1}^{s}e^{it\lambda_{j}}(E_{j})_{u,u}\right\rvert$. If
$a>\frac{1}{2}$, then $u$ is sedentary whenever (i) $F(t)-(1-a)>b$ for all $t$
or (ii) $F(t)\geq 1-a$ for all $t$, $u$ is periodic and $U(t_{1})_{u,u}\neq 0$
for all $t_{1}$ with $F(t_{1})=1-a$.

###### Proof.

The statement in (1) follows from Theorem 11. To prove (1a), let
$a<\frac{1}{2}$. Then $u$ is clearly $2a$-sedentary. Since (9) holds by
default, the sedentariness is tight by Theorem 11 whenever (10) holds. If the
premise of Lemma 12 holds, then $\inf_{t>0}\lvert U_{M}(t)_{u,u}\rvert=2a-1$,
and so $u$ is tightly sedentary. If we add that $u$ is periodic, then
$\min_{t>0}\lvert U_{M}(t)_{u,u}\rvert=C\geq 2a-1$, where $\min_{t>0}\lvert
U_{M}(t)_{u,u}\rvert$ is attained at some $t_{1}\in(o,\rho)$. Thus, $u$ is
$C$-sedentary. For (1b), if $a=\frac{1}{2}$, then $\lvert
U_{M}(t)_{u,u}\rvert\geq 0$. If (9) and (10) hold, then $\lvert
U_{M}(t_{1})_{u,u}\rvert=0$ at some $t_{1}\in(o,\rho)$, while if Lemma 12
holds, then $\inf_{t>0}\lvert U_{M}(t)_{u,u}\rvert=0$. This proves (1b).
Finally, let $\lvert S\rvert\geq 2$ and $a>\frac{1}{2}$. If (2i) holds, then
$u$ is sedentary by (8). If (2ii) holds, then $\lvert U_{M}(t)_{u,u}\rvert>0$
for all $t$, and so Lemma 9 implies that $u$ is tightly sedentary. This proves
(2). ∎

As we will see, Corollary 13(1) will be useful in the later sections. We end
this section with an example that illustrates Corollary 13.

######  Example 14.

Consider the path $P_{3}$ with end vertex $u$. Then
$\sigma_{u}(A)=\\{\pm\sqrt{2},0\\}$ with associated eigenvectors
$(1,\pm\sqrt{2},1)$ and $(1,0,-1)$, while $\sigma_{u}(L)=\\{3,1,0\\}$ with
associated eigenvectors $(1,-2,1)$, $(1,0,-1)$ and 1. Note that $u$ is
periodic in both cases. Moreover,

$U_{A}(t)_{u,u}=\frac{1}{4}e^{i\sqrt{2}t}+\frac{1}{4}e^{-i\sqrt{2}t}+\frac{1}{2}\quad\text{and}\quad
U_{L}(t)_{u,u}=\frac{1}{6}e^{i3t}+\frac{1}{2}e^{it}+\frac{1}{3}$.

For $A$, let $S=\\{0\\}$. Then $(E_{0})_{u,u}=1/2$ and one checks that (9) and
(10) hold at $t_{1}=\pi/\sqrt{2}$. By Corollary 13(1b), $u$ is not sedentary,
which is consistent with the fact that adjacency PST occurs between end
vertices of $P_{3}$ at $t_{1}$. For $L$, take $S=\\{3,1\\}$ so that
$(E_{3})_{u,u}+(E_{1})_{u,u}=2/3$. Applying Theorem 11 with $a=2/3$, we get
that
$F(t)=\left\lvert\frac{1}{6}e^{i3t}+\frac{1}{2}e^{it}\right\rvert-\frac{1}{3}$.
Now, $F(t_{1})=0$ if and only if $t_{1}=j\pi/2$ for any odd $j$. Since
$U_{L}(t_{1})_{u,u}\neq 0$ and $u$ is periodic, Corollary 13(2ii) implies that
$u$ is tightly sedentary.

##  5 Twin vertices

In this section, we show that a vertex with at least two twins is sedentary.
Unless otherwise stated, all results in this section apply to both A and
$\mathcal{A}$.

Two vertices $u$ and $v$ of $X$ are twins if (i)
$N_{X}(u)\backslash\\{u,v\\}=N_{X}(v)\backslash\\{u,v\\}$, (ii) the edges
$(u,w)$ and $(v,w)$ have the same weight for each $w\in
N_{X}(u)\backslash\\{u,v\\}$, and (iii) the loops on $u$ and $v$ have the same
weight if they exist. We say that a subset $T=T(\omega,\eta)$ of $V(X)$ with
at least two vertices is a set of twins in $X$ if each pair of vertices in $T$
are twins, where each vertex in $T$ has a loop of weight $\omega$ whenever
$\omega\neq 0$ and every pair of vertices in $T$ are connected by an edge with
weight $\eta$ whenever $\eta\neq 0$. Since there exists an automorphism that
switches any pair of twins, it follows that all vertices in $T$ are pairwise
cospectral. For a more extensive treatment of the role of twin vertices in
quantum state transfer, see [Monterde].

We now restate a spectral characterization of twin vertices [MonterdeELA,
Lemma 2.9].

######  Lemma 15.

Let $T=T(\omega,\eta)$ be a set of twins in $X$. Then $u,v\in T$ if and only
if $\textbf{e}_{u}-\textbf{e}_{v}$ is an eigenvector of $M$ corresponding to
the eigenvalues $\theta$ given by

$\theta=\begin{cases}\alpha\operatorname{deg}(u)+\omega-\eta,&\text{if
$M=\textbf{A}$}\\\ \frac{\omega-\eta}{\operatorname{deg}(u)},&\text{if
$M=\mathcal{A}$}.\end{cases}$ (11)

If $u\in T$, then $\theta\in\sigma_{u}(M)$ by Lemma 15. We use this to prove
our main result.

######  Theorem 16.

Let $T$ be a set of twins in $X$. If $u\in T$ with
$\sigma_{u}(M)=\\{\theta,\lambda_{2},\ldots,\lambda_{r}\\}$, then

$\lvert U_{M}(t)_{u,u}\rvert\geq 1-\frac{2}{\lvert T\rvert}\quad\text{for all
$t$},$ (12)

with equality whenever (10) holds with $S=\\{\theta\\}$. Further, if $\lvert
T\rvert\geq 3$, then $u$ is $(1-\frac{2}{\lvert T\rvert})$-sedentary.

###### Proof.

Let $T$ be a set of twins in $X$. If we index the first $\lvert T\rvert$ rows
of $M$ by the elements of $T$, then for a fixed $u\in T$, Lemma 15 implies
that $\textbf{e}_{u}-\textbf{e}_{v}$ is an eigenvector for $M$ for all $v\in
T\backslash\\{u\\}$ corresponding to the eigenvalue $\theta$ in (11). Assuming
$u$ is the first row of $M$, we get $E_{\theta}=\left(I_{\lvert
T\rvert}-\frac{1}{\lvert T\rvert}\textbf{J}_{\lvert T\rvert}\right)+F$ for
some matrix $F$. Taking $S=\\{\theta\\}$, we get $1-a=1-\frac{1}{\lvert
T\rvert}\geq\frac{1}{2}$. Applying Corollary 13(1) yields the desired result.
∎

######  Remark 17.

If $X$ is simple and unweighted, and $T$ is a set of twins in $X$, then $T$ is
also a set of twins in the complement $X^{c}$ of $X$. Thus, if $X^{c}$ is
connected, then Theorem 16 also holds for $X^{c}$.

Theorem 16 reveals that twin vertices in quantum walks behave like vertices in
a complete graph, which is an interesting observation because the underlying
graph induced by a set of twins is either complete or empty. But unlike
complete graphs, equality in (12) may not be attained for other graphs.

Figure 1: The complete multipartite graph $K_{2,3,3}$ (left) and the threshold
graph $((O_{3}\vee K_{2})\cup O_{4})\vee K_{1}$ (right) with sedentary
vertices marked blue

### Joins

Since the property of being twins is preserved under joins, Theorem 16 yields
the following results.

######  Corollary 18.

Let $T$ be a set of twins in $Y$. If $\lvert T\rvert\geq 3$, then the vertices
in $T$ are $(1-\frac{2}{\lvert T\rvert})$-sedentary in $Y\vee X$ for any
weighted graph $X$ with possible loops.

######  Corollary 19.

Let $X$ be a weighted graph with possible loops. For each $m\geq 3$, the
vertices of $K_{m}$ and $O_{m}$ resp. are $(1-\frac{2}{m})$-sedentary in
$K_{m}\vee X$ and $O_{m}\vee X$.

By Corollary 19, a degree $m-1$ vertex of $K_{m}\backslash e=K_{m-2}\vee
O_{2}$ is $(1-\frac{2}{m-1})$-sedentary for all $m\geq 5$.

We now examine sedentariness in two well known classes of graphs obtained
using the join operation.

######  Corollary 20.

Let $n_{1},\ldots,n_{2k}$ be integers such that $n_{j}\geq 3$ for some
$j\in\\{1,\ldots,k\\}$.

  1. 1.

Each vertex of $K_{n_{1},n_{2},\ldots,n_{k}}$ in the partite set of size
$n_{j}$ is $(1-\frac{2}{n_{j}})$-sedentary. Moreover, if
$\lvert\\{\ell:n_{\ell}=1\\}\rvert=p\geq 3$, then each vertex in a singleton
partite set of $K_{n_{1},n_{2},\ldots,n_{k}}$ is $(1-\frac{2}{p})$-sedentary.

  2. 2.

Each vertex of $Z\in\\{K_{n_{j}},O_{n_{j}}\\}$ is
$(1-\frac{2}{n_{j}})$-sedentary in the threshold graph

$((((O_{n_{1}}\vee K_{n_{2}})\cup O_{n_{3}})\vee K_{n_{4}})\cdots)\vee
K_{n_{2k}}\quad\text{or}\quad((((K_{n_{1}}\cup O_{n_{2}})\vee K_{n_{3}})\cup
O_{n_{4}})\cdots)\vee K_{n_{2k+1}}.$ (13)

###### Proof.

If $n_{j}\geq 3$, then each partite set of $K_{n_{1},n_{2},\ldots,n_{k}}$ and
those vertices in each $K_{n_{j}}$ and $O_{n_{j}}$ in (13) form a set of
twins. If $\lvert\\{\ell:n_{\ell}=1\\}\rvert=p\geq 3$, then the singleton
partite sets also form a set of twins size $p$. Applying Theorem 16 yields the
desired result. ∎

Threshold graphs with form given in (13), where $n_{1}\geq 2$ and $n_{j}\geq
1$ for $j\geq 2$, are precisely all the connected threshold graphs as
characterized by Kirkland and Severini (see [Severini, Lemma 1]).

Next, we have the following immediate consequence of Theorem 16.

######  Corollary 21.

Let $\mathscr{F}$ be a family of graphs with a set of twins $T$ with $\lvert
T\rvert\geq 3$.

  1. 1.

If $\lvert T\rvert$ is fixed for all $X\in\mathscr{F}$, then $\mathscr{F}$ is
$(1-\frac{2}{\lvert T\rvert})$-sedentary at every vertex in $T$.

  2. 2.

If $\lvert V(X)\backslash T\rvert$ is fixed for all $X\in\mathscr{F}$, then
$\mathscr{F}$ is sedentary at every vertex of $T$.

For the family $\mathscr{K}_{1}$ of complete graphs on $m\geq 5$ vertices
minus an edge, all vertices in $K_{m}\backslash e$, except for the non-
adjacent pair, form a set of twins $T$ with $\lvert T\rvert=m-2$. Thus,
$\lvert V(X)\backslash T\rvert=2$ is fixed, and so by Corollary 21(2),
$\mathscr{K}_{1}$ is a family that is sedentary at all vertices except for the
non-adjacent pair.

The next result follows immediately from Corollaries 19, 20 and 21.

######  Corollary 22.

The following hold.

  1. 1.

Let $\mathscr{F}_{1}$ and $\mathscr{F}_{2}$ be families of graphs resp. of the
form $O_{m}\vee X$ and $K_{m}\vee X$. Let $Z\in\\{O_{m},K_{m}\\}$. If $X$ has
fixed number of vertices, then each $\mathscr{F}_{i}$ is sedentary at every
vertex of $Z$. If $m\geq 3$ is fixed, then each $\mathscr{F}_{i}$ is
$(1-\frac{2}{m})$-sedentary at every vertex of $Z$.

  2. 2.

Let $\mathscr{F}_{1}$ and $\mathscr{F}_{2}$ resp. be families of complete
multipartite graphs $K_{n_{1},\ldots,n_{k}}=\bigvee_{\ell=1}^{k}O_{n_{\ell}}$
and threshold graphs in (13). Let $Z\in\\{K_{n_{j}},O_{n_{j}}\\}$. If
$n_{j}\geq 3$ is fixed, then each $\mathscr{F}_{i}$ is
$(1-\frac{2}{n_{j}})$-sedentary at every vertex in $Z$. If $k\geq 1$ is fixed
and $n_{j}\rightarrow\infty$, then each $\mathscr{F}_{i}$ is sedentary at
every vertex of $Z$.

Figure 2: The lollipop graph $L_{4,2}$ (left), the graph $X_{3,4,2}$ (center)
and the graph $Y_{3,4,2}$ (right) with sedentary vertices marked blue

### Graphs with tails

For $n\geq 4$ and $k\geq 1$, let $L_{n,k}$ be a lollipop graph, which is a
graph obtained after attaching a path $P_{k}$ to a vertex $u$ of $K_{n}$. The
vertices $v\neq u$ of $K_{n}$ in $L_{n,k}$ form a set of twins of size
$n-1\geq 3$, and so each of them is $(1-\frac{2}{n-1})$-sedentary by Theorem
16. More generally, if $n\geq k+3$, then attaching $k$ paths (possibly with
different lengths) to $k$ vertices of $K_{n}$ leaves the remaining $n-k$
vertices of $K_{n}$ $(1-\frac{2}{n-k})$-sedentary in the resulting graph. The
same holds in the complement of $L_{n,k}$.

For $n,m\geq 3$ and $k\geq 0$, let $X_{n,m,k}$ and $Y_{n,m,k}$ be graphs
obtained from $K_{n}\vee O_{m}$ by attaching copies of $P_{k}$ resp. to the
vertices of $O_{m}$ and $K_{n}$. The vertices of $K_{n}$ form a set of twins
in $X_{n,m,k}$ of size $n$, while those of $O_{m}$ form a set of twins in
$Y_{n,m,k}$ of size $m$. Thus, the vertices of $K_{n}$ and $O_{m}$ resp. are
$(1-\frac{2}{n})$\- and $(1-\frac{2}{m})$-sedentary in $X_{n,m,k}$ and
$Y_{n,m,k}$. This remains true even if we vary the lengths of paths attached
to $O_{m}$ and $K_{n}$. This also holds in the complements of $X_{n,m,k}$ and
$Y_{n,m,k}$.

The above considerations combined with Corollary 21 yield the following
result.

######  Corollary 23.

Let $\mathscr{F}$ be a family of simple unweighted lollipop graphs $L_{n,k}$,
$\mathscr{F}_{1}$ be a family of graphs $X_{n,m,k}$, and $\mathscr{F}_{2}$ be
a family of graphs $Y_{n,m,k}$.

  1. 1.

Let $n\geq 4$ and $k\geq 1$. If $k$ is fixed, then $\mathscr{F}$ is sedentary
at every vertex of $v\neq u$ of $K_{n}$. If $n$ is fixed, then $\mathscr{F}$
is $(1-\frac{2}{n-1})$-sedentary at every vertex $v\neq u$ of $K_{n}$

  2. 2.

Let $n,m\geq 3$ and $k\geq 0$. If $m$ and $k$ are fixed, then
$\mathscr{F}_{1}$ (resp., $\mathscr{F}_{2}$) is sedentary at every vertex of
$K_{n}$ (resp., $O_{n}$). If $n$ is fixed, then $\mathscr{F}_{1}$ is
$(1-\frac{2}{n})$-sedentary at every vertex of $K_{n}$, while if $m$ is fixed,
then $\mathscr{F}_{2}$ is $(1-\frac{2}{m})$-sedentary at every vertex of
$O_{m}$.

For $n\geq 4$, let $L_{n}$ be an infinite lollipop, which is the resulting
graph after attaching an infinite path to a vertex $u$ of a complete graph
$K_{n}$. In [Tamon, Proposition 3], Bernard et al. showed that the family of
infinite lollipops is sedentary at each vertex $v\neq u$ of $K_{n}$. This
complements Corollary 23(1) which states that the family of lollipop graphs
$L_{n,k}$ with $k$ fixed is sedentary at every vertex $v\neq u$ of $K_{n}$.
They also showed that attaching infinite paths to the vertices of $O_{m}$ in
$K_{n}\vee O_{m}$ yields a family that is sedentary at every vertex of $K_{n}$
[Tamon, Theorem 4], which again, complements our result in Corollary 23(2a),
which states that the family of graphs $X_{n,m,k}$ is sedentary at every
vertex of $K_{n}$ whenever $m$ and $k$ are fixed.

Similar to lollipop graphs, one may construct barbell-type graphs with
sedentary vertices. Barbell-type graphs are obtained by joining corresponding
vertices of two copies of complete graphs with a path. For instance, if
$m,n\geq 4$ and $k\geq 1$ then the barbell-type graph $L_{n,k,m}$ formed by
joining vertices $u$ of $K_{n}$ and $w$ of $K_{m}$ by a path $P_{k}$ is
$(1-\frac{2}{n-1})$\- and $(1-\frac{2}{m-1})$-sedentary resp. at any vertex
$v\neq u$ of $K_{n}$ and $v\neq w$ of $K_{m}$. One can then derive results
about sedentary families of barbell-type graphs similar to Corollary 23.

Figure 3: Blow-ups of $C_{4}$: $C_{4}^{2}(V)$ (left), $C_{4}(2,3,2,3)(V)$
(center), and $C_{4}^{3}(E)$ (right) with sets of twins filled with the same
color, all members of which are sedentary

### Blow-ups

Let $X$ be a weighted graph with possible loops with vertices
$v_{1},\ldots,v_{n}$ and edges with distinct endpoints (i.e., non-loops)
$e_{1},\ldots,e_{m}$. Let $(k_{1},\ldots,k_{n})$ and $(k_{1},\ldots,k_{m})$ be
$n$\- and $m$-tuples of positive integers.

A $(k_{1},\ldots,k_{n})$-vertex blow-up of $X$, denoted
$X(k_{1},\ldots,k_{n})(V)$, is the graph obtained by replacing every vertex
$v_{j}$ of $X$ by the graph $X_{j}\in\\{O_{k_{j}},K_{k_{j}}\\}$ such that a
vertex in $X_{j}$ is adjacent to a vertex in $X_{\ell}$ in the resulting graph
if and only if $v_{j}$ and $v_{\ell}$ are adjacent in $X$, and the weight of
each edge between $X_{j}$ and $X_{\ell}$ is the same as the weight of the edge
$[v_{j},v_{\ell}]$ in $X$. If $k_{1}=\cdots=k_{m}=k$, then we call the
resulting graph a $k$-vertex blow-up of $X$, denoted $X^{k}(V)$. Vertex blow-
ups in the literature typically mean replacing each vertex by an empty graph,
but in our definition, we have the freedom to choose between an empty or a
complete graph. For example, $K_{m,n}$ and $K_{m+n}$ are $(m,n)$-vertex blow-
ups of $K_{2}$, where each vertex of $K_{2}$ was replaced by an empty graph
for the former, and by a complete graph for the latter.

A $(k_{1},\ldots,k_{m})$-edge blow-up of $X$, denoted
$X(k_{1},\ldots,k_{n})(E)$, is a graph obtained by replacing every edge
$e_{j}=[u_{j},v_{j}]$ of $X$ by $X_{j}\in\\{O_{k_{j}},K_{k_{j}}\\}$ and adding
edges $[u_{j},w]$ and $[v_{j},w]$ for all vertices $w$ of $X_{j}$, each with
weight equal to that of $e_{j}$. If $k_{1}=\cdots=k_{m}=k$, then we call the
resulting graph a $k$-edge blow-up of $X$, denoted $X^{k}(E)$. A 1-edge blow-
up of $X$ is obtained by subdividing every edge of $X$.

######  Theorem 24.

Let $X$ be a weighted graph with possible loops with vertices
$v_{1},\ldots,v_{n}$ and edges $e_{1},\ldots,e_{m}$ with distinct endpoints.
Let $(k_{1},\ldots,k_{n})$ and $(k_{1},\ldots,k_{m})$ be $n$ and $m$-tuples of
positive integers.

  1. 1.

If $k_{j}\geq 3$ for some $j$, then the vertices of
$X_{j}\in\\{O_{k_{j}},K_{k_{j}}\\}$ added in place of $v_{j}$ (resp., $e_{j}$)
are $(1-\frac{2}{k_{j}})$-sedentary in $X(k_{1},\ldots,k_{n})(V)$ (resp.,
$X(k_{1},\ldots,k_{m})(E)$). If $k\geq 3$, then each vertex in $X^{k}(V)$ is
$(1-\frac{2}{k})$-sedentary, while each vertex in $\bigcup_{j=1}^{m}X_{j}$ is
$(1-\frac{2}{k})$-sedentary in $X^{k}(E)$.

  2. 2.

Let $T$ be a set of twins in $X$ with $k_{j}\geq 2$ for some $v_{j}\in T$.
Suppose $W_{1}=\bigcup_{v_{j}\in T,X_{j}=K_{K_{j}}}X_{j}$ and
$W_{2}=\bigcup_{v_{j}\in T,X_{j}=O_{k_{j}}}X_{j}$. If the vertices in $T$ are
pairwise adjacent, then $W_{1}$ is a set of twins in
$X(k_{1},\ldots,k_{n})(V)$. Otherwise, $W_{2}$ is a set of twins in
$X(k_{1},\ldots,k_{n})(V)$. If $\lvert W_{i}\rvert\geq 3$ for some
$i\in\\{1,2\\}$, then each vertex in $W_{i}$ is $(1-\frac{2}{\lvert
W_{i}\rvert})$-sedentary.

###### Proof.

Since the vertices of $X_{j}$ form a set of twins of size $k_{j}\geq 3$, (1)
follows directly from Theorem 16. Now, let $T$ be a set of twins in $X$ such
that $k_{j}\geq 2$ for some $v_{j}\in T$. Note that the vertices in $T$ are
either all pairwise adjacent, or all pairwise non-adjacent. Suppose the former
holds. If $v_{1},v_{2}\in T$ are distinct, and we replace $v_{1}$ by
$O_{k_{1}}$ with $k_{1}\geq 2$ and $v_{2}$ by
$Z_{2}\in\\{O_{k_{2}},K_{k_{2}}\\}$, then a vertex $u_{1}$ in $O_{k_{1}}$ and
a vertex $u_{2}$ in $Z$ are not twins in $X(k_{1},\ldots,k_{n})(V)$, because
$u_{1}$ is not adjacent to least one vertex $w\neq u_{1}$ in $O_{k_{1}}$ while
$u_{2}$ is adjacent to this $w$. The same holds if reverse the roles of
$v_{1}$ and $v_{2}$. Thus, we are left with the case when $v_{1}$ and $v_{2}$
are replaced by $K_{k_{1}}$ and $K_{k_{2}}$. In this case, any two vertices in
$K_{k_{1}}\cup K_{k_{2}}$ are adjacent twins, and so the first statement in
(2) holds. The second follows by using the same argument, and the third is a
direct consequence of Theorem 16. ∎

Theorem 24(2) tells us that if $T$ is a set of non-adjacent (resp., adjacent)
twins in $X$ and each vertex in $T$ is replaced by an empty (resp., complete)
graph, one of which has size at least two, then $W=\bigcup_{v_{j}\in
T}V(X_{j})$ is a set of twins in $X(k_{1},\ldots,k_{n})(V)$ and each vertex in
$W$ is $(1-\frac{2}{\lvert W\rvert})$-sedentary.

######  Example 25.

Figure 3 depicts three blow-ups of $C_{4}$: $C_{4}^{2}(V)$ is obtained by
replacing each vertex of $C_{4}$ by $O_{2}$, $C_{4}(2,3,2,3)(V)$ by replacing
two vertices of $C_{4}$ by two copies of $K_{2}$ and the rest by $O_{3}$, and
$C_{4}^{3}(E)$ by replacing all edges of $C_{4}$ by copies of $O_{3}$. By
Theorem 24(1), the vertices in the two copies of $O_{3}$ are
$\frac{1}{3}$-sedentary in $C_{4}(2,3,2,3)(V)$, while the 12 coloured vertices
are $\frac{1}{3}$-sedentary in $C_{4}^{3}(E)$. By Theorem 24(2), a set of two
twins in $C_{4}$ becomes a set of four in $C_{4}^{2}(V)$, all members of each
set are $\frac{1}{2}$-sedentary.

##  6 Cones

A graph of the form $K_{1}\vee X$ is called a cone on $X$ with apex $u$, where
$V(K_{1})=\\{u\\}$. A graph of the form $Z\vee X$, where
$Z\in\\{K_{2},O_{2}\\}$ is called a double cone on $X$, and any vertex of $Z$
is called an apex. In particular, $K_{2}\vee X$ and $O_{2}\vee X$ are resp.
called connected and disconnected double cones.

For cones over $d$-regular graphs on $n$ vertices, Godsil showed that $\lvert
U_{A}(t)_{u,u}\rvert\geq\frac{d^{2}}{d^{2}+4n}$, with equality if and only if
$t=\frac{\pi}{\sqrt{d^{2}+4n}}$ [SedQW]. This yields the following.

######  Proposition 26.

Let $d>0$ and $\mathscr{C}$ be a family of cones over weighted $d$-regular
graphs on $n$ vertices.

  1. 1.

If $d^{2}/n\rightarrow\infty$ as $n$ increases, then $\mathscr{C}$ is tightly
sedentary at the apex.

  2. 2.

If $\gamma$ is a constant such that $d^{2}/n\rightarrow\gamma$ as $n$
increases, then $\mathscr{F}$ is $\frac{\gamma}{\gamma+4}$-sedentary at the
apex. In particular, if $d$ is fixed, then $\mathscr{C}$ is quasi-sedentary at
the apex.

######  Remark 27.

If $d=0$, then $\lvert U(\frac{\pi}{2\sqrt{n}})_{u,u}\rvert=0$, and so the
apex in this case is not sedentary.

######  Theorem 28.

For each $0\leq C\leq 1$, there exists a $C$-sedentary family with respect to
the adjacency matrix.

###### Proof.

If $0\leq C<1$, then $\mathscr{C}$ is $C$-sedentary at the apex by Proposition
26(2) whenever $d^{2}/n\rightarrow\frac{4C}{1-C}$. If
$d^{2}/n\rightarrow\infty$, then $\mathscr{C}$ is sedentary at the apex by
Proposition 26(1). ∎

For the Laplacian case, we prove a more general result for cones.

######  Theorem 29.

Let $m\geq 1$ and $X$ be a simple positively weighted graph on $n\geq 2$
vertices. For any vertex $u$ of $K_{m}$ in $K_{m}\vee X$, $\lvert
U_{L}(t)_{u,u}\rvert\geq 1-\frac{2}{m+n}$ for all $t$ with equality if and
only if $t=\frac{j\pi}{m+n}$ for some odd $j$. Thus, the family of joins
$K_{m}\vee X$ is tightly sedentary at any vertex of $K_{m}$.

###### Proof.

Let $u$ be a vertex of $K_{m}$ in $K_{m}\vee X$. By [Alvir2016, Equation 31],
$U_{L}(t)_{u,u}=\frac{1}{m+n}+\frac{m+n-1}{m+n}e^{it(m+n)}$. Thus, $\lvert
U_{L}(t)_{u,u}\rvert^{2}\geq\frac{(m+n-2)^{2}}{(m+n)^{2}}$ for all $t$ and the
result is immediate. ∎

By Corollary 22(1), if $m\geq 3$ is fixed, then $K_{m}\vee X$ is
$(1-\frac{2}{m})$-sedentary at every vertex of $K_{m}$ with respect to $M$.
But since $\lvert U_{L}(t)_{u,u}\rvert\geq 1-\frac{2}{m+n}>1-\frac{2}{m}$,
this family of joins is, in fact, sedentary with respect to $L$. This also
implies that Theorem 29 yields a sharper bound than Theorem 16, which suggests
that the bound obtained in Theorem 16 can be improved if we take a more
specific Hamiltonian.

Taking $m\in\\{1,2\\}$ in Theorem 29 yields the following result.

######  Corollary 30.

The families of cones and connected double cones on simple positively weighted
graphs are tightly sedentary at the apexes with respect to the Laplacian
matrix.

##  7 Strongly cospectral vertices

We say that two vertices $u$ and $v$ are strongly cospectral if
$E_{j}\textbf{e}_{u}=\pm E_{j}\textbf{e}_{v}$ for all
$\lambda_{j}\in\sigma_{u}(M)$. In this case,

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

partition $\sigma_{u}(M)$. The interest in the study of strongly cospectrality
is motivated by the fact that it is a requirement for two vertices to exhibit
PGST [Godsil2012a, Lemma 13.1]. In this section, we show that there are
infinitely many graphs with strongly cospectral vertices that are sedentary.
But as the next result shows, the machinery that we have developed in Section
4 has limitations for strongly cospectral vertices.

######  Proposition 31.

Let $u$ and $v$ are strongly cospectral. If $S=\sigma_{uv}^{\pm}(M)$, then
$a=\frac{1}{2}$ in (7).

###### Proof.

Assume $\sigma_{uv}^{+}(M)=\\{\lambda_{1},\ldots,\lambda_{s}\\}$ and
$\sigma_{uv}^{-}(M)=\\{\lambda_{s+1},\ldots,\lambda_{r}\\}$. Then we have
$(E_{j})_{u,u}=(E_{j})_{u,v}$ for $j=1,\ldots,s$, while
$(E_{j})_{u,u}=-(E_{j})_{u,v}$ for $j=s+1,\ldots,r$. As the $E_{j}$’s sum to
identity, we get
$\sum_{j=1}^{s}(E_{j})_{u,u}=\sum_{k=s+1}^{r}(E_{k})_{u,u}=\frac{1}{2}$. ∎

Let $u$ and $v$ be strongly cospectral. If we take
$S\in\\{\sigma_{uv}^{+}(M),\sigma_{uv}^{-}(M)\\}$, then $a=\frac{1}{2}$ from
Proposition 31. In this case, Theorem 11 is not very useful, as (8) yields the
trivial statement $\lvert U(t)_{u,u}\rvert\geq 0$ for all $t$. If we add that
either (9) and (10) hold or Corollary 12(2) holds, then Corollary 13(1b)
implies that $u$ is not sedentary. Indeed, this holds because strong
cospectrality together with either (9) and (10) or Corollary 12(2) resp. yield
PST or PGST between $u$ and $v$. Hence, in order for Theorem 11 to work for
strongly cospectral vertices, one may avoid taking
$S\in\\{\sigma_{uv}^{+}(M),\sigma_{uv}^{-}(M)\\}$. For the case of strongly
cospectral twin vertices, it is known that $\lvert\sigma_{uv}^{-}(M)\rvert=1$
[MonterdeELA, Theorem 3.4]), and so the only viable option is to choose $S$
such that $\sigma_{uv}^{-}(M)$ is a proper subset of $S$, in which case,
$\lvert S\rvert\geq 2$ and $\theta\in S$, where $\theta$ is given in (11).
However, we shall see in Remark 33 of the next subsection that, unlike the
case $S=\\{\theta\\}$ which yields Theorem 16, the case $\lvert S\rvert\geq 2$
with $\theta\in S$ requires more work in order to establish sedentariness of
$u$.

To achieve our goal of showing that there are infinitely many graphs with
strongly cospectral vertices that are sedentary, we consider disconnected
double cones. Indeed, the apexes of such graphs are strongly cospectral with
respect to $A$ and $L$ by [MonterdeELA, Corollary 6.9]. Our main motivation
for considering these graphs is that their apexes form a set of twins of size
two, and our results in Corollary 19(1) prompt us to investigate whether the
apexes of $O_{2}\vee X$ are also sedentary. Results in the literature indicate
that the apexes of disconnected double cones are excellent sources of PST and
PGST (see for instance [Alvir2016] for the Laplacian case and [Monterde2022]
for the adjacency and signless Laplacian case), and so one might be inclined
to speculate that these apexes are not sedentary. But as it turns out, the
apexes of disconnected double cones are sedentary whenever they do not exhibit
PST or PGST.

### Laplacian case

Let $X$ be a simple positively weighted graph on $n$ vertices. Then

$U_{L}(t)_{u,u}=\frac{1}{m+n}+\frac{(m-1)e^{itn}}{m}+\frac{ne^{it(m+n)}}{m(m+n)}$
(14)

for each vertex $u$ of $O_{m}$ in $O_{m}\vee X$ (see [Alvir2016, Equation
33]).

######  Theorem 32.

Let $X$ be a simple positively weighted graph on $n$ vertices, and let $u$ be
an apex of $O_{2}\vee X$.

  1. 1.

If $n\equiv 2$ (mod 4), then $O_{2}\vee X$ is not sedentary at $u$.

  2. 2.

If $n\equiv 0$ (mod 4), then $\lvert U_{L}(t)_{u,u}\rvert\geq 1-\frac{n}{n+2}$
with equality if and only if $t=\frac{j\pi}{2}$ for any odd $j$.

  3. 3.

Let $n$ be odd. If $n=1$, then $\lvert U_{L}(t)_{u,u}\rvert\geq\frac{1}{3}$
with equality if and only if $t=\ell\pi$ for any odd $\ell$, while $n\geq 3$,
then $\lvert U_{L}(t)_{u,u}\rvert\geq 1-\frac{n+2-\sqrt{2}}{n+2}$ with
equality if and only if $t=\frac{j\pi}{2}$ for any odd $j$.

###### Proof.

Let $u$ be an apex of $O_{2}\vee X$. From (14),
$U_{L}(t)_{u,u}=\frac{1}{n+2}+\frac{e^{itn}}{2}+\frac{ne^{it(n+2)}}{2(n+2)}$,
and so

$\begin{split}\lvert
U_{L}(t)_{u,u}\rvert^{2}&=\frac{n^{2}+2n+4+h(t)}{2(n+2)^{2}},\end{split}$ (15)

where $h(t)=n(n+2)\cos(2t)+2(n+2)\cos(tn)+2n\cos(t(n+2))$. Note that $\lvert
U_{L}(t)_{u,u}\rvert^{2}$ is maximum (resp., minimum) if and only if $h(t)$ is
maximum (resp., minimum). One can then verify that

$h^{\prime}(t)\stackrel{{\scriptstyle(*)}}{{=}}-2n(n+2)\left[\sin(2t)+\sin(tn)+\sin(t(n+2))\right]=-8n(n+2)\cos\left(t\right)\cos\left(tn/2\right)\sin\left(t(n+2)/2\right)$.

Thus, $h^{\prime}(t)=0$ if and only if either (i) $t=j\pi/2$ for some odd $j$,
(ii) $t=\ell\pi/n$ for some odd $\ell$, or (iii) $t=\frac{k\pi}{n+2}$ for some
even $k$. We now differentiate $h^{\prime}(t)$ in $(*)$ to get

$h^{\prime\prime}(t)=-4n(n+2)\left[2\cos\left(t(n+4)/2\right)\cos\left(tn/2\right)+n\cos\left(t(n+1)\right)\cos(t)\right]$
(16)

If $t=\frac{j\pi}{2}$ for some odd $j$, then $\cos(t)=0$ and
$\cos\left(t(n+4)/2\right)=-\cos\left(tn/2\right)$ because $j$ is odd. While
if $t=\ell\pi/n$ for some odd $\ell$, then $\cos\left(tn/2\right)=0$ and
$\cos(t(n+1))=-\cos(t)$ because $\ell$ is odd. In both cases, (16) yields
$h^{\prime\prime}(t)>0$, and so $\lvert U_{L}(t)_{u,u}\rvert^{2}$ has a
relative minimum. Now, if $t=\frac{k\pi}{n+2}$ for some even $k$, then
$\cos\left(t(n+4)/2\right)=\cos\left(tn/2\right)$ and
$\cos\left(t(n+1)\right)=\cos(t)$. Using (16), one checks that
$h^{\prime\prime}(t)<0$, and so $\lvert U_{L}(t)_{u,u}\rvert^{2}$ has a
relative maximum. From these three cases, it suffices to compare the values of
$\lvert U_{L}(t)_{u,u}\rvert^{2}$ at $t=j\pi/2$ for odd $j$ and $t=\ell\pi/n$
for odd $\ell$ to get the absolute minimum. We begin with $t=j\pi/2$ for some
odd $j$. In this case, $\cos(2t)=-1$ and $\cos(t(n+2))=-\cos(tn)$, and so (15)
yields

$\begin{split}\lvert
U_{L}(t)_{u,u}\rvert^{2}=\frac{4\left[1+\cos\left(tn\right)\right]}{2(n+2)^{2}}.\end{split}$
(17)

If $n\equiv 2$ (mod 4), then $u$ exhibits PST [Alvir2016, Corollary 4], and so
it is not sedentary. This proves (a). Thus, we have two remaining cases.

  * •

If $n=4m$, then $\cos(tn)=\cos\left(2jm\pi\right)=1$, and so (17) yields
$\lvert U_{L}(t)_{u,u}\rvert^{2}=\frac{4}{(n+2)^{2}}$.

  * •

If $n$ is odd, then $\cos(tn)=0$, and so (17) gives us $\lvert
U_{L}(t)_{u,u}\rvert^{2}=\frac{2}{(n+2)^{2}}$.

Next, let $t=\frac{\ell\pi}{n}$ for odd $\ell$. Then $\cos(tn)=-1$ and
$\cos(t(n+2))=-\cos(2t)$. From (15), we obtain

$\lvert U_{L}(t)_{u,u}\rvert^{2}=\frac{n^{2}(1+\cos(2t))}{2(n+2)^{2}}.$ (18)

  * •

Let $n=4m$. Then $2t=\frac{\ell\pi}{2m}$ cannot be an odd multiple of $\pi$,
and so $\cos(2t)>-1$. The closest that $2t$ will be from an odd multiple of
$\pi$ is when $\ell=2ms\pm 1$ for some odd $s$, in which case,
$2t=\left(s\pm\frac{1}{2m}\right)\pi$. From (18), we get $\lvert
U_{L}(t)_{u,u}\rvert^{2}\geq\frac{n^{2}\left(1-\cos\left(2\pi/n\right)\right)}{2(n+2)^{2}}=\frac{n^{2}\sin^{2}\left(\pi/n\right)}{(n+2)^{2}}\geq\frac{8}{(n+2)^{2}}$.

  * •

Let $n$ be odd. If $n=1$, then $2t=2\ell\pi$ and so $\lvert
U_{L}(t)_{u,u}\rvert^{2}=\frac{1}{9}$. If $n>1$, then using (18) and the same
argument in the case $n=4m$ yields $\lvert
U_{L}(t)_{u,u}\rvert^{2}\geq\frac{n^{2}\left(1-\cos\left(\pi/n\right)\right)}{2(n+2)^{2}}=\frac{n^{2}\sin^{2}\left(\pi/2n\right)}{(n+2)^{2}}\geq\frac{2.25}{(n+2)^{2}}$.

Finally, comparing the above subcases yields the desired result. ∎

######  Remark 33.

In the above proof, if we take $S=\\{0,n\\}$, then $\sigma_{u}(L)\backslash
S=\\{n+2\\}$ and $(E_{n})_{u,u}+(E_{0})_{u,u}=a$, where
$a=1-\frac{n}{2(n+2)}>\frac{1}{2}$. If
$F(t):=\lvert\frac{1}{n+2}+\frac{1}{2}e^{itn}\rvert-\frac{n}{2(n+2)}$, then
Theorem 11 yields $|U(t)_{u,u}\rvert\geq F(t)\geq 0$ for all $t$. If $n\equiv
0$ (mod 4), then one checks that (10) holds at $t_{1}=j\pi/2$ for odd $j$,
i.e., the first inequality is an equality, in which case
$U(t_{1})=2a-1=1-\frac{n}{n+2}$. As $F(t)$ is not a constant function, we are
not guaranteed that $\lvert U(t)_{u,u}\rvert\geq 1-\frac{n}{n+2}$ for all $t$.
To show that this is indeed the case, we need to establish that $\lvert
U(t)_{u,u}\rvert$ is minimum at $t_{1}$, which was the approach taken in the
proof of Theorem 32.

The following is an immediate consequence of Theorem 32.

######  Corollary 34.

The family of disconnected double cones on simple positively weighted graphs
on $n$ vertices, where $n\not\equiv 2$ (mod 4), is quasi-sedentary at the
apexes with respect to the Laplacian matrix.

In keeping with Theorem 29, we show that the bound in Theorem 16 is tight for
the vertices of $O_{m}$ in $O_{m}\vee X$ for some values of $m$ and $n$.

######  Theorem 35.

Let $m\geq 3$ and $X$ be a simple positively weighted graph on $n\geq 1$
vertices. If $\nu_{2}(m)=\nu_{2}(n)$, then for any vertex $u$ of $O_{m}$ in
$O_{m}\vee X$, $\lvert U_{L}(t)_{u,u}\rvert=1-\frac{2}{m}$ whenever
$t_{1}=\frac{j\pi}{\operatorname{gcd}(m,n)}$ for any odd $j$.

###### Proof.

By Theorem 16, $\lvert U_{L}(t)_{u,u}\rvert\geq 1-\frac{2}{m}$ for all $t$.
Letting $t_{1}=\frac{j\pi}{g}$, where $j$ is odd and
$g=\operatorname{gcd}(m,n)$, one can check that using (14) that $\lvert
U_{L}(t)_{u,u}\rvert=1-\frac{2}{m}$. ∎

Let $m$ be fixed. By Theorem 35, the family of joins $O_{m}\vee X$ such that
$\nu_{2}(m)=\nu_{2}(n)$ is tightly $(1-\frac{2}{m})$-sedentary at the vertices
of $O_{m}$. Our numerical observations indicate that if
$\nu_{2}(m)\neq\nu_{2}(n)$ with $m$ fixed, then any vertex of $O_{m}$ is
$C(n)$-sedentary in $O_{m}\vee X$, where $C(n)$ satisfies $C(n)<1-\frac{2}{m}$
for all $n$ and $C(n)\rightarrow 1-\frac{2}{m}$ as $n$ increases. This
suggests that in general, the family of graphs of the form $O_{m}\vee X$ with
fixed $m$ is sharply $(1-\frac{2}{m})$-sedentary at the vertices of $O_{m}$.

### Adjacency case

Next, we examine the adjacency case for disconnected double cones.

######  Theorem 36.

Let $X$ be a simple uweighted $d$-regular graph on $n$ vertices, and let $u$
be an apex of $O_{2}\vee X$.

  1. 1.

If either (i) $d^{2}+8n$ is not a perfect square, (ii) $d=0$, or (iii)
$d^{2}+8n$ is a perfect square and
$\nu_{2}(d+\sqrt{d^{2}+8n})=\nu_{2}(d-\sqrt{d^{2}+8n})$, then $O_{2}\vee X$ is
not sedentary at $u$.

  2. 2.

Let $d>0$ and $n=\frac{1}{2}s(d+s)$ for some integer $s>0$ such that $s(d+s)$
is even and $\nu_{2}(d+s)\neq\nu_{2}(s)$ (i.e., $d^{2}+8n$ is a perfect square
and $\nu_{2}(d+\sqrt{d^{2}+8n})\neq\nu_{2}(d-\sqrt{d^{2}+8n})$). Suppose
$d_{1}=d/g$ and $s_{1}=s/g$, where $g=\operatorname{gcd}(d,s)$. The following
hold.

    1. (a)

If $s_{1}=1$, then $\lvert U_{A}(t)_{u,u}\rvert\geq\frac{1}{d_{1}+2}$ with
equality if and only if $t=\frac{j\pi}{g}$ for any odd $j$.

    2. (b)

If $s_{1}\geq 2$, then $\lvert
U_{A}(t)_{u,u}\rvert\geq\frac{\sqrt{2}}{d_{1}+2s_{1}}$ with equality if and
only if $t=\frac{j\pi}{g}$ for any odd $j$.

###### Proof.

Let $Y$ be a double cone on $X$ with apexes $u$ and $v$. From [Monterde2022,
Lemma 3(2)], we know that $\sigma_{u}(A)=\\{0,\lambda^{\pm}\\}$, where
$\lambda^{\pm}=\frac{1}{2}\left(d\pm\sqrt{d^{2}+8n}\right)$. Applying
[Coutinho2021, Lemma 12.3.1], we obtain

$U_{A}(t)_{u,u}=\frac{1}{2}+\frac{n}{2n+(\lambda^{+})^{2}}e^{it\lambda^{+}}+\frac{n}{2n+(\lambda^{-})^{2}}e^{it\lambda^{-}}$
(19)

If $d^{2}+8n$ is not a perfect square, then PGST occurs between $u$ and $v$
[Monterde2022, Theorem 11(1)], while if $d=0$, or $d^{2}+8n$ is a perfect
square and $\nu_{2}(d+\sqrt{d^{2}+8n})=\nu_{2}(d-\sqrt{d^{2}+8n})$, then PST
occurs between $u$ and $v$ [Monterde2022, Theorem 14(2a)]. Invoking
Proposition 2(1c) yields (1a). To prove (1b), suppose $d^{2}+8n$ is a perfect
square, $d>0$, and $\nu_{2}(d+\sqrt{d^{2}+8n})\neq\nu_{2}(d-\sqrt{d^{2}+8n})$.
This is equivalent to the existence of an integer $s>0$ such that
$n=\frac{1}{2}s(d+s)$ with $s(d+s)$ is even and

$\nu_{2}(s)\geq\nu_{2}(d).$ (20)

One also checks that $2n+(\lambda^{+})^{2}=(d+s)(d+2s)$ and
$2n+(\lambda^{-})^{2}=s(d+2s)$. Thus, we obtain
$\frac{n}{2n+(\lambda^{+})^{2}}=\frac{s(d+s)}{2(d+s)(d+2s)}=\frac{s}{2(d+2s)}$
and
$\frac{n}{2n+(\lambda^{-})^{2}}=\frac{s(d+s)}{2s(d+2s)}=\frac{d+s}{2(d+2s)}$.
Combining this with (19), we get that
$U_{A}(t)_{u,u}=\frac{1}{2(d+2s)}\left((d+2s)+se^{it(d+s)}+(d+s)e^{-its}\right)$,
and so

$\lvert U_{A}(t)_{u,u}\rvert^{2}=\frac{d^{2}+3ds+3s^{2}+h(t)}{2(d+2s)^{2}}$
(21)

where $h(t)=s(d+s)\cos(t(d+2s))+s(d+2s)\cos(t(d+s))+(d+s)(d+2s)\cos(ts)$.
Following the Laplacian case, $\lvert U_{A}(t)_{u,u}\rvert^{2}$ is maximum
(resp., minimum) if and only if $h(t)$ is, and we have

$\begin{split}h^{\prime}(t)&=-4s(d+s)(d+2s)\cos\left(ts/2\right)\cos\left(t(d+s)/2\right)\sin\left(t(d+2s)/2\right).\end{split}$
(22)

and

$h^{\prime\prime}(t)=-2s(d+s)(d+2s)\left[(d+s)\cos\left(t(2d+3s)/2\right)\cos\left(ts/2\right)+s\cos\left(t(d+3s)/2\right)\cos\left(t(d+s)/2\right)\right].$
(23)

From (22), $h^{\prime}(t)=0$ if and only if either $t=j\pi/s$ for some odd
$j$, $t=\frac{\ell\pi}{d+s}$ for some odd $\ell$, and $t=\frac{k\pi}{d+2s}$
for even $k\neq 0$. Among these three values, one can use (23) to show that
$h^{\prime\prime}(t)>0$ if and only if $t=j\pi/s$ for odd $j$ and
$t=\frac{\ell\pi}{d+s}$ for odd $\ell$. Thus, it suffices to compare the
values of $\lvert U_{A}(t)_{u,u}\rvert^{2}$ at the points $t=j\pi/s$ for some
odd $j$ and $t=\frac{\ell\pi}{d+s}$ for some odd $\ell$ to obtain the absolute
minimum. Let’s start with $t=j\pi/s$ for some odd $j$. In this case,
$\cos(ts)=-1$, $\cos(t(d+2s))=\cos\left(jd\pi/s\right)$ and
$\cos(t(d+s))=-\cos\left(jd\pi/s\right)$. Thus,
$h(t)=-s^{2}\cos\left(jd\pi/s\right)-(d+s)(d+2s)$, and (21) gives us

$\lvert
U_{A}(t)_{u,u}\rvert^{2}=\frac{s^{2}\left[1-\cos\left(jd\pi/s\right)\right]}{2(d+2s)^{2}}$
(24)

Let $d=gd_{1}$ and $s=gs_{1}$, where $g=\operatorname{gcd}(d,s)$. Then we can
write $jd\pi/s=jd_{1}\pi/s_{1}$, where $d_{1}$ is odd by (20). We proceed with
two subcases.

  * •

Let $s_{1}=1$. Then (24) yields $\lvert
U_{A}(t)_{u,u}\rvert^{2}=\frac{1-\cos\left(jd_{1}\pi\right)}{2(d_{1}+2)^{2}}=\frac{1}{(d_{1}+2)^{2}}$.

  * •

Let $s_{1}\geq 2$. As $j$ is odd, $jd_{1}\pi/s_{1}$ is not an even multiple of
$\pi$. Hence,
$\cos\left(jd_{1}\pi/s_{1}\right)\leq\cos\left(\pi/s_{1}\right)$, and making
use of (24) then yields $\lvert
U_{A}(t)_{u,u}\rvert^{2}\geq\frac{s_{1}^{2}\left[1-\cos\left(\pi/s_{1}\right)\right]}{2(d_{1}+2s_{1})^{2}}=\frac{s_{1}^{2}\sin^{2}\left(\pi/2s_{1}\right)}{(d_{1}+2s_{1})^{2}}\geq\frac{2}{(d_{1}+2s_{1})^{2}}$.

Next, let $t=\frac{\ell\pi}{d+s}$ for some odd $\ell$. In this case,
$\cos(t(d+s))=-1$ and $\cos(t(d+2s))=\cos\left(\frac{\ell s\pi}{d+s}\right)$.
Thus, $h(t)=-s(d+2s)+(d+s)^{2}\cos(ts)$, and making use of (21) gives us

$\lvert U_{A}(t)_{u,u}\rvert^{2}=\frac{(d+s)^{2}\left[1+\cos\left(\frac{\ell
s\pi}{d+s}\right)\right]}{2(d+2s)^{2}}$ (25)

Note that we can write $\frac{\ell s\pi}{d+s}=\frac{\ell
s_{1}\pi}{d_{1}+s_{1}}$. We proceed with two subcases.

  * •

Let $s_{1}=1$. The same argument in the proof of Theorem 32 for the case
$t=\frac{\ell\pi}{n}$ for odd $\ell$ and $n=4m$ yields $\lvert
U_{A}(t)_{u,u}\rvert^{2}\geq\frac{(d_{1}+1)^{2}\left[1-\cos\left(\frac{\pi}{d_{1}+1}\right)\right]}{2(d_{1}+2)^{2}}=\frac{(d_{1}+1)^{2}\sin^{2}\left(\frac{\pi}{2(d_{1}+1)}\right)}{(d_{1}+2)^{2}}\geq\frac{2}{(d_{1}+2)^{2}}$.

  * •

Let $s_{1}\geq 2$. Since $d_{1}+s_{1}$ and $s_{1}$ must have opposite
parities, $\frac{\ell s_{1}\pi}{d_{1}+s_{1}}$ is not an odd multiple of
$2\pi$. Thus, (25) yields $\lvert
U_{A}(t)_{u,u}\rvert^{2}\geq\frac{(d_{1}+s_{1})^{2}\left[1-\cos\left(\frac{\pi}{d_{1}+s_{1}}\right)\right]}{2(d_{1}+2s_{1})^{2}}=\frac{(d_{1}+s_{1})\sin^{2}\left(\frac{\pi}{2(d_{1}+s_{1})}\right)}{(d_{1}+2s_{1})^{2}}\geq\frac{2.25}{(d_{1}+2s_{1})^{2}}$.

Comparing the minima for each subcases above yields the desired conclusion. ∎

The following is a straightforward consequence of Theorem 36(2).

######  Corollary 37.

With the assumption in Theorem 36(2), let $\mathscr{F}$ be a family of
disconnected double cones on simple unweighted $d$-regular graphs on $n$
vertices. If $s_{1}$ and $d_{1}$ are fixed, then $\mathscr{F}$ is
$\frac{1}{d_{1}+2}$\- and $\frac{\sqrt{2}}{d_{1}+2s_{1}}$-sedentary at the
apexes resp. whenever $s_{1}=1$ and $s_{1}\geq 2$. On the other hand, if
either $d_{1}\rightarrow\infty$ or $s_{1}\rightarrow\infty$ as $n$ increases,
then $\mathscr{F}$ is quasi-sedentary at the apexes.

We end this section with the following remark.

######  Remark 38.

Let $u$ and $v$ be the apexes of $O_{2}\vee X$, where $X$ is regular whenever
$M=A$.

  1. 1.

Suppose the assumption Theorem 32 holds. If $n\equiv 0$ (mod 4), then $\lvert
U_{L}(t)_{u,v}\rvert\leq 1-\delta$ for all $t$, where $\delta=\frac{2}{n+2}$,
while if $n\geq 3$ is odd, then $\lvert U_{L}(t)_{u,v}\rvert\leq 1-\delta$ for
all $t$, where $\delta=\frac{\sqrt{2}}{n+2}$.

  2. 2.

Suppose the assumption in Theorem 36(2) holds. If $s_{1}=1$, then $\lvert
U_{A}(t)_{u,v}\rvert\leq 1-\delta$ for all $t$, where
$\delta=\frac{1}{d_{1}+2}$, while if $s_{1}\geq 2$, then $\rvert
U_{A}(t)_{u,v}\rvert\leq 1-\delta$ for all $t$, where
$\delta=\frac{\sqrt{2}}{d_{1}+2s_{1}}$.

In [Godsil2017], Godsil and Smith asked to find examples of strongly
cospectral vertices $u$ and $v$ such that for some constant $\delta>0$,
$\lvert U(t)_{u,v}\rvert\leq 1-\delta$ for all $t$. Mirror symmetric vertices
in paths without PGST and antipodal vertices in even cycles without PGST are
infinite families that answer this question. However, we do not know whether
paths and cycles are sedentary. Thus, the families in (1) and (2) are the
first examples that answer Godsil and Smith’s question, whereby the vertices
involved are sedentary.

##  8 Trees

Our first result in this section is a direct consequence of Theorem 16.

######  Proposition 39.

Let $T$ be a set of leaves of a tree $X$ that share a common neighbour. Then
$T$ is a set of twins in $X$, and for each $u\in T$, we have $\lvert
U_{M}(t)_{u,u}\rvert\geq 1-\frac{2}{\lvert T\rvert}$ for all $t$.

Next, we examine whether the central vertex of a star is sedentary.

######  Proposition 40.

Let $T$ be the set of leaves of $K_{1,n}$ and $u\in T$. Then $\lvert
U_{M}(t)_{u,u}\rvert\geq 1-\frac{2}{n}$ for all $t$. Hence, the family
$\mathscr{S}$ of stars $K_{1,n}$ on $n\geq 3$ vertices is sedentary at every
leaf vertex. The following also hold.

  1. 1.

For all $n\geq 2$, $\lvert U_{A}(t)_{u,u}\rvert=1-\frac{2}{n}$ if and only if
$t=\frac{j\pi}{\sqrt{n}}$ for any odd $j$. Moreover, the central vertex of
$K_{1,n}$ is not sedentary with respect to $A$.

  2. 2.

If $n$ is odd, then $\lvert U_{L}(t)_{u,u}\rvert=1-\frac{2}{n}$ whenever
$t=j\pi$ for any odd $j$. For all $n\geq 2$, the central vertex $w$ of
$K_{1,n}$ satisfies $\lvert U_{L}(t)_{w,w}\rvert\geq 1-\frac{2}{n+1}$ with
equality if and only if $t=\frac{j\pi}{n+1}$ for any odd $j$.

###### Proof.

The first statement follows from Proposition 39, while the second one is
obtained by by noting that $\lvert V(X)\backslash T\rvert=1$ and applying
Corollary 21(2). To prove (1a), one can use the fact that
$\sigma_{u}(A)=\\{0,\pm\sqrt{n}\\}$ to show that (9) and (10) hold if and only
if $t_{1}=\frac{j\pi}{\sqrt{n}}$ for odd $j$. Thus, equality holds in (12) in
Theorem 16, which yields the first statement of (1a). As $K_{1,n}$ is a cone
on a $0$-regular graph, the second statement follows from Remark 27. Finally,
since $K_{1,n}=O_{n}\vee K_{1}$ with $T=V(O_{n})$, Theorem 35 yields the first
statement of (2), while the second follows by noting that $K_{1,n}=O_{n}\vee
K_{1}$ and applying Theorem 29. ∎

Proposition 40(2) implies that the apex of a cone on any simple positively
weighted graph $X$ on $n$ vertices is Laplacian tightly $(1-\frac{2}{n+1})$\-
sedentary. Next, we use Proposition 40 to create more sedentary families using
Cartesian products.

######  Example 41.

Let $k\geq 1$, $n\geq 3$ and $Z_{k,n}$ be the Cartesian product of $K_{1,n}$
with itself $k$ times. Let $u$ be a leaf of $K_{1,n}$. By Proposition 40,
$\lvert
U_{Z_{k,n}}(t)_{(u,\ldots,u),(u,\ldots,u)}\rvert\geq(1-\frac{2}{n})^{k}$ for
all $t$ with respect to $M$. Consider the families
$\mathscr{F}_{1}=\\{Z_{k,n}:k\ \text{fixed}\\}$,
$\mathscr{F}_{2}=\\{Z_{k,n}:n=\lfloor mk\rfloor\ \text{for some}\ m>0\\}$ and
$\mathscr{F}_{3}=\\{Z_{k,n}:n\ \text{fixed}\\}$.

  1. 1.

As $n$ increases, $(1-\frac{2}{n})^{k}\rightarrow 1$ in $\mathscr{F}_{1}$ and
$(1-\frac{2}{n})^{k}\rightarrow 1/\sqrt[m]{e^{2}}$ in $\mathscr{F}_{2}$. Thus,
$\mathscr{F}_{1}$ is sedentary while $\mathscr{F}_{2}$ is
$1/\sqrt[m]{e^{2}}$-sedentary at $(u,\ldots,u)$. If $M=A$, then the
sedentariness of $\mathscr{F}_{1}$ and $\mathscr{F}_{2}$ is tight by
Proposition 40(1). If $M=L$, then the sedentariness of the subfamilies of
$\mathscr{F}_{1}$ and $\mathscr{F}_{2}$ is tight whenever $n$ in each
$Z_{k,n}$ is odd by virtue of Proposition 40(2).

  2. 2.

Since $(1-\frac{2}{n})^{k}\rightarrow 0$ as $k\rightarrow\infty$,
$\mathscr{F}_{3}$ is quasi-sedentary by Proposition 40.

If $u$ in Example 41 is instead the degree $n$ vertex of $K_{1,n}$, then
$\lvert
U_{Z_{k,n}}(t)_{(u,\ldots,u),(u,\ldots,u)}\rvert\geq(1-\frac{2}{n+1})^{k}$ for
all $t$ with respect to $L$ by Proposition 40(2). Thus, $\mathscr{F}_{1}$ is
sedentary, $\mathscr{F}_{2}$ is $1/\sqrt[m]{e^{2}}$-sedentary, and
$\mathscr{F}_{3}$ is quasi-sedentary at $(u,\ldots,u)$ with respect to $L$.
Now, this does not hold for $A$ by Proposition 40, and so we get a family that
is sedentary with respect to $L$ but not to $A$. By Theorems 32 and 36, one
may construct a family that is sedentary with respect to $A$ but not to $L$ by
taking the family of disconnected double cones on $d$-regular graphs on
$n\equiv 2$ (mod 4) vertices satisfying condition (2) of Theorem 36.

We also note that if we replace the $Z_{k,n}$’s in the above example by the
Hamming graphs $H(k,n)$, then $\mathscr{F}_{1}$ is tightly sedentary (which we
already know by Corollary 8), $\mathscr{F}_{2}$ is tightly
$1/\sqrt[m]{e^{2}}$-sedentary at $(u,\ldots,u)$, and $\mathscr{F}_{3}$ is
quasi-sedentary. Moreover, this applies to any $M$ because each $H(k,n)$ is
regular.

A double star $S_{k,\ell}$ is a tree resulting from attaching $k$ and $\ell$
pendent vertices to the vertices of $K_{2}$. Like the central vertex of
$K_{1,n}$, we show that an internal vertex of $S_{k,k}$ is not sedentary
whenever $M=A$.

######  Theorem 42.

Let $k\geq 1$, and consider a double star $S_{k,k}$ with internal vertices $u$
and $v$.

  1. 1.

Let $4k+1$ be a perfect square, and let $w$ be a leaf of $S_{k,k}$. If $k=2$,
then $\lvert U_{A}(t)_{w,w}\rvert\geq\frac{1}{4}$, with equality whenever
$t=\frac{j\pi}{3}$, where $j\equiv 2,4$ (mod 6). If $k>2$, then $\lvert
U_{A}(t)_{w,w}\rvert\geq 1-\frac{2}{k}$, with equality whenever $t=j\pi$ for
an integer $j$ such that $j\sqrt{4k+1}\equiv 3$ (mod 4).

  2. 2.

For all $k\geq 1$, $u$ and $v$ are not sedentary in $S_{k,k}$ with respect to
the adjacency matrix.

###### Proof.

Suppose we index the vertices of $A(S_{k,\ell})$ starting with the $k$ leaves
attached to $u$, followed by $u$ and $v$, and then the $k$ leaves attached to
$v$. Then we can write $A(S_{k,k})=\left[\begin{array}[]{cccc}A(K_{1,k})&Y\\\
Y^{T}&A(K_{1,k})\end{array}\right]$, where
$A(K_{1,k})=\left[\begin{array}[]{cccc}\textbf{0}_{k}&\textbf{1}\\\
\textbf{1}^{T}&0\end{array}\right]$ and
$Y=\left[\begin{array}[]{cccc}\textbf{0}&\textbf{0}_{k}\\\
1&\textbf{0}\end{array}\right]$. Thus, $\textbf{e}_{1}-\textbf{e}_{j}$ for
$j=1,\ldots,k$ and $\textbf{e}_{k+2}-\textbf{e}_{j}$ for $j=k+3,\ldots,2k+2$
are eigenvectors for $A(S_{k,k})$ corresponding to the eigenvalue 0. Moreover,

$\lambda_{1}=-\frac{1}{2}(1+\sqrt{4k+1}),\lambda_{2}=\frac{1}{2}(-1+\sqrt{4k+1}),\lambda_{3}=\frac{1}{2}(1-\sqrt{4k+1})\
\ \text{and}\ \ \lambda_{4}=\frac{1}{2}(1+\sqrt{4k+1})$ (26)

are simple eigenvalues of $A(S_{k,k})$ resp. with eigenvectors
$\textbf{v}_{1}=\left[-\textbf{1},\frac{1+\sqrt{4k+1}}{2},-\frac{1+\sqrt{4k+1}}{2},\textbf{1}\right]$,
$\textbf{v}_{2}=\left[-\textbf{1},\frac{1-\sqrt{4k+1}}{2},\frac{-1+\sqrt{4k+1}}{2},\textbf{1}\right]$,
$\textbf{v}_{3}=\left[\textbf{1},\frac{1-\sqrt{4k+1}}{2},\frac{1-\sqrt{4k+1}}{2},\textbf{1}\right]$
and
$\textbf{v}_{4}=\left[\textbf{1},\frac{1+\sqrt{4k+1}}{2},\frac{1+\sqrt{4k+1}}{2},\textbf{1}\right]$.
Thus, $E_{0}=I_{k}-\frac{1}{k}\textbf{J}_{k}\oplus O_{2}\oplus
I_{k}-\frac{1}{k}\textbf{J}_{k}$, where $A\oplus B$ is the direct sum of
matrices $A$ and $B$, and
$E_{\lambda_{j}}=\frac{1}{\|\textbf{v}_{j}\|^{2}}\textbf{v}_{j}^{T}\textbf{v}_{j}$
for $j\in\\{1,2,3,4\\}$. Thus,
$\sigma_{u}(A)=\\{\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}\\}$ and
$\sigma_{w}(A)=\\{0\\}\cup\sigma_{u}(A)$ for any leaf $w$ of $S_{k,k}$. Using
spectral decomposition and the fact that $\lambda_{1}=-\lambda_{4}$ and
$\lambda_{2}=-\lambda_{3}$ yields

$\begin{split}U_{A}(t)_{u,u}&=\frac{(1+\sqrt{4k+1})^{2}\cos(t\lambda_{1})}{2(4k+1+\sqrt{4k+1})}+\frac{(1-\sqrt{4k+1})^{2}\cos(t\lambda_{2})}{2(4k+1-\sqrt{4k+1})}\end{split}$
(27)

and

$\begin{split}U_{A}(t)_{w,w}&=\frac{k-1}{k}+\frac{2\cos(t\lambda_{1})}{4k+1+\sqrt{4k+1}}+\frac{2\cos(t\lambda_{2})}{4k+1-\sqrt{4k+1}}.\end{split}$
(28)

Let $4k+1$ be a perfect square. Note that the first statement of (1) can be
easily verified using (28). Since Theorem 16 yields $\lvert
U_{A}(t)_{w,w}\rvert\geq 1-\frac{2}{k}$ for all $t$, one can check using (28)
that indeed, $\lvert U_{A}(t_{1})_{w,w}\rvert=1-\frac{2}{k}$ whenever
$t_{1}=j\pi$, where $j$ is an integer such that $j\ell\equiv 3$ (mod 4). This
proves (1). To prove (2), it suffices to check the case when $4k+1$ is a
perfect square by Proposition 2(1c) and [Fan2013, Theorem 5.3]. Observe that
$U_{A}(t)_{u,u}$ in (27) is a real valued continuous function that has
positive and negative values as $t$ ranges across $[0,2\pi]$. By IVT, there
exists a $t\in[0,2\pi]$ such that $U_{A}(t)_{u,u}=0$, i.e., $u$ is not
sedentary. ∎

######  Corollary 43.

Let $u$ be a vertex of $S_{k,\ell}$ with $\operatorname{deg}(u)=k+1$. The
following hold for $M=A$.

  1. 1.

If $k=2$, then the leaves attached to $u$ in $S_{2,\ell}$ are sedentary if and
only if $\ell=2$. In particular, if $k=\ell=2$, then all leaves in $S_{2,2}$
are tightly $\frac{1}{4}$-sedentary.

  2. 2.

If $k\geq 3$ and the nonzero eigenvalues of $A(S_{k,\ell})$ are linearly
independent over $\mathbb{Q}$, then the leaves attached to $u$ are sharply
$(1-\frac{2}{k})$-sedentary. In particular, if $4k+1$ is not a perfect square,
then all leaves of $S_{k,k}$ are sharply $(1-\frac{2}{k})$-sedentary.

###### Proof.

If $k=2$, then PGST occurs between the leaves attached to $u$ if and only if
$\ell=2$ [Fan2013, Theorem 5.2]. By Proposition 2(1c), we only need to check
$S_{2,2}$. By Theorem 42(1), a leaf attached to $u$ is tightly
$\frac{1}{4}$-sedentary. As all leaves in $S_{2,2}$ are cospectral,
Proposition 2(1b) implies they are tightly $\frac{1}{4}$-sedentary. This
proves (1). Now, assume the premise of (2). Take $S=\\{0\\}$ in Theorem 11 so
that $(E_{0})_{u,u}=a\geq\frac{1}{2}$, where $a=1-\frac{1}{k}$. If $m_{j}$ and
$\ell_{j}$ are integers such that $\sum_{\lambda_{j}\neq
0}\ell_{j}\lambda_{j}=0$ and $m_{j}+\sum_{\lambda_{j}\neq 0}\ell_{j}=0$, then
$m_{j}=0$. Invoking Lemma 12 yields sharp $(1-\frac{2}{k})$-sedentariness at
$u$. The last statement follows from the linear independence of the nonzero
eigenvalues of $A(S_{k,k})$ when $4k+1$ is not a perfect square. ∎

It is natural to ask whether the internal vertices of $S_{k,\ell}$ are in
general sedentary. We leave this as an open question.

##  9 Other types of state transfer

By Proposition 2(1c), a sedentary vertex cannot be involved in PGST. Hence, we
ask, which types of state transfer can a sedentary vertex exhibit? Here, we
show that there are sedentary families where each member graph exhibits proper
fractional revival and local uniform mixing at a sedentary vertex.

Proper $(\alpha,\beta)$-fractional revival (FR) occurs between $u$ and $v$ at
time $t_{1}$ if $\alpha^{2}+\beta^{2}=1$, where $\alpha=\lvert
U(t_{1})_{u,u}\rvert$ and $\beta=\lvert U(t_{1})_{u,v}\rvert\neq 0$. In
[Chan2020, Theorem 11], Chan et al. showed that proper Laplacian FR occurs
between the apexes of $O_{2}\vee X$, where $X$ is a simple unweighted graph on
$n$ vertices. Meanwhile, in [Chan2019, Example 6.3], Chan et al. showed that
$O_{2}\vee X$ exhibits proper Laplacian FR between its apexes, where $X$ is a
simple unweighted $d$-regular graph on $n$ vertices. If $X$ is a simple
positively weighted graph, it is shown in [Monterde2023] that the apexes of
$K_{2}\vee X$ do not admit proper Laplacian FR. Combining these facts with
Corollaries 30, 34 and 37, we obtain families where each member graph exhibits
(resp., does not exhibit) proper FR involving a sedentary vertex. This tells
us that, unlike PGST, FR and sedentariness can occur together, although they
do not always happen together.

######  Example 44.

The following hold.

  1. 1.

Each graph in the quasi-sedentary family of disconnected double cones in
Corollary 34 exhibits proper Laplacian FR between apexes. Moreover, each graph
in the $C$-sedentary families of disconnected double cones in Corollary 37
(1a-c) exhibits proper adjacency FR between apexes.

  2. 2.

Each graph in the sedentary family of complete graphs on $n\geq 3$ vertices
does not exhibit proper FR between any two vertices with respect to $A$ and
$\mathcal{A}$. Moreover, each graph in the sedentary family of connected
double cones in Corollary 30 does not exhibit proper Laplacian FR between
apexes.

We say that $u$ admits (instantaneous) local uniform mixing in $X$ at time
$t_{1}$ if $\lvert U(t_{1})_{u,v}\rvert=1/\sqrt{\lvert V(X)\rvert}$ for each
vertex $v$ in $X$. We say that $X$ admits (instantaneous) uniform mixing in
$X$ at time $t_{1}$ if each vertex in $X$ admits local uniform mixing at time
$t_{1}$.

######  Proposition 45.

Let $0<C\leq 1$ and $\mathscr{F}$ be a $C$-sedentary family of graphs.

  1. 1.

Almost all graphs in $\mathscr{F}$ do not exhibit local uniform mixing.

  2. 2.

If the function $f$ in Definition 3 satisfies $f(\lvert
V(X)\rvert)>\frac{1}{\sqrt{\lvert V(X)\rvert}}$ for all $X\in\mathscr{F}$,
then each $X\in\mathscr{F}$ does not exhibit local uniform mixing at a
sedentary vertex.

###### Proof.

By assumption, for each $X\in\mathscr{F}$ and some vertex $u$ of $X$, we have
$\lvert U_{M}(t)_{u,u}\rvert\geq f(\lvert V(X)\rvert)$ for all $t$, where
$f(s)\rightarrow C>0$ as $s$ increases. Now, if $X\in\mathscr{F}$ admits local
uniform mixing, then $\lvert U_{M}(t_{1})_{u,u}\rvert=1/\sqrt{\lvert
V(X)\rvert}$ for some time $t_{1}$. But since $C>0$ and $1/\sqrt{s}\rightarrow
0$ as $s$ increases, only finitely many graphs in $\mathscr{F}$ can exhibit
local uniform mixing. This proves (1), and (2) is straightforward. ∎

If $\mathscr{K}^{\prime}$ is the family of complete graphs on $n\geq 5$
vertices, then from (6), we may take $f$ such that $f(n)=1-\frac{2}{n}$. Since
$f(n)>\frac{1}{\sqrt{n}}$ for all $n\geq 5$, no member of
$\mathscr{K}^{\prime}$ exhibits local uniform mixing by Proposition 45(2).
Proposition 45(1), on the other hand, implies that only quasi-sedentary
families exhibit local uniform mixing at a sedentary vertex as illustrated by
our next examples.

######  Example 46.

Let $\mathscr{F}$ be a family of cones on weighted $d$-regular graphs, where
$0<d\leq 2$. Combining Proposition 26(2) and [SedQW, Lemma 7.1], we conclude
that $\mathscr{F}$ is quasi-sedentary at the apex and each $X\in\mathscr{F}$
admits local uniform mixing at the apex with respect to $A$.

######  Example 47.

Consider $Z_{k,3}$ in Example 41, which is a Cartesian power of $K_{1,3}$.
This graph admits uniform mixing at $t_{1}=\frac{\pi}{3\sqrt{3}}$
[Godsil2015UniformMO, Section 11], and so Example 41(2) implies that each
graph in the quasi-sedentary family $\mathscr{F}=\\{Z_{k,3}:k\geq 1\\}$ admits
adjacency uniform mixing.

Since Cartesian powers of $K_{3}$ admit uniform mixing at time
$t_{1}=\frac{\pi}{9}$, the same result holds if we replace $Z_{k,3}$ in the
previous example by $H(k,3)$. Moreover, this result applies to any $M$ because
$H(k,3)$ is regular.

## Acknowledgements

I thank the University of Manitoba Faculty of Science and Faculty of Graduate
Studies for the support. I thank Steve Kirkland, Sarah Plosker, Chris Godsil
and Cristino Tamon for the helpful comments and useful discussions. I am also
grateful to the referees for their suggestions that helped improve this paper.

## References