Distance matrix of enhanced power graphs of finite groups

Anita Arora, Hiranya Kishore Dey and Shivani Goel

###### Abstract

The enhanced power graph of a group $G$ is the graph $\mathcal{G}_{E}(G)$ with
vertex set $G$ and edge set $\\{(u,v):u,v\in\langle w\rangle,~{}\mbox{for
some}~{}w\in G\\}$. In this paper, we compute the spectrum of the distance
matrix of the enhanced power graph of non-abelian groups of order $pq$,
dihedral groups, dicyclic groups, elementary abelian groups
$\mathrm{El}(p^{n})$ and the non-cyclic abelian groups
$\mathrm{El}(p^{n})\times\mathrm{El}(q^{m})$ and
$\mathrm{El}(p^{n})\times\mathbb{Z}_{m}$, where $p$ and $q$ are distinct
primes.

For the non-cyclic abelian group $\mathrm{El}(p^{n})\times\mathrm{El}(q^{m})$,
we also compute the spectrum of the adjacency matrix of its enhanced power
graph and the spectrum of the adjacency and the distance matrix of its power
graph.

Keywords. Enhanced power graphs, power graphs, distance matrix, finite groups,
spectrum

AMS CLASSIFICATION. 05C50, 05C25, 20K01

##  1 Introduction

The study of graphs associated with various algebraic structures has been a
topic of increasing interest during the last couple of decades. Cayley graphs
are one of the well-known classes of graphs arising from algebraic structures.
Many different types of graphs, such as power graph [1, 2] and enhanced power
graph [3] of a group has been introduced to explore the properties of
algebraic structures using graph theory. It is important to notice that these
graphs represent some combinatorial properties of the underlying group.
Research on combinatorial properties of groups and semigroups originates from
Bernhard Neumann’s theorem [4] answering a question of Paul Erdós. In [5], a
brief theory of applications of combinatorial properties of groups is given.
We are interested in studying power graphs and enhanced power graphs as they
represent some combinatorial properties of groups. Also, these graphs help us
to characterize the algebraic structures with isomorphic graphs and realize
the interdependence between the algebraic structures and the corresponding
graphs. The concept of power graph was introduced in the context of semigroup
theory by Kelarev and Quin [2].

######  Definition 1.

The _power graph_ $\mathcal{P}(G)$ of a group $G$ is a simple graph with
vertex set $G$ and two vertices $a$ and $b$ are adjacent if and only if one of
them is some power of the other, i.e., either $b^{k}=a$ or $a^{k}=b$, for some
$k\in\mathbb{N}$.

Another closely related graph in this context is the enhanced power graph of a
group which is introduced by Alipour et al. [3].

######  Definition 2.

The _enhanced power graph_ $\mathcal{G}_{E}(G)$ of a group $G$ is the graph
with vertex set $G,$ and two vertices $a$ and $b$ are adjacent if and only if
$a,b\in\langle c\rangle$, for some $c\in G$.

Various properties of the enhanced power graph of finite groups have been
studied in detail. Aalipour et al. [3] characterized finite groups for which
the power and the enhanced power graphs are equal. Besides, Zahirovic et al.
[6] proved that finite groups with isomorphic enhanced power graphs have
isomorphic directed power graphs. Bera et al. [7] studied the completeness,
dominatability, and many other interesting properties of the enhanced power
graph. The metric dimension of enhanced power graphs of finite groups was
studied by Ma and She [8].

In the last decade, many researchers have studied matrices related to the
power and enhanced power graph of finite groups. Chattopadhyay and Panigrahi
[9] studied the Laplacian spectra of power graphs of finite cyclic groups and
the dihedral groups. Mehranian et al. [10] computed the adjacency spectrum of
the power graph of cyclic groups, dihedral groups, elementary abelian groups
of prime power order and the Mathieu group. Hamzeh and Ashrafi [11]
investigated adjacency and Laplacian spectra of power graphs of the cyclic,
dihedral and quaternion groups.

Wani and Srivastav [12] studied the spectrum of the distance matrix of power
graphs of cyclic groups, dihedral groups of order $2p^{n}$, dicyclic groups of
order $4p^{n}$, and groups of order $pq$, where $p$ and $q$ are distinct
primes. However, they have not described the full spectrum explicitly for some
of these graphs. In this paper, we give the full spectrum of the distance
matrix of the power graph of these groups. The main aim of this paper is to
study the distance spectra of the enhanced power graphs of finite groups. In
Section 2, we develop some backgrounds and state some earlier known results,
which are crucial for the forthcoming sections. In Section 3, we study the
distance spectra of enhanced power graphs of groups of order $pq$, dihedral
groups, and dicyclic groups. We conclude the paper by considering certain
classes of finite abelian groups viz.
$\mathrm{El}(p^{n})\times\mathrm{El}(q^{m})$ and
$\mathrm{El}(p^{n})\times\mathbb{Z}_{m}$ in Section 4 and 5.

##  2 Preliminaries

For the convenience of the reader, we start by recalling some basic
definitions and notations.

  1. 1.

Throughout the paper, we consider $G$ as a finite group and denote its
identity element by $e$. $|G|$ denotes the cardinality of the set $G.$ For a
prime $p,$ a group $G$ is said to be a $p$-group if $|G|=p^{r}$ for some
$r\in\mathbb{N}.$ If every element of $G$ is of order $p$ we call $G$ to be an
_elementary abelian $p$-group_ and the elementary abelian group of order
$p^{n}$ is denoted by $\mathrm{El}(p^{n})$. We use $G^{*}$ to denote the set
$G\backslash\\{e\\}$.

  2. 2.

Let $\Gamma$ be a graph with vertex set $V(\Gamma)$ and edge set $E(\Gamma)$.
For a vertex $u$, we denote by $N(u)$, the set of vertices adjacent to $u$. A
vertex is called a _dominating vertex_ if it is adjacent to every other vertex
in the graph. A _path_ in a graph is a finite sequence of edges which joins a
sequence of distinct vertices. A graph is said to be _connected_ if there
exists a path between for any pair of vertices $u$ and $v.$ The distance
between two vertices $u$ and $v$ is the length of the shortest path between
them and it is denoted by $d(u,v).$ For a graph $\Gamma,$
$\text{diam}(\Gamma)=\max_{u,v\in V}d(u,v)$ is called the _diameter_ of
$\Gamma$.

  3. 3.

All vectors are considered to be column vectors unless stated otherwise. We
use $\mathbf{0}_{n}$ and $\mathbf{1}_{n}$ to denote the zero vector and the
all ones vector in $\mathbb{R}^{n}$. The identity and the all ones matrix of
order $n$ are denoted by $I_{n}$ and $J_{n}$, respectively. We use $J_{m,n}$
to denote the all ones matrix of order $m\times n$ and $O$ to denote the zero
matrix of appropriate order. The Kronecker product of matrices is represented
by $\otimes$.

  4. 4.

Suppose $\Gamma$ is a simple graph with vertex set $V=\\{1,2,\dots,n\\}$. The
adjacency matrix $A:=(a_{ij})$ of $\Gamma$ is an $n\times n$ matrix with
$a_{ij}:=1$ if $i$ and $j$ are adjacent and zero otherwise. Moreover, the
matrix $D(\Gamma):=(d_{ij})$ is called the distance matrix of $\Gamma$, where

$d_{ij}:=\begin{cases}0&~{}\mbox{if}~{}i=j\\\
d(i,j)&~{}\mbox{otherwise}.\end{cases}$

  5. 5.

A partition of a set $V$ is a set of non-empty subsets of $V$ such that every
element of $V$ is in exactly one of these subsets. A partition
$V(\Gamma)=V_{1}\cup\dotsc\cup V_{m}$ of the vertex set of $\Gamma$ is called
_equitable_ if for each $i\in[m]$ and for all $u,v\in V_{i}$, $|N(u)\cap
V_{j}|=|N(v)\cap V_{j}|$, for all $j\in[m]$. The matrix $T:=(t_{ij})$, where
$t_{ij}:=|N(u)\cap V_{j}|$, for $i,j\in[m]$ and $u\in V_{i}$ is called the
_quotient matrix_ of $\Gamma$ corresponding to the equitable partition
$\\{V_{1},\dotsc,V_{m}\\}$.

  6. 6.

Suppose $\\{V_{1},\dotsc,V_{m}\\}$ is an equitable partition of a graph
$\Gamma$. For each $i,j\in[m]$ and $u\in V_{i}$, we define

$t^{D}_{ij}:=\begin{cases}2|V_{i}|-2-|N(u)\cap V_{i}|&~{}\mbox{if}~{}i=j\\\
2|V_{j}|-|N(u)\cap V_{j}|&~{}\mbox{otherwise.}\end{cases}$ (1)

The matrix $T^{D}:=(t^{D}_{ij})$ is called the _distance quotient matrix_ of
$\Gamma$. Note that $T^{D}$ is the same as the quotient matrix defined in
Andries [13] for the distance matrix of a graph.

  7. 7.

For a matrix $B$, the characteristic polynomial is denoted by $\phi(B,x)$. We
denote the characteristic polynomial of the adjacency and distance matrix of a
graph $\Gamma$ by $\phi_{A}(\Gamma,x)$ and $\phi_{D}(\Gamma,x)$, respectively.

  8. 8.

For a graph $\Gamma$ with vertex set $V(\Gamma)=\\{1,\dotsc,n\\}$, the
$\Gamma$-join of a set of graphs $\\{\Gamma_{i}:1\leq i\leq n\\}$ is the graph
$\Gamma[\Gamma_{1},\Gamma_{2},\dots,\Gamma_{n}]$ with vertex set $V$ and edge
set $E$, where

$V:=\\{(i,u):1\leq i\leq n,u\in V(\Gamma_{i})\\}$
$E:=\\{(i,u)(j,v):\mbox{if}~{}ij\in E(\Gamma)~{}\mbox{or
else}~{}i=j~{}\mbox{and}~{}uv\in E(\Gamma_{i})\\}.$

Note that the graph $\Gamma[\Gamma_{1},\dotsc,\Gamma_{n}]$ is obtained by
replacing each vertex $i\in V(\Gamma)$ by $\Gamma_{i}$. For the sake of
convenience, we assume
$V(\Gamma[\Gamma_{1},\Gamma_{2},\dots,\Gamma_{n}])=\cup_{i=1}^{n}V(\Gamma_{i})$.

We now state some basic results, which are useful in proving the main results
of the paper.

######  Theorem 1.

[13, Chapter 2] Suppose $\Gamma$ is a graph on $n$ vertices with diameter at
most $2$ and let $V_{1},\dotsc,V_{m}$ be an equitable partition of
$V(\Gamma)$. Let $T^{D}$ be the matrix defined in (1). Then $\phi(T^{D},x)$
divides $\phi_{D}(\Gamma,x)$.

The following result appears in [14, Theorem 2.1]. Since it is presented
differently than what is required for our results, we state it below in a
different way.

######  Theorem 2.

Suppose $\Gamma$ is a connected graph with vertex set $\\{1,\dotsc,p\\}$. If
$\Gamma_{i}$, $1\leq i\leq p$ are all $r_{i}$ regular with diameter at most 2,
then $V(\Gamma_{1}),\dotsc,V(\Gamma_{p})$ is an equitable partition of
$\Gamma[\Gamma_{1},\dotsc,\Gamma_{p}]$. Let $T^{D}$ denote the matrix
associated with this partition as defined in (1). Then the characteristic
polynomial of the distance matrix of $\Gamma[\Gamma_{1},\dotsc,\Gamma_{p}]$ is

$\phi_{D}(\Gamma[\Gamma_{1},\dotsc,\Gamma_{p}],x)=\phi(T^{D},x)\prod_{i=1}^{p}\frac{\phi_{D}(\Gamma_{i},x)}{(x-2n_{i}+2+r_{i})}.$

Here, for each $i$, $n_{i}$ is the number of vertices in $\Gamma_{i}$.

##  3 Groups of order $pq$, Dihedral groups, and Dicyclic groups

In this section, we study the distance spectra of enhanced power graphs of
groups of order $pq$, dihedral groups, and dicyclic groups. We start with the
non-abelian groups of order $pq$, where $p$ and $q$ are distinct primes. This
is because the enhanced power graph of any abelian group of order $pq$ is
complete.

######  Theorem 3.

Let $p,q$ be primes with $p<q$ and $G_{pq}$ denote the unique non-abelian
group of order $pq$, if it exists. Then, the characteristic polynomial of the
distance matrix of $\mathcal{G}_{E}(G_{pq})$ is

$\displaystyle\phi_{D}(\mathcal{G}_{E}(G_{pq}),x)$
$\displaystyle=(x+1)^{pq-q-2}(x+p)^{q-1}\bigg{[}x^{3}+x^{2}(-2pq+p+q+2)+$
$\displaystyle\hskip 17.07164ptx(-2pq^{2}-2pq+2q^{2}+2p+1)-(pq^{2}+pq-
p-q^{2})\bigg{]}.$

###### Proof.

By Sylow Theorem, there is a unique $q$-Sylow subgroup of $G_{pq}$, say
$H_{1}$. As $G_{pq}$ is non-cyclic, the number of p-Sylow subgroups of
$G_{pq}$ is $q$. Suppose $H_{2},\dotsc,H_{q+1}$ are the p-Sylow subgroups of
$G_{pq}$. It is easy to see that $H_{j}$’s, $2\leq j\leq q+1$ have a trivial
intersection and hence no two non-identity elements of distinct $H_{j}$’s are
adjacent in $\mathcal{G}_{E}(G_{pq})$. Moreover, since $G_{pq}$ is non-cyclic,
no non-identity element of $H_{1}$ is adjacent to a non-identity element of
$H_{j}$ in $\mathcal{G}_{E}(G_{pq})$, for all $2\leq j\leq q+1$. Finally,
since each $H_{j}$, $1\leq j\leq q+1$ induces a complete subgraph in
$\mathcal{G}_{E}(G_{pq})$, we have

$\mathcal{G}_{E}(G_{pq})\cong
K_{1,q+1}[K_{1},K_{q-1},\underbrace{K_{p-1},\dotsc,K_{p-1}}_{q}].$ (2)

Suppose $T^{D}$ is the distance quotient matrix of $\mathcal{G}_{E}(G_{pq})$
corresponding to the equitable partition given in (2). Then

$T^{D}=\left[\begin{array}[]{cccc}0&q-1&(p-1)\mathbf{1}_{q}^{\prime}\\\
1&q-2&(2p-2)\mathbf{1}_{q}^{\prime}\\\
\mathbf{1}_{q}&(2q-2)\mathbf{1}_{q}&2(p-1)J_{q}-pI_{q}\end{array}\right].$

Using Theorem 2, we get

$\displaystyle\phi_{D}(\mathcal{G}_{E}(G_{pq}),x)$
$\displaystyle=\phi(T^{D},x)\frac{\phi_{D}(K_{q-1},x)}{(x-(q-2))}\prod_{i=1}^{q}\frac{\phi_{D}(K_{p-1},x)}{(x-(p-2))}$
(3) $\displaystyle=\phi(T^{D},x)(x+1)^{q-2}(x+1)^{(p-2)q}$
$\displaystyle=\phi(T^{D},x)(x+1)^{pq-q-2}.$

We now determine the characteristic polynomial of $T^{D}$. Using elementary
row and column operations, we have

$\displaystyle\phi(T^{D},x)$ $\displaystyle={\rm det}(xI_{q+2}-T^{D})$ (4)
$\displaystyle=\left|\begin{array}[]{cccc}x&1-q&(1-p)\mathbf{1}_{q}^{\prime}\\\
-1&x-(q-2)&(2-2p)\mathbf{1}_{q}^{\prime}\\\
-\mathbf{1}_{q}&(2-2q)\mathbf{1}_{q}&(x+p)I_{q}-2(p-1)J_{q}\end{array}\right|$
$\displaystyle=\left|\begin{array}[]{cccc}x&1-q&(1-p)\mathbf{1}_{q}^{\prime}\\\
-1&x-(q-2)&(2-2p)\mathbf{1}_{q}^{\prime}\\\
0&-(x+q)\mathbf{1}_{q}&(x+p)I_{q}\end{array}\right|$
$\displaystyle=x\left|\begin{array}[]{cc}x-q+2&(2-2p)\mathbf{1}_{q}^{\prime}\\\
-(x+q)\mathbf{1}_{q}&(x+p)I_{q}\end{array}\right|+\left|\begin{array}[]{cc}1-q&(1-p)\mathbf{1}_{q}^{\prime}\\\
-(x+q)\mathbf{1}_{q}&(x+p)I_{q}\end{array}\right|$
$\displaystyle=x(x-q+2)(x+p)^{q}+xq(2-2p)(x+q)(x+p)^{q-1}+(1-q)(x+p)^{q}$
$\displaystyle\hskip 11.38109pt+q(x+q)(1-p)(x+p)^{q-1}$
$\displaystyle=(x+p)^{q-1}[x^{3}+x^{2}(-2pq+p+q+2)+x(-2pq^{2}-2pq+2q^{2}+2p+1)-$
$\displaystyle\hskip 17.07164pt(pq^{2}+pq-p-q^{2})].$

From (3) and (4), the proof is complete. ∎

By [3, Theorem 28] the power graph and the enhanced power graph for a finite
group $G$ are same if and only if every cyclic subgroup of $G$ has prime power
order. Thus, for a non-cyclic group $G$ of order $pq$, the power graph and the
enhanced power graph are the same and hence $\mathcal{P}(G_{pq})$ and
$\mathcal{G}_{E}(G_{pq})$ have identical distance spectra. Next, we compute
the distance spectra of the enhanced power graph of the dihedral group
$D_{2n}$. For $n\geq 3$, the _dihedral group_ $D_{2n}$ of order $2n$ is
defined as

$D_{2n}=\langle a,b:a^{n}=b^{2}=e,ab=ba^{-1}\rangle.$

######  Theorem 4.

For any positive integer $n$, the characteristic polynomial of the distance
matrix of the enhanced power graph of $D_{2n}$ is

$\phi_{D}(\mathcal{G}_{E}(D_{2n}),x)=(x+2)^{n-1}(x+1)^{n-2}(x^{3}-(3n-4)x^{2}-(2n^{2}+4n-5)x-n^{2}-2n+2).$

###### Proof.

It is easy to verify that

$V(\mathcal{G}_{E}(D_{2n}))=\langle e\rangle\cup\langle
a\rangle^{*}\cup\langle ab\rangle^{*}\cup\dotsc\cup\langle
a^{n-1}b\rangle^{*}$

is an equitable partition for the vertex set of $\mathcal{G}_{E}(D_{2n})$.
This yields,

$\mathcal{G}_{E}(D_{2n})\cong
K_{1,n+1}[K_{1},K_{n-1},\underbrace{K_{1},\dotsc,K_{1}}_{n}].$

By Theorem 2, we get

$\phi_{D}(\mathcal{G}_{E}(D_{2n}),x)=\phi(T^{D},x)(x+1)^{2n-n-2},$

where

$T^{D}=\left[\begin{array}[]{cccc}0&n-1&\mathbf{1}^{\prime}_{n}\\\
1&n-2&\mathbf{1}^{\prime}_{n}\\\
\mathbf{1}_{n}&(2n-2)\mathbf{1}_{n}&2J_{n}-2I_{n}\end{array}\right].$

Proceeding similarly as in the proof of Theorem 3, we compute

$\phi(T^{D},x)=(x+2)^{n-1}\big{(}x^{3}-(3n-4)x^{2}-(2n^{2}+4n-5)x-n^{2}-2n+2\big{)}.$

This completes the proof. ∎

By $\mathcal{P}^{*}(G)$, we denote the induced subgraph of $\mathcal{P}(G)$ on
$G\setminus\\{e\\}.$ This is also known as _proper power graph_ of $G$.
Suppose $\mathbb{Z}_{n}$ denotes the cyclic group of order $n$. In the
following result, we compute the characteristic polynomial of the distance
matrix of $\mathcal{P}(D_{2n})$ in terms of the characteristic polynomials of
the distance matrix of $\mathcal{P}(\mathbb{Z}_{n})$ and
$\mathcal{P}^{*}(\mathbb{Z}_{n})$. This is analogous to [15, Theorem 2.2].

######  Theorem 5.

For any positive integer $n$, the characteristic polynomial of the distance
matrix of the power graph of $D_{2n}$ is

$\phi_{D}(\mathcal{P}(D_{2n}),x)=(x+2)^{n-1}(((4n+1)x+2(n+1))\phi_{D}(\mathcal{P}(\mathbb{Z}_{n}),x)-n(2x+1)^{2}\phi_{D}(\mathcal{P}^{*}(\mathbb{Z}_{n}),x)).$

###### Proof.

From [16], we first note that $\mathcal{P}(D_{2n})$ is a union of
$\mathcal{P}(\mathbb{Z}_{n})$ and $n$ copies of $K_{2}$ that share in the
identity element of $D_{2n}$. Thus, we have

$\mathcal{P}(D_{2n})\cong
K_{1,n+1}[K_{1},\underbrace{K_{1},\dotsc,K_{1}}_{n},\mathcal{P}^{*}(\mathbb{Z}_{n})].$
(5)

From [17, Theorem 2.2], we know that

$\mathcal{P}^{*}(\mathbb{Z}_{n})\cong\Gamma_{1}[K_{\phi(n)},K_{\phi(d_{1})},\dots,K_{\phi(d_{t})}],$
(6)

where $\Gamma_{1}:=K_{1}+\Delta_{n}$ and $\Delta_{n}$ is the graph with vertex
set $V(\Delta_{n})=\\{d_{i}:1,n\neq d_{i},d_{i}|n,1\leq i\leq t\\}$ and edge
set $E(\Delta_{n})=\\{d_{i}d_{j}:d_{i}|d_{j},1\leq i<j\leq t\\}$. Combining
(5) and (6), we have

$\displaystyle\mathcal{P}(D_{2n})$ $\displaystyle\cong
K_{1,n+1}[K_{1},\underbrace{K_{1},\dotsc,K_{1}}_{n},\Gamma_{1}[K_{\phi(n)},K_{\phi(d_{1})},\dots,K_{\phi(d_{t})}]]$
(7)
$\displaystyle\cong\Gamma_{2}[K_{1},\underbrace{K_{1},\dotsc,K_{1}}_{n},K_{\phi(n)},K_{\phi(d_{1})},\dots,K_{\phi(d_{t})}],$

where $\Gamma_{2}$ is the graph obtained by identifying the lower last pendant
vertex of $K_{1,n+1}$ with the vertex corresponding to $K_{1}$ in
$\Gamma_{1}$. Here, we are looking at $K_{1,n+1}$ as a graph with a vertex on
the left attached to $n+1$ vertices on its right. Now, by Theorem 2

$\displaystyle\phi_{D}(\mathcal{P}(D_{2n}),x)$
$\displaystyle=\phi(T^{D},x)(x+1)^{\phi(n)-1}\prod_{i=1}^{t}(x+1)^{\phi(d_{i})-1}$
(8) $\displaystyle=\phi(T^{D},x)(x+1)^{\phi(n)-1}(x+1)^{n-\phi(1)-\phi(n)-t}$
$\displaystyle=\phi(T^{D},x)(x+1)^{n-t-2}.$

Let $T^{D}_{\mathbb{Z}}$ be the distance quotient matrix of
$\mathcal{P}^{*}(\mathbb{Z}_{n})$ corresponding to the equitable partition
given by
$\mathcal{P}^{*}(\mathbb{Z}_{n})\cong\Gamma_{1}[K_{\phi(n)},K_{\phi(d_{1})},\dots,K_{\phi(d_{t})}]$.
Suppose $T^{D}_{\mathbb{Z}}(1)$ denotes the first row of $T^{D}_{\mathbb{Z}}$.
Then
$T^{D}_{\mathbb{Z}}(1)=(\phi(n)-1,\phi(d_{1}),\phi(d_{2}),\cdots,\phi(d_{t}))$.
Let $T^{D}$ be the distance quotient matrix of $\mathcal{P}(D_{2n})$ for the
partition given in (7). Then

$T^{D}=\left[\begin{array}[]{ccccccccccc}0&\mathbf{1}_{n}^{\prime}&T^{D}_{\mathbb{Z}}(1)+e_{1}^{\prime}\\\
\mathbf{1}_{n}&2(J_{n}-I_{n})&2\mathbf{1}_{n}(T^{D}_{\mathbb{Z}}(1)+e_{1}^{\prime})\\\
\mathbf{1}_{t+1}&2J_{t+1,n}&T^{D}_{\mathbb{Z}}\\\ \end{array}\right],$

where $e_{1}=(1,0,\dotsc,0)^{\prime}\in\mathbb{R}^{t+1}$. Now, the
characteristic polynomial of $T^{D}$ is

$\displaystyle\phi(T^{D},x)$ $\displaystyle={\rm det}(xI_{n+t+2}-T^{D})$
$\displaystyle=\left|\begin{array}[]{ccccccccccc}x&-\mathbf{1}_{n}^{\prime}&-T^{D}_{\mathbb{Z}}(1)-e_{1}^{\prime}\\\
-\mathbf{1}_{n}&(x+2)I_{n}-2J_{n}&-2\mathbf{1}_{n}(T^{D}_{\mathbb{Z}}(1)+e_{1}^{\prime})\\\
-\mathbf{1}_{t+1}&-2J_{t+1,n}&xI_{t+1}-T^{D}_{\mathbb{Z}}\\\
\end{array}\right|.$

Subtracting twice the first row from the second to $(n+1)^{\rm th}$ row and
then subtracting twice the first column from the second to $(n+1)^{\rm th}$
column, we get

$\displaystyle\phi(T^{D},x)$
$\displaystyle=\left|\begin{array}[]{ccccccccccc}x&-(2x+1)\mathbf{1}_{n}^{\prime}&-T^{D}_{\mathbb{Z}}(1)-e_{1}^{\prime}\\\
-(2x+1)\mathbf{1}_{n}&(x+2)I_{n}+(4x+2)J_{n}&O\\\
-\mathbf{1}_{t+1}&O&xI_{t+1}-T^{D}_{\mathbb{Z}}\\\ \end{array}\right|$
$\displaystyle=\small\left|\begin{array}[]{ccccccccccc}x&-(2x+1)\mathbf{1}_{n-1}^{\prime}&-(2x+1)&-T^{D}_{\mathbb{Z}}(1)-e_{1}^{\prime}\\\
-(2x+1)\mathbf{1}_{n-1}&(x+2)I_{n-1}+(4x+2)J_{n-1}&(4x+2)\mathbf{1}_{n-1}&O\\\
-(2x+1)&(4x+2)\mathbf{1}^{\prime}_{n-1}&(5x+4)&\mathbf{0}_{t+1}^{\prime}\\\
-\mathbf{1}_{t+1}&O&\mathbf{0}_{t+1}&xI_{t+1}-T^{D}_{\mathbb{Z}}\\\
\end{array}\right|.$

Subtracting the $(n+1)^{\rm th}$ row from the second to $n^{\rm th}$ row and
then adding the sum of the second to $n^{\rm th}$ column to the $(n+1)^{\rm
th}$ column, we have

$\displaystyle\phi(T^{D},x)$
$\displaystyle=\left|\begin{array}[]{ccccccccccc}x&-(2x+1)\mathbf{1}_{n-1}&-n(2x+1)&-T^{D}_{\mathbb{Z}}(1)-e_{1}^{\prime}\\\
\mathbf{0}_{n-1}&(x+2)I_{n-1}&\mathbf{0}_{n-1}&O\\\
-(2x+1)&(4x+2)\mathbf{1}^{\prime}_{n-1}&(4n+1)x+2(n+1)&\mathbf{0}^{\prime}_{t+1}\\\
-\mathbf{1}_{t+1}&O&\mathbf{0}_{t+1}&xI_{t+1}-T^{D}_{\mathbb{Z}}\\\
\end{array}\right|$
$\displaystyle=((4n+1)x+2(n+1))\left|\begin{array}[]{ccccccccccc}x&-(2x+1)\mathbf{1}_{n-1}&-T^{D}_{\mathbb{Z}}(1)-e_{1}^{\prime}\\\
\mathbf{0}_{n-1}&(x+2)I_{n-1}&O\\\
-\mathbf{1}_{t+1}&O&xI_{t+1}-T^{D}_{\mathbb{Z}}\\\ \end{array}\right|$
$\displaystyle~{}~{}~{}-n(2x+1)^{2}(x+2)^{n-1}\phi(T^{D}_{\mathbb{Z}},x).$

In the first determinant, subtracting the $(n+1)^{\rm th}$ row from the first
row and then adding the first column to the $(n+1)^{\rm th}$ column, we get

$\displaystyle\phi(T^{D},x)$
$\displaystyle=((4n+1)x+2(n+1))\small\left|\begin{array}[]{ccccccccccc}x+1&-(2x+1)\mathbf{1}_{n-1}&\mathbf{0}_{t+1}^{\prime}\\\
\mathbf{0}_{n-1}&(x+2)I_{n-1}&O\\\
-\mathbf{1}_{t+1}&O&xI_{t+1}-(T^{D}_{\mathbb{Z}}+\mathbf{1}_{t+1}e_{1}^{\prime})\\\
\end{array}\right|$ (9)
$\displaystyle~{}~{}~{}-n(2x+1)^{2}(x+2)^{n-1}\phi(T^{D}_{\mathbb{Z}},x)$
$\displaystyle=(x+2)^{n-1}((x+1)((4n+1)x+2(n+1)){\rm
det}(xI-(T^{D}_{\mathbb{Z}}+\mathbf{1}_{t+1}e_{1}^{\prime}))$
$\displaystyle~{}~{}~{}-n(2x+1)^{2}\phi(T^{D}_{\mathbb{Z}},x)).$

Note that, $T^{D}_{\mathbb{Z}}+\mathbf{1}_{t+1}e_{1}^{\prime}$ is the quotient
matrix of $\mathcal{P}(\mathbb{Z}_{n})$ corresponding to the equitable
partition given by the expression
$\mathcal{P}(\mathbb{Z}_{n})\cong\Gamma_{1}[K_{\phi(n)+1},K_{\phi(d_{1})},\dots,K_{\phi(d_{t})}]$.
Thus, from (8) and (9), we conclude that

$\phi_{D}(\mathcal{P}(D_{2n}),x)\\\
=(x+2)^{n-1}(((4n+1)x+2(n+1))\phi_{D}(\mathcal{P}(\mathbb{Z}_{n}),x)-n(2x+1)^{2}\phi_{D}(\mathcal{P}^{*}(\mathbb{Z}_{n}),x)).$

The proof is complete. ∎

We now compute the distance spectra of the enhanced power graph of the
dicyclic group. For $n\geq 3$, the dicyclic group $Dic_{4n}$ of order $4n$ is
defined as

$Dic_{4n}=\langle a,x:a^{2n}=e,x^{2}=a^{n},ax=xa^{-1}\rangle.$

######  Theorem 6.

For any positive integer $n$, the characteristic polynomial of the distance
matrix of the enhanced power graph of $Dic_{4n}$ is

$\phi_{D}(\mathcal{G}_{E}(Dic_{4n}),x)=(x+1)^{3n-2}(x+3)^{n-1}(x^{3}-(6n-5)x^{2}-(8n^{2}+4n-7)x-(6n-3)).$

###### Proof.

There are two dominating vertices in $\mathcal{G}_{E}(Dic_{4n})$, one is the
identity $e$ and the other is element $a^{n}$. Since $a^{2n}=e$ and
$(a^{i}x)^{2}=x^{2}$, for $1\leq i\leq n$

$\mathcal{G}_{E}(Dic_{4n})\cong
K_{1,n+1}[K_{2},K_{2n-2},\underbrace{K_{2},\dotsc,K_{2}}_{n}].$

Suppose $T^{D}$ is the distance quotient matrix of $\mathcal{G}_{E}(Dic_{4n})$
corresponding to the equitable partition given in (2). Then

$T^{D}=\left[\begin{array}[]{cccc}1&2n-2&2\mathbf{1}_{n}^{\prime}\\\
2&2n-3&4\mathbf{1}_{n}^{\prime}\\\
2\mathbf{1}_{n}&2(2n-2)\mathbf{1}_{n}&4J_{n}-3I_{n}\end{array}\right].$

By Theorem 2, we get

$\phi_{D}(\mathcal{G}_{E}(Dic_{4n}),x)=\phi(T^{D},x)(x+1)^{3n-2}.$

Proceeding similar to the proof of Theorem 3, the characteristic polynomial of
$T^{D}$ can be easily computed. This completes the proof. ∎

Suppose $n$ is a power of $2$. From Theorem 6, we immediately have the
following:

$\phi_{D}(\mathcal{P}(Dic_{4n}),x)=(x+1)^{3n-2}(x+3)^{n-1}(x^{3}-(6n-5)x^{2}-(8n^{2}+4n-7)x-(6n-3)).$

Wani and Srivastsav [12, Proposition 2.12] also found out the distance
quotient matrix of the graph $\mathcal{P}(Dic_{4n})$ but the $(n+1,n+2)$-th
entry of their matrix is $2$, which is not correct, as seen from the proof of
Theorem 6.

##  4 The finite abelian group $\mathrm{El}(p^{n})\times\mathrm{El}(q^{m})$

In this section, we are interested in studying the spectral properties of the
power graph and the enhanced power graph of certain non-cyclic finite abelian
groups. Kelarev and Quinn gave the structure of the power graphs of all finite
abelian groups in [2]. We further study the structure of power and enhanced
power graphs of these groups and obtain a precise representation in terms of
equitable partitions. Suppose $p$ and $q$ are distinct primes. We give two
equitable partitions for the power graph of
$\mathrm{El}(p^{n})\times\mathrm{El}(q^{m})$. These partitions are equitable
partitions for the enhanced power graph of
$\mathrm{El}(p^{n})\times\mathrm{El}(q^{m})$ as well. The later is proved
before section 4.3. We recall that $e$ represents the identity element of a
group.

######  Lemma 1.

Define $V_{1}:=\\{(e,e)\\}$, $V_{2}:=\\{(a,e):a\in\mathrm{El}(p^{n})^{*}\\}$,
$V_{3}:=\\{(a,b):a\in\mathrm{El}(p^{n})^{*},b\in\mathrm{El}(q^{m})^{*}\\}$,
and $V_{4}:=\\{(e,b):b\in\mathrm{El}(q^{m})^{*}\\}$. Then
$\\{V_{1},V_{2},V_{3},V_{4}\\}$ is an equitable partition for
$\mathcal{P}(\mathrm{El}(p^{n})\times\mathrm{El}(q^{m}))$.

###### Proof.

Let $G=\mathrm{El}(p^{n})\times\mathrm{El}(q^{m})$. Since the sets
$V_{1},V_{2},V_{3}$ and $V_{4}$ are mutually exclusive with cardinalities $1$,
$p^{n}-1$, $(p^{n}-1)(q^{m}-1)$ and $q^{m}-1$, respectively,
$\\{V_{1},V_{2},V_{3},V_{4}\\}$ forms a partition for $V(\mathcal{P}(G))$.
Suppose $1\leq i\leq 4$. We need to show that for each $i$ and for all $u,v\in
V_{i}$,

$|N(u)\cap V_{j}|=|N(v)\cap V_{j}|,~{}\mbox{for}~{}j=1,2,3,4.$

For $i=1$, as $(e,e)$ is a dominating vertex, this readily holds. Observe that
for any $(a_{1},b_{1}),(a_{2},b_{2})\in G$,

$(a_{2},b_{2})\in\langle(a_{1},b_{1})\rangle,~{}\mbox{if and only
if}~{}a_{2}\in\langle a_{1}\rangle~{}\mbox{and}~{}b_{2}\in\langle
b_{1}\rangle.$ (10)

Using this observation, we have the following

  1. 1.

For any $u\in V_{2}$,

$|N(u)\cap V_{1}|=1,~{}|N(u)\cap V_{2}|=p-2,$ $|N(u)\cap
V_{3}|=(p-1)(q^{m}-1)~{}\mbox{and}~{}|N(u)\cap V_{4}|=0.$

  2. 2.

For any $u\in V_{3}$,

$|N(u)\cap V_{1}|=1,~{}|N(u)\cap V_{2}|=p-1,$ $|N(u)\cap
V_{3}|=(p-1)(q-1)-1~{}\mbox{and}~{}|N(u)\cap V_{4}|=q-1.$

  3. 3.

For any $u\in V_{4}$,

$|N(u)\cap V_{1}|=1,~{}|N(u)\cap V_{2}|=0,$ $|N(u)\cap
V_{3}|=(p^{n}-1)(q-1)~{}\mbox{and}~{}|N(u)\cap V_{4}|=q-2.$

Hence, $\\{V_{1},V_{2},V_{3},V_{4}\\}$ is an equitable partition for
$\mathcal{P}(G)$. The proof is complete. ∎

######  Lemma 2.

Suppose $\alpha=\frac{p^{n}-1}{p-1}$, and $\beta=\frac{q^{m}-1}{q-1}$. Let
$\langle(a_{1},e)\rangle,\dots,\langle(a_{\alpha},e)\rangle$ be distinct
subgroups of order $p$ and
$\langle(e,b_{1})\rangle,\dots,\langle(e,b_{\beta})\rangle$ be distinct
subgroups of order $q$ in $\mathrm{El}(p^{n})\times\mathrm{El}(q^{m})$. Define
$U_{1}:=\\{(e,e)\\}$, $V_{i}:=\langle(a_{i},e)\rangle^{*}$,
$W_{i,j}:=\langle(a_{i},b_{j})\rangle\setminus(\langle(a_{i},e)\rangle\cup\langle(e,b_{j})\rangle)$
and $X_{i}:=\langle(e,b_{i})\rangle^{*}$, for $i\in[\alpha]$ and
$j\in[\beta]$. Then

$P:=\\{U_{1},V_{1},\dotsc,V_{\alpha},W_{1,1},\dotsc,W_{1,\beta},\dotsc,W_{\alpha,1},\dotsc,W_{\alpha,\beta},X_{1},\dotsc,X_{\beta}\\}$

forms an equitable partition for
$\mathcal{P}(\mathrm{El}(p^{n})\times\mathrm{El}(q^{m}))$.

###### Proof.

Let $G=\mathrm{El}(p^{n})\times\mathrm{El}(q^{m})$. Note that the sets in $P$
are mutually exclusive with $|U_{1}|=1$, $|V_{i}|=p-1$, $|W_{i,j}|=(p-1)(q-1)$
and $|X_{i}|=q-1$, for $i\in[\alpha]$ and $j\in[\beta]$. Therefore, $P$ forms
a partition for $V(\mathcal{P}(G))$. We need to show that $P$ is an equitable
partition. Since, $(e,e)$ is a dominating vertex,

$|N((e,e))\cap Y|=|Y|,~{}\mbox{for}~{}Y\in P\setminus\\{U_{1}\\}.$

Using (10), we have the following

  1. 1.

Fix $i\in[\alpha]$ and let $u\in V_{i}$. Then for any $i^{\prime}\in[\alpha]$
and $j\in[\beta],$

$|N(u)\cap U_{1}|=1,~{}|N(u)\cap
V_{i^{\prime}}|=\begin{cases}p-2&~{}\mbox{if}~{}i^{\prime}=i,\\\
0&~{}\mbox{otherwise,}\end{cases}$ $|N(u)\cap
X_{j}|=0~{}\mbox{and}~{}|N(u)\cap
W_{i^{\prime},j}|=\begin{cases}(p-1)(q-1)&~{}\mbox{if}~{}i^{\prime}=i,\\\
0&~{}\mbox{otherwise.}\end{cases}$

  2. 2.

Fix $i\in[\alpha]$, $j\in[\beta]$ and let $u\in W_{i,j}$. Then for any
$i^{\prime}\in[\alpha]$ and $j^{\prime}\in[\beta],$

$|N(u)\cap U_{1}|=1,~{}|N(u)\cap
V_{i^{\prime}}|=\begin{cases}p-1&~{}\mbox{if}~{}i^{\prime}=i,\\\
0&~{}\mbox{otherwise,}\end{cases}$ $|N(u)\cap
W_{i^{\prime},j^{\prime}}|=\begin{cases}(p-1)(q-1)-1&~{}\mbox{if}~{}i^{\prime}=i~{}\mbox{and}~{}j^{\prime}=j,\\\
0&~{}\mbox{otherwise,}\end{cases}$

and

$|N(u)\cap X_{j^{\prime}}|=\begin{cases}q-1&~{}\mbox{if}~{}j^{\prime}=j,\\\
0&~{}\mbox{otherwise.}\end{cases}$

  3. 3.

Fix $j\in[\beta]$ and let $u\in X_{j}$. Then for any $i\in[\alpha]$ and
$j^{\prime}\in[\beta],$

$|N(u)\cap U_{1}|=1,~{}|N(u)\cap V_{i}|=0,$ $|N(u)\cap
W_{i,j^{\prime}}|=\begin{cases}(p-1)(q-1)&~{}\mbox{if}~{}j^{\prime}=j,\\\
0&~{}\mbox{otherwise,}\end{cases}$

and

$|N(u)\cap X_{j^{\prime}}|=\begin{cases}q-2&~{}\mbox{if}~{}j^{\prime}=j,\\\
0&~{}\mbox{otherwise.}\end{cases}$

Hence, $P$ is an equitable partition for $\mathcal{P}(G)$ and the proof is
complete. ∎

From Lemma 2, the following result is immediate.

######  Lemma 3.

Let $G=\mathrm{El}(p^{n})\times\mathrm{El}(q^{m})$. Suppose
$\alpha=\frac{p^{n}-1}{p-1}$ and $\beta=\frac{q^{m}-1}{q-1}$. Then,

$\mathcal{P}(G)\cong(K_{1}+\Gamma)[K_{1},\underbrace{K_{(p-1)},\dots,K_{(p-1)}}_{\alpha},\underbrace{K_{(p-1)(q-1)},\dots,K_{(p-1)(q-1)}}_{\alpha\beta},\underbrace{K_{(q-1)},\dots,K_{(q-1)}}_{\beta}],$

where $\Gamma$ is the graph given in Figure 1.

$1$$2$$\bf{\alpha}$${\alpha+1}$$\alpha+2$$\alpha+\beta$$\alpha+\beta+1$$\alpha+\beta+2$$\alpha+2\beta$$\alpha+(\alpha-1)\beta+1$$\alpha+(\alpha-1)\beta+2$$\alpha+\alpha\beta$$\alpha+\alpha\beta+1$$\alpha+\alpha\beta+2$$\alpha+\alpha\beta+\beta$
Figure 1: The graph $\Gamma$ related to the power graph of
$\mathrm{El}(p^{n})\times\mathrm{El}(q^{m})$.

We now illustrate Lemma 3 by the following example.

######  Example 1.

Suppose $p=2$, $q=3$ and $n=m=2$. Then

$V_{1}=\langle(1,0,0,0)\rangle^{*},V_{2}=\langle(0,1,0,0)\rangle^{*},V_{3}=\langle(1,1,0,0)\rangle^{*},$

and

$X_{1}=\langle(0,0,0,1)\rangle^{*},X_{2}=\langle(0,0,1,0)\rangle^{*},X_{3}=\langle(0,0,1,1)\rangle^{*},X_{4}=\langle(0,0,1,2)\rangle^{*}.$

Therefore

$W_{1,1}=\\{(1,0,0,1),(1,0,0,2)\\},W_{1,2}=\\{(1,0,1,0),(1,0,2,0)\\},$
$W_{1,3}=\\{(1,0,1,1),(1,0,2,2)\\},W_{1,4}=\\{(1,0,1,2),(1,0,2,1)\\},$
$W_{2,1}=\\{(0,1,0,1),(0,1,0,2)\\},W_{2,2}=\\{(0,1,1,0),(0,1,2,0)\\},$
$W_{2,3}=\\{(0,1,1,1),(0,1,2,2)\\},W_{2,4}=\\{(0,1,1,2),(0,1,2,1)\\},$
$W_{3,1}=\\{(1,1,0,1),(1,1,0,2)\\},W_{3,2}=\\{(1,1,1,0),(1,1,2,0)\\},$
$W_{3,3}=\\{(1,1,1,1),(1,1,2,2)\\},~{}\mbox{and}~{}W_{3,4}=\\{(1,1,1,2),(1,1,2,1)\\}.$

Hence the power graph of $\mathrm{El}(4)\times\mathrm{El}(9)$ is the graph in
Figure 2. In Figure 2, each vertex represents either the graph $K_{1}$ or
$K_{2}$ and an edge between two graphs implies all the vertices of one graph
are connected to all the vertices of the other graph.

$K_{1}$$K_{1}$$K_{1}$$K_{2}$$K_{2}$$K_{2}$$K_{2}$$K_{2}$$K_{2}$$K_{2}$$K_{2}$$K_{2}$$K_{2}$$K_{2}$$K_{2}$$K_{2}$$K_{2}$$K_{2}$$K_{2}$
Figure 2: Proper power graph of $\mathrm{El}(4)\times\mathrm{El}(9)$.

###  4.1 Adjacency spectrum of
$\mathcal{P}(\mathrm{El}(p^{n})\times\mathrm{El}(q^{m}))$

In the next lemma, we compute the quotient matrix of
$\mathcal{P}(\mathrm{El}(p^{n})\times\mathrm{El}(q^{m}))$ corresponding to the
equitable partitions given in Lemma 1 and Lemma 2. As the proof of the lemma
is direct, we omit it.

######  Lemma 4.

The quotient matrices $T_{1}$ and $T_{2}$ of
$\mathcal{P}(\mathrm{El}(p^{n})\times\mathrm{El}(q^{m}))$ corresponding to the
equitable partition given in Lemma 1 and Lemma 2, respectively are

$T_{1}=\left[\begin{array}[]{cccc}0&p^{n}-1&(p^{n}-1)(q^{m}-1)&q^{m}-1\\\
1&p-2&(p-1)(q^{m}-1)&0\\\ 1&p-1&(p-1)(q-1)-1&q-1\\\ 1&0&(p^{n}-1)(q-1)&q-2\\\
\end{array}\right],$

and

$T_{2}=\left[\begin{array}[]{cccc}0&(p-1)\mathbf{1}^{\prime}_{\alpha}&(p-1)(q-1)\mathbf{1}^{\prime}_{\alpha\beta}&(q-1)\mathbf{1}^{\prime}_{\beta}\\\
\mathbf{1}_{\alpha}&(p-2)I_{\alpha}&(p-1)(q-1)I_{\alpha}\otimes\mathbf{1}^{\prime}_{\beta}&O\\\
\mathbf{1}_{\alpha\beta}&(p-1)I_{\alpha}\otimes\mathbf{1}_{\beta}&((p-1)(q-1)-1)I_{\alpha\beta}&(q-1)\mathbf{1}_{\alpha}\otimes
I_{\beta}\\\
\mathbf{1}_{\beta}&O&(p-1)(q-1)\mathbf{1}^{\prime}_{\alpha}\otimes
I_{\beta}&(q-2)I_{\beta}\end{array}\right],$ (11)

where $\alpha=\frac{p^{n}-1}{p-1}$, and $\beta=\frac{q^{m}-1}{q-1}$.

We now compute the characteristic polynomial of $T_{2}$.

######  Lemma 5.

Suppose $T_{2}$ is the matrix given in (11). Then

$\phi(T_{2},x)=\phi(T_{1},x)(x-pq+p+q)^{(\alpha-1)(\beta-1)}f(x)^{\alpha-1}g(x)^{\beta-1},$

where $\alpha=\frac{p^{n}-1}{p-1}$, $\beta=\frac{q^{m}-1}{q-1}$, and $f(x)$
and $g(x)$ are the characteristic polynomials of the matrices
$\left[\begin{array}[]{cc}p-2&(p-1)(q^{m}-1)\\\
p-1&(p-1)(q-1)-1\end{array}\right]$ and
$\left[\begin{array}[]{cc}(p-1)(q-1)-1&q-1\\\
(p^{n}-1)(q-1)&q-2\end{array}\right]$, respectively.

###### Proof.

Suppose $(x_{1},x_{2},x_{3},x_{4})^{\prime}\in\mathbb{R}^{4}$ is an
eigenvector of $T_{1}$ corresponding to an eigenvalue $\lambda$. Consider the
vector
$y=(x_{1},x_{2}\mathbf{1}_{\alpha}^{\prime},x_{3}\mathbf{1}_{\alpha\beta}^{\prime},x_{4}\mathbf{1}_{\beta}^{\prime})^{\prime}\in\mathbb{R}^{1+\alpha+\alpha\beta+\beta}$.
Then, $y$ is an eigenvector of $T_{2}$ corresponding to the eigenvalue
$\lambda$. Let $S_{1}$ be the set of eigenvectors of $T_{2}$ formed this way
from the linear independent eigenvectors of $T_{1}$. Note that any linear
dependence among the vectors in $S_{1}$ must provide an analogous dependence
among the eigenvectors of $T_{1}$. Thus, we conclude that $S_{1}$ is linearly
independent. Hence $\phi(T_{1},x)$ divides $\phi(T_{2},x)$. For
$i\in\\{1,\dotsc,\alpha-1\\}$ and $j\in\\{1,\dotsc,\beta-1\\}$, we define the
vectors $v^{i}:=(v^{i}_{k})^{\prime}\in\mathbb{R}^{\alpha}$ and
$w^{j}:=(w^{j}_{k})^{\prime}\in\mathbb{R}^{\beta}$ as follows:

$v^{i}_{k}:=\begin{cases}~{}~{}1&~{}\mbox{if}~{}k=i\\\
-1&~{}\mbox{if}~{}k=\alpha\\\
~{}~{}0&~{}\mbox{otherwise}\end{cases}~{}~{}~{}~{}~{}\mbox{and}~{}~{}~{}~{}~{}w^{j}_{k}:=\begin{cases}~{}~{}1&~{}\mbox{if}~{}k=j\\\
-1&~{}\mbox{if}~{}k=\beta\\\ ~{}~{}0&~{}\mbox{otherwise.}\end{cases}$

Now, for each $i\in\\{1,\dotsc,\alpha-1\\}$ and $j\in\\{1,\dotsc,\beta-1\\}$,

$\displaystyle T_{2}\left[\begin{array}[]{c}0\\\ \mathbf{0}_{\alpha}\\\
v^{i}\otimes w^{j}\\\ \mathbf{0}_{\beta}\end{array}\right]$
$\displaystyle=\left[\begin{array}[]{c}(p-1)(q-1)\mathbf{1}^{\prime}_{\alpha\beta}(v^{i}\otimes
w^{j})\\\
((p-1)(q-1)I_{\alpha}\otimes\mathbf{1}^{\prime}_{\beta})(v^{i}\otimes
w^{j})\\\ ((p-1)(q-1)-1)I_{\alpha\beta}(v^{i}\otimes w^{j})\\\
((p-1)(q-1)\mathbf{1}^{\prime}_{\alpha}\otimes I_{\beta})(v^{i}\otimes
w^{j})\end{array}\right]=(pq-p-q)\left[\begin{array}[]{c}0\\\
\mathbf{0}_{\alpha}\\\ v^{i}\otimes w^{j}\\\
\mathbf{0}_{\beta}\end{array}\right].$

Since $S_{2}=\\{(0,\mathbf{0}^{\prime}_{\alpha},v^{i}\otimes
w^{j},\mathbf{0}_{\beta}^{\prime})^{\prime}:1\leq i\leq\alpha-1,1\leq
j\leq\beta-1\\}$ is a linearly independent set, $pq-p-q$ is an eigenvalue of
$T_{2}$ with multiplicity $(\alpha-1)(\beta-1)$. Clearly, $S_{1}\cup S_{2}$ is
a set of linearly independent eigenvectors of $T_{2}$. Now consider the
following matrices

$B=\left[\begin{array}[]{cc}p-2&(p-1)(q^{m}-1)\\\
p-1&(p-1)(q-1)-1\end{array}\right],~{}\mbox{and}~{}C=\left[\begin{array}[]{cc}(p-1)(q-1)-1&q-1\\\
(p^{n}-1)(q-1)&q-2\end{array}\right].$

Suppose $x=(x_{1},x_{2})$ is an eigenvector of $B$ corresponding to the
eigenvalue $\lambda$. For each $i\in\\{1,\dotsc,\alpha-1\\}$, define
$z^{i}:=(0,x_{1}v^{i},x_{2}(v^{i}\otimes\mathbf{1}_{\beta}),\mathbf{0}_{\beta})^{\prime}$.
Using the fact that $Bx=\lambda x$, we have for each $i$

$\displaystyle T_{2}z^{i}$
$\displaystyle=\left[\begin{array}[]{cc}x_{1}(p-1)\mathbf{1}^{\prime}_{\alpha}v^{i}+x_{2}(p-1)(q-1)\mathbf{1}^{\prime}_{\alpha\beta}(v^{i}\otimes\mathbf{1}_{\beta})\\\
x_{1}(p-2)I_{\alpha}v^{i}+x_{2}((p-1)(q-1)I_{\alpha}\otimes\mathbf{1}^{\prime}_{\beta})(v^{i}\otimes\mathbf{1}_{\beta})\\\
x_{1}((p-1)I_{\alpha}\otimes\mathbf{1}_{\beta})v^{i}+x_{2}((p-1)(q-1)-1)I_{\alpha\beta}(v^{i}\otimes\mathbf{1}_{\beta})\\\
x_{2}((p-1)(q-1)\mathbf{1}^{\prime}_{\alpha}\otimes
I_{\beta})(v^{i}\otimes\mathbf{1}_{\beta})\end{array}\right]=\lambda z^{i}.$

Since $S_{3}:=\\{z^{i}:i=1,\dotsc,\alpha-1\\}$ is a linearly independent set
of vectors, every eigenvalue of $B$ is an eigenvalue of $T_{2}$ with
multiplicity $\alpha-1$. Similarly, suppose $x=(x_{1},x_{2})$ is an
eigenvector of $C$ corresponding to the eigenvalue $\mu$. For each
$j\in\\{1,\dotsc,\beta-1\\}$, define
$\widetilde{z}^{j}:=(0,\mathbf{0}_{\alpha},x_{1}(\mathbf{1}_{\alpha}\otimes
w^{j}),x_{2}w^{j})^{\prime}$. Then

$\displaystyle T_{2}\widetilde{z}^{j}=\mu\widetilde{z}^{j}.$

Noting that $S_{4}=\\{\widetilde{z}^{j}:j=1,\dotsc,\beta-1\\}$ is a linearly
independent set, we can conclude that every eigenvalue of $C$ is an eigenvalue
of $T_{2}$ with multiplicity $\beta-1$. Suppose $f(x)$ and $g(x)$ are the
characteristic polynomials of the matrices $B$ and $C$, respectively. Since
$S_{1}\cup S_{2}\cup S_{3}\cup S_{4}$ is linearly independent, we have

$\phi(T_{2},x)=\phi(T_{1},x)(x-pq+p+q)^{(\alpha-1)(\beta-1)}f(x)^{\alpha-1}g(x)^{\beta-1}.$

This completes the proof. ∎

We are now ready to compute the characteristic polynomial of the adjacency
matrix of the power graph of $\mathrm{El}(p^{n})\times\mathrm{El}(q^{m})$.

######  Theorem 7.

. Let $G=\mathrm{El}(p^{n})\times\mathrm{El}(q^{m})$. If $A$ is the adjacency
matrix of $\mathcal{P}(G)$, then

$\phi_{A}(\mathcal{P}(G),x)=\phi(T_{1},x)(x+1)^{(p^{n}q^{m}-(\alpha+1)(\beta+1))}(x-pq+p+q)^{(\alpha-1)(\beta-1)}f(x)^{\alpha-1}g(x)^{\beta-1},$

where $\alpha=\frac{p^{n}-1}{p-1}$, $\beta=\frac{q^{m}-1}{q-1}$, and $f(x)$
and $g(x)$ are the characteristic polynomials of the matrices
$\left[\begin{array}[]{cc}p-2&(p-1)(q^{m}-1)\\\
p-1&(p-1)(q-1)-1\end{array}\right]$ and
$\left[\begin{array}[]{cc}(p-1)(q-1)-1&q-1\\\
(p^{n}-1)(q-1)&q-2\end{array}\right]$, respectively.

###### Proof.

Since the partition corresponding to $T_{2}$ is an equitable partition for
$\mathcal{P}(G)$, by Theorem 4 in [18], $\phi(T_{2},x)$ divides
$\phi_{A}(\mathcal{P}(G),x)$. From Lemma 3, we know that

$\mathcal{P}(G)\cong(K_{1}+\Gamma)[K_{1},\underbrace{K_{(p-1)},\dots,K_{(p-1)}}_{\alpha},\underbrace{K_{(p-1)(q-1)},\dots,K_{(p-1)(q-1)}}_{\alpha\beta},\underbrace{K_{(q-1)},\dots,K_{(q-1)}}_{\beta}],$

where $\alpha=\frac{p^{n}-1}{p-1}$ and $\beta=\frac{q^{m}-1}{q-1}$. By Theorem
7 in [18], we have

$\displaystyle\phi_{A}(\mathcal{P}(G),x)$
$\displaystyle=\phi(T_{2},x)(x+1)^{(p-2)\alpha}(x+1)^{((p-1)(q-1)-1)\alpha\beta}(x+1)^{(q-2)\beta}$
$\displaystyle=\phi(T_{2},x)(x+1)^{p^{n}q^{m}-(\alpha+1)(\beta+1)}.$

By using Lemma 5, the proof is complete. ∎

###  4.2 Distance spectrum of
$\mathcal{P}(\mathrm{El}(p^{n})\times\mathrm{El}(q^{m}))$

Suppose $T:=[t_{ij}]$ is the quotient matrix for a graph $\Gamma$
corresponding to the equitable partition $V(\Gamma)=V_{1}\cup\dotsc\cup
V_{m}$. Let $T^{D}:=[t^{D}_{ij}]$ be the corresponding distance quotient
matrix. From (1), it is easy to see that for each $i,j\in[m]$

$t^{D}_{ij}:=\begin{cases}2|V_{i}|-2-t_{ii}&~{}\mbox{if}~{}i=j,\\\
2|V_{j}|-t_{ij}&~{}\mbox{otherwise.}\end{cases}$

In the following lemma, we compute the distance quotient matrices of
$\mathcal{P}(\mathrm{El}(p^{n})\times\mathrm{El}(q^{m}))$ corresponding to the
equitable partitions given in Lemma 1 and Lemma 2.

######  Lemma 6.

The distance quotient matrices $T^{D}_{1}$ and $T^{D}_{2}$ of
$\mathcal{P}(\mathrm{El}(p^{n})\times\mathrm{El}(q^{m}))$ corresponding to the
equitable partition given in Lemma 1 and Lemma 2 are

$\left[\begin{array}[]{cccc}0&p^{n}-1&(p^{n}-1)(q^{m}-1)&q^{m}-1\\\
1&2p^{n}-p-2&(2p^{n}-p-1)(q^{m}-1)&2(q^{m}-1)\\\
1&2p^{n}-p-1&2(p^{n}-1)(q^{m}-1)-(p-1)(q-1)-1&2q^{m}-q-1\\\
1&2(p^{n}-1)&(p^{n}-1)(2q^{m}-q-1)&2q^{m}-q-2\\\ \end{array}\right]$

and

$\small\left[\begin{array}[]{cccc}0&(p-1)\mathbf{1}^{\prime}_{\alpha}&(p-1)(q-1)\mathbf{1}^{\prime}_{\gamma}&(q-1)\mathbf{1}^{\prime}_{\beta}\\\
\mathbf{1}_{\alpha}&p(2J_{\alpha}-I_{\alpha})-2J_{\alpha}&(p-1)(q-1)(2J_{\alpha}-I_{\alpha})\otimes\mathbf{1}^{\prime}_{\beta}&2(q-1)J_{\alpha,\beta}\\\
\mathbf{1}_{\gamma}&(p-1)(2J_{\alpha}-I_{\alpha})\otimes\mathbf{1}_{\beta}&(p-1)(q-1)(2J_{\gamma}-I_{\gamma})-I_{\gamma}&(q-1)\mathbf{1}_{\alpha}\otimes(2J_{\beta}-I_{\beta})\\\
\mathbf{1}_{\beta}&2(p-1)J_{\beta,\alpha}&(p-1)(q-1)\mathbf{1}^{\prime}_{\alpha}\otimes(2J_{\beta}-I_{\beta})&q(2J_{\beta}-I_{\beta})-2J_{\beta}\end{array}\right],$
(12)

respectively, where $\alpha=\frac{p^{n}-1}{p-1}$, and
$\beta=\frac{q^{m}-1}{q-1}$ and $\gamma=\alpha\beta$.

Next, we compute the characteristic polynomial of $T^{D}_{2}$.

######  Lemma 7.

Suppose $T^{D}_{2}$ is the matrix given in (12), then

$\phi(T^{D}_{2},x)=\phi(T_{1}^{D},x)(x+(p-1)(q-1)+1)^{(\alpha-1)(\beta-1)}f(x)^{\alpha-1}g(x)^{\beta-1},$

where $\alpha=\frac{p^{n}-1}{p-1}$, $\beta=\frac{q^{m}-1}{q-1}$, and $f(x)$
and $g(x)$ are the characteristic polynomials of the matrices
$\left[\begin{array}[]{cc}-p&(1-p)(q^{m}-1)\\\
1-p&(1-p)(q-1)-1\end{array}\right]$ and
$\left[\begin{array}[]{cc}(p-1)(1-q)-1&1-q\\\
(p^{n}-1)(1-q)&-q\end{array}\right]$, respectively.

###### Proof.

As in Lemma 5, it is easy to see that $\phi(T^{D}_{1},x)$ divides
$\phi(T^{D}_{2},x)$. Let $S_{1}$ be the set of eigenvectors of $T^{D}_{2}$
formed from the linear independent eigenvectors of $T^{D}_{1}$. For
$i\in\\{1,\dotsc,\alpha-1\\}$ and $j\in\\{1,\dotsc,\beta-1\\}$, let
$v^{i}\in\mathbb{R}^{\alpha}$ and $w^{j}\in\mathbb{R}^{\beta}$ be the vectors
defined in (4.1). Now, for each $i\in\\{1,\dotsc,\alpha-1\\}$ and
$j\in\\{1,\dotsc,\beta-1\\}$

$\displaystyle T_{2}^{D}\left[\begin{array}[]{c}0\\\ \mathbf{0}_{\alpha}\\\
v^{i}\otimes w^{j}\\\ \mathbf{0}_{\beta}\end{array}\right]$
$\displaystyle=\left[\begin{array}[]{c}(p-1)(q-1)\mathbf{1}^{\prime}_{\gamma}(v^{i}\otimes
w^{j})\\\
((p-1)(q-1)(2J_{\alpha}-I_{\alpha})\otimes\mathbf{1}^{\prime}_{\beta})(v^{i}\otimes
w^{j})\\\ ((p-1)(q-1)(2J_{\gamma}-I_{\gamma})-I_{\gamma})(v^{i}\otimes
w^{j})\\\
((p-1)(q-1)\mathbf{1}^{\prime}_{\alpha}\otimes(2J_{\beta}-I_{\beta}))(v^{i}\otimes
w^{j})\end{array}\right]$
$\displaystyle=-((p-1)(q-1)+1)\left[\begin{array}[]{c}0\\\
\mathbf{0}_{\alpha}\\\ v^{i}\otimes w^{j}\\\
\mathbf{0}_{\beta}\end{array}\right].$

Since $S_{2}=\\{(0,\mathbf{0}^{\prime}_{\alpha},v^{i}\otimes
w^{j},\mathbf{0}_{\beta}^{\prime})^{\prime}:1\leq i\leq\alpha-1,1\leq
j\leq\beta-1\\}$ is a linearly independent set, $-((p-1)(q-1)+1)$ is an
eigenvalue of $T_{2}^{D}$ with multiplicity $(\alpha-1)(\beta-1)$. Clearly,
$S_{1}\cup S_{2}$ is a set of linearly independent eigenvectors of $T_{2}$.
Now consider the following matrices

$B=\left[\begin{array}[]{cc}-p&(1-p)(q^{m}-1)\\\
1-p&(1-p)(q-1)-1\end{array}\right]~{}\mbox{and}~{}C=\left[\begin{array}[]{cc}(p-1)(1-q)-1&1-q\\\
(p^{n}-1)(1-q)&-q\end{array}\right].$

Suppose $f(x)$ and $g(x)$ are the characteristic polynomials of the matrices
$B$ and $C$, respectively. As in Lemma 5, it is easy to prove that $f(x)$ and
$g(x)$ divides $\phi(T^{D}_{2},x)$ such that

$\phi(T_{2}^{D},x)=\phi(T_{1}^{D},x)(x+(p-1)(q-1)+1)^{(\alpha-1)(\beta-1)}f(x)^{\alpha-1}g(x)^{\beta-1}.$

This completes the proof. ∎

In the following theorem, we compute the characteristic polynomial of the
distance matrix of $\mathcal{P}(\mathrm{El}(p^{n})\times\mathrm{El}(q^{m}))$.

######  Theorem 8.

Suppose $G=\mathrm{El}(p^{n})\times\mathrm{El}(q^{m})$. If $D$ is the distance
matrix of $\mathcal{P}(G)$, then

$\phi_{D}(\mathcal{P}(G),x)=\phi(T_{1}^{D},x)(x+1)^{p^{n}q^{m}-(\alpha+1)(\beta+1)}(x+(p-1)(q-1)+1)^{(\alpha-1)(\beta-1)}f(x)^{\alpha-1}g(x)^{\beta-1},$

where $\alpha=\frac{p^{n}-1}{p-1}$, $\beta=\frac{q^{m}-1}{q-1}$, and $f(x)$
and $g(x)$ are the characteristic polynomials of the matrices
$\left[\begin{array}[]{cc}-p&(1-p)(q^{m}-1)\\\
1-p&(1-p)(q-1)-1\end{array}\right]$ and
$\left[\begin{array}[]{cc}(p-1)(1-q)-1&1-q\\\
(p^{n}-1)(1-q)&-q\end{array}\right]$, respectively.

###### Proof.

Since the adjacency and distance matrix of a complete graph is the same, using
Theorem 2 along with Theorem 7, we get the desired result. ∎

We now discuss the adjacency and distance spectrum of the enhanced power graph
of $\mathrm{El}(p^{n})\times\mathrm{El}(q^{m})$. We first need the following
lemmas.

######  Lemma 8.

The partition given in Lemma 1 is an equitable partition for the enhanced
power graph of $\mathrm{El}(p^{n})\times\mathrm{El}(q^{m})$.

###### Proof.

We recall that $u$ is adjacent to $v$ in the enhanced power graph of a group
$G$ (see definition 2) if $u,v\in\langle w\rangle$ for some $w\in G$. Let
$G=\mathrm{El}(p^{n})\times\mathrm{El}(q^{m})$. Observe that $(a_{1},b_{1})$
and $(a_{2},b_{2})$ are adjacent in $\mathcal{G}_{E}(G)$, if and only if
$\langle a_{1},a_{2}\rangle$ and $\langle b_{1},b_{2}\rangle$ are cyclic in
$\mathrm{El}(p^{n})$ and $\mathrm{El}(q^{m})$, respectively. Thus,

$|N(u)\cap V_{4}|=q^{m}-1~{}\mbox{for}~{}u\in V_{2}$

and

$|N(u)\cap V_{2}|=p^{n}-1~{}\mbox{for}~{}u\in V_{4}.$

Moreover, except the edges between the elements of $V_{2}$ and $V_{4}$, there
is no edge (other than the edges of $\mathcal{P}(G)$) in $\mathcal{G}_{E}(G)$.
Now, by using Lemma 1, the proof is complete. ∎

######  Lemma 9.

The partition given in Lemma 2 is an equitable partition for the enhanced
power graph of $\mathrm{El}(p^{n})\times\mathrm{El}(q^{m})$.

###### Proof.

The proof follows from Lemma 2 and Lemma 8. ∎

The next result follows directly from Lemma 9.

######  Lemma 10.

Let $G=\mathrm{El}(p^{n})\times\mathrm{El}(q^{m})$. Suppose
$\alpha=\frac{p^{n}-1}{p-1}$ and $\beta=\frac{q^{m}-1}{q-1}$. Then,

$\mathcal{G}_{E}(G)\cong(K_{1}+\Gamma^{\prime})[K_{1},\underbrace{K_{(p-1)},\dots,K_{(p-1)}}_{\alpha},\underbrace{K_{(p-1)(q-1)},\dots,K_{(p-1)(q-1)}}_{\alpha\beta},\underbrace{K_{(q-1)},\dots,K_{(q-1)}}_{\beta}],$

where $\alpha=\frac{p^{n}-1}{p-1}$, $\beta=\frac{q^{m}-1}{q-1}$ and
$\Gamma^{\prime}$ is the graph with vertex set $V(\Gamma)$ and edge set
$E(\Gamma)\cup\\{\\{i,\alpha+\alpha\beta+j\\}:1\leq i\leq\alpha,1\leq
j\leq\beta\\}$, where $\Gamma$ is given in Figure 1.

###  4.3 Adjacency spectrum of
$\mathcal{G}_{E}(\mathrm{El}(p^{n})\times\mathrm{El}(q^{m}))$

From Lemma 8, and Lemma 9, the following result is immediate.

######  Lemma 11.

The quotient matrices $T_{1}$ and $T_{2}$ of
$\mathcal{G}_{E}(\mathrm{El}(p^{n})\times\mathrm{El}(q^{m}))$ corresponding to
the equitable partitions given in Lemma 1 and Lemma 2, respectively are

$T_{1}=\left[\begin{array}[]{cccc}0&p^{n}-1&(p^{n}-1)(q^{m}-1)&q^{m}-1\\\
1&p-2&(p-1)(q^{m}-1)&q^{m}-1\\\ 1&p-1&(p-1)(q-1)-1&q-1\\\
1&p^{n}-1&(p^{n}-1)(q-1)&q-2\\\ \end{array}\right]$

and

$T_{2}=\left[\begin{array}[]{cccc}0&(p-1)\mathbf{1}^{\prime}_{\alpha}&(p-1)(q-1)\mathbf{1}^{\prime}_{\alpha\beta}&(q-1)\mathbf{1}^{\prime}_{\beta}\\\
\mathbf{1}_{\alpha}&(p-2)I_{\alpha}&(p-1)(q-1)I_{\alpha}\otimes\mathbf{1}^{\prime}_{\beta}&(q-1)\mathbf{1}_{\alpha}\mathbf{1}^{\prime}_{\beta}\\\
\mathbf{1}_{\alpha\beta}&(p-1)I_{\alpha}\otimes\mathbf{1}_{\beta}&((p-1)(q-1)-1)I_{\alpha\beta}&(q-1)\mathbf{1}_{\alpha}\otimes
I_{\beta}\\\
\mathbf{1}_{\beta}&(p-1)\mathbf{1}_{\beta}\mathbf{1}^{\prime}_{\alpha}&(p-1)(q-1)\mathbf{1}^{\prime}_{\alpha}\otimes
I_{\beta}&(q-2)I_{\beta}\end{array}\right],$ (13)

where $\alpha=\frac{p^{n}-1}{p-1}$, and $\beta=\frac{q^{m}-1}{q-1}$.

######  Lemma 12.

Suppose $T_{2}$ is the matrix given in (13). Then

$\phi(T_{2},x)=\phi(T_{1},x)(x-pq+p+q)^{(\alpha-1)(\beta-1)}f(x)^{\alpha-1}g(x)^{\beta-1},$

where $\alpha=\frac{p^{n}-1}{p-1}$, $\beta=\frac{q^{m}-1}{q-1}$, and $f(x)$
and $g(x)$ are the characteristic polynomials of the matrices
$\left[\begin{array}[]{cc}p-2&(p-1)(q^{m}-1)\\\
p-1&(p-1)(q-1)-1\end{array}\right]$ and
$\left[\begin{array}[]{cc}(p-1)(q-1)-1&q-1\\\
(p^{n}-1)(q-1)&q-2\end{array}\right]$, respectively.

###### Proof.

At this point, it is evidently clear that $\phi(T_{1},x)$ divides
$\phi(T_{2},x)$. Let $S_{1}$ be the set of eigenvectors of $T_{2}$ formed from
the linear independent eigenvectors of $T_{1}$. For
$i\in\\{1,\dotsc,\alpha-1\\}$ and $j\in\\{1,\dotsc,\beta-1\\}$, let
$v^{i}\in\mathbb{R}^{\alpha}$ and $w^{j}\in\mathbb{R}^{\beta}$ be the vectors
defined in (4.1). From Lemma 5, it is obvious that

$\displaystyle T_{2}\left[\begin{array}[]{c}0\\\ \mathbf{0}_{\alpha}\\\
v^{i}\otimes w^{j}\\\ \mathbf{0}_{\beta}\end{array}\right]=(pq-
p-q)\left[\begin{array}[]{c}0\\\ \mathbf{0}_{\alpha}\\\ v^{i}\otimes w^{j}\\\
\mathbf{0}_{\beta}\end{array}\right],$

for each $i\in\\{1,\dotsc,\alpha-1\\}$ and $j\in\\{1,\dotsc,\beta-1\\}$ and
hence $pq-p-q$ is an eigenvalue of $T_{2}$ with multiplicity
$(\alpha-1)(\beta-1)$. Also, these $(\alpha-1)(\beta-1)$ eigenvectors and the
vectors in $S_{1}$ are linearly independent. Suppose $f(x)$ and $g(x)$ are the
characteristic polynomials of the matrices

$B=\left[\begin{array}[]{cc}p-2&(p-1)(q^{m}-1)\\\
p-1&(p-1)(q-1)-1\end{array}\right]~{}\mbox{and}~{}C=\left[\begin{array}[]{cc}(p-1)(q-1)-1&q-1\\\
(p^{n}-1)(q-1)&q-2\end{array}\right],$

respectively. Similar to the proof of Lemma 5, it is easy to see that $f(x)$
and $g(x)$ divides $\phi(T_{2},x)$ such that

$\phi(T_{2},x)=\phi(T_{1},x)(x-pq+p+q)^{(\alpha-1)(\beta-1)}f(x)^{\alpha-1}g(x)^{\beta-1}.$

The proof is complete. ∎

Now, we are ready to state the result analogous to Theorem 7 for the
characteristic polynomial of the adjacency matrix of the
$\mathcal{G}_{E}(\mathrm{El}(p^{n})\times\mathrm{El}(q^{m}))$. The proof is
omitted since it is similar to the proof of Theorem 7.

######  Theorem 9.

Let $G=\mathrm{El}(p^{n})\times\mathrm{El}(q^{m})$. If $A$ is the adjacency
matrix of $\mathcal{G}_{E}(G)$, then

$\phi_{A}(\mathcal{G}_{E}(G),x)=\phi(T_{1},x)(x+1)^{(p^{n}q^{m}-(\alpha+1)(\beta+1))}(x-pq+p+q)^{(\alpha-1)(\beta-1)}f(x)^{\alpha-1}g(x)^{\beta-1},$

where $\alpha=\frac{p^{n}-1}{p-1}$, $\beta=\frac{q^{m}-1}{q-1}$, and $f(x)$
and $g(x)$ are the characteristic polynomials of the matrices
$\left[\begin{array}[]{cc}p-2&(p-1)(q^{m}-1)\\\
p-1&(p-1)(q-1)-1\end{array}\right]$ and
$\left[\begin{array}[]{cc}(p-1)(q-1)-1&q-1\\\
(p^{n}-1)(q-1)&q-2\end{array}\right]$, respectively.

###  4.4 Distance spectrum of
$\mathcal{G}_{E}(\mathrm{El}(p^{n})\times\mathrm{El}(q^{m}))$

Using (4.2) and Lemma 11, we can compute the distance quotient matrices of
$\mathcal{G}_{E}(\mathrm{El}(p^{n})\times\mathrm{El}(q^{m}))$ corresponding to
the equitable partition given in Lemma 1 and Lemma 2. These matrices are
described in the following lemma.

######  Lemma 13.

The distance quotient matrices $T^{D}_{1}$ and $T^{D}_{2}$ of
$\mathcal{G}_{E}(\mathrm{El}(p^{n})\times\mathrm{El}(q^{m}))$ corresponding to
the equitable partition given in Lemma 1 and Lemma 2 are

$\left[\begin{array}[]{cccc}0&p^{n}-1&(p^{n}-1)(q^{m}-1)&q^{m}-1\\\
1&2p^{n}-p-2&(2p^{n}-p-1)(q^{m}-1)&q^{m}-1\\\
1&2p^{n}-p-1&(p-1)(q-1)(2\gamma-1)-1&2q^{m}-q-1\\\
1&p^{n}-1&(p^{n}-1)(2q^{m}-q-1)&2q^{m}-q-2\\\ \end{array}\right]$

and

$\small\left[\begin{array}[]{cccc}0&(p-1)\mathbf{1}^{\prime}_{\alpha}&(p-1)(q-1)\mathbf{1}^{\prime}_{\gamma}&(q-1)\mathbf{1}^{\prime}_{\beta}\\\
\mathbf{1}_{\alpha}&p(2J_{\alpha}-I_{\alpha})-2J_{\alpha}&(p-1)(q-1)(2J_{\alpha}-I_{\alpha})\otimes\mathbf{1}^{\prime}_{\beta}&(q-1)J_{\alpha,\beta}\\\
\mathbf{1}_{\gamma}&(p-1)(2J_{\alpha}-I_{\alpha})\otimes\mathbf{1}_{\beta}&(p-1)(q-1)(2J_{\gamma}-I_{\gamma})-I_{\gamma}&(q-1)\mathbf{1}_{\alpha}\otimes(2J_{\beta}-I_{\beta})\\\
\mathbf{1}_{\beta}&(p-1)J_{\beta,\alpha}&(p-1)(q-1)\mathbf{1}^{\prime}_{\alpha}\otimes(2J_{\beta}-I_{\beta})&q(2J_{\beta}-I_{\beta})-2J_{\beta}\end{array}\right],$
(14)

respectively, where $\alpha=\frac{p^{n}-1}{p-1}$, and
$\beta=\frac{q^{m}-1}{q-1}$ and $\gamma=\alpha\beta$.

In the following lemma, we compute the characteristic polynomial of
$T_{2}^{D}$.

######  Lemma 14.

Suppose $T^{D}_{2}$ is the matrix given in (14), then

$\phi(T^{D}_{2},x)=\phi(T_{1}^{D},x)(x+(p-1)(q-1)+1)^{(\alpha-1)(\beta-1)}f(x)^{\alpha-1}g(x)^{\beta-1},$

where $\alpha=\frac{p^{n}-1}{p-1}$, $\beta=\frac{q^{m}-1}{q-1}$, and $f(x)$
and $g(x)$ are the characteristic polynomials of the matrices
$\left[\begin{array}[]{cc}-p&(1-p)(q^{m}-1)\\\
1-p&(1-p)(q-1)-1\end{array}\right]$ and
$\left[\begin{array}[]{cc}(p-1)(1-q)-1&1-q\\\
(p^{n}-1)(1-q)&-q\end{array}\right]$, respectively.

###### Proof.

It is easy to see that $\phi(T_{1}^{D},x)$ divides $\phi(T_{2}^{D},x)$. Also,
it follows directly from Lemma 7, that for each $i\in\\{1,\dotsc,\alpha-1\\}$
and $j\in\\{1,\dotsc,\beta-1\\}$

$\displaystyle T_{2}^{D}\left[\begin{array}[]{c}0\\\ \mathbf{0}_{\alpha}\\\
v^{i}\otimes w^{j}\\\
\mathbf{0}_{\beta}\end{array}\right]=-((p-1)(q-1)+1)\left[\begin{array}[]{c}0\\\
\mathbf{0}_{\alpha}\\\ v^{i}\otimes w^{j}\\\
\mathbf{0}_{\beta}\end{array}\right],$

where $v^{i}\in\mathbb{R}^{\alpha}$ and $w^{j}\in\mathbb{R}^{\beta}$ are
defined in (4.1). Thus, we can conclude that $-(1+(p-1)(q-1))$ is an
eigenvalue of $T_{2}^{D}$ with multiplicity $(\alpha-1)(\beta-1)$. Now,
consider the following matrices

$B=\left[\begin{array}[]{cc}-p&(1-p)(q^{m}-1)\\\
1-p&(1-p)(q-1)-1\end{array}\right]~{}\mbox{and}~{}C=\left[\begin{array}[]{cc}(p-1)(1-q)-1&1-q\\\
(p^{n}-1)(1-q)&-q\end{array}\right].$

Similar to the proof of Lemma 7, we can conclude that every eigenvalue of $B$
is an eigenvalue of $T_{2}^{D}$ with multiplicity $\alpha-1$, and every
eigenvalue of $C$ is an eigenvalue of $T_{2}^{D}$ with multiplicity $\beta-1$.
Also, if $f(x)$ and $g(x)$ are the characteristic polynomials of the matrices
$B$ and $C$, respectively, then

$\phi(T_{2}^{D},x)=\phi(T_{1}^{D},x)(x+(p-1)(q-1)+1)^{(\alpha-1)(\beta-1)}f(x)^{\alpha-1}g(x)^{\beta-1}.$

This completes the proof. ∎

The next theorem is analogous to Theorem 8 and computes the characteristic
polynomial of the distance matrix of
$\mathcal{G}_{E}(\mathrm{El}(p^{n})\times\mathrm{El}(q^{m}))$. The proof is
similar to the proof of Theorem 8.

######  Theorem 10.

Suppose $G=\mathrm{El}(p^{n})\times\mathrm{El}(q^{m})$. If $D$ is the distance
matrix of $\mathcal{G}_{E}(G)$, then the characteristic polynomial
$\phi_{D}(\mathcal{G}_{E}(G),x)$ of $D$ is

$\phi(T_{1}^{D},x)(x+1)^{(p^{n}q^{m}-(\alpha+1)(\beta+1))}(x+(p-1)(q-1)+1)^{(\alpha-1)(\beta-1)}f(x)^{\alpha-1}g(x)^{\beta-1},$

where $\alpha=\frac{p^{n}-1}{p-1}$, $\beta=\frac{q^{m}-1}{q-1}$, and $f(x)$
and $g(x)$ are the characteristic polynomials of the matrices
$\left[\begin{array}[]{cc}-p&(1-p)(q^{m}-1)\\\
1-p&(1-p)(q-1)-1\end{array}\right]$ and
$\left[\begin{array}[]{cc}(p-1)(1-q)-1&1-q\\\
(p^{n}-1)(1-q)&-q\end{array}\right]$, respectively.

##  5 The finite abelian group $\mathrm{El}(p^{n})\times\mathbb{Z}_{m}$

We conclude the paper by determining the distance spectra of the finite
abelian group $\mathrm{El}(p^{n})\times\mathbb{Z}_{m}$, where $n\geq 2$ and
$\text{gcd}(p,m)=1$.

######  Theorem 11.

Let $G=\mathrm{El}(p^{n})\times\mathbb{Z}_{m}$, where $m$ is a positive
integer such that $\text{gcd}(m,p)=1$. Suppose $\alpha=\frac{p^{n}-1}{p-1}$,
then $\phi_{D}(\mathcal{G}_{E}(G),x)$ is

$(x+mp-m+1)^{\alpha-1}(x+1)^{(mp-m-1)\alpha+m-1}(x^{2}+(mp+2-2mp^{n})x+m^{2}(p^{n}-p)-2mp^{n}+mp+1).$

###### Proof.

By [19, Corollary 4.2], the dominating vertices of the graph
$\mathcal{G}_{E}(G)$ are precisely the vertices of the form $(e,a)$, where $e$
is the identity of $\mathrm{El}(p^{n})$ and $a\in\mathbb{Z}_{m}.$ From Theorem
3.2 of [20], we know that the number of connected components of
$\mathcal{G}_{E}(\mathrm{El}(p^{n})\setminus\\{e\\})$ is
$\frac{p^{n}-1}{p-1}$. Let $C_{1},\dots,C_{\alpha}$ be the components of
$\mathcal{G}_{E}(\mathrm{El}(p^{n})\setminus\\{e\\})$. Since
$\text{gcd}(p,m)=1$, the sets

$V_{1}:=\\{(e,a):a\in\mathbb{Z}_{m}\\},~{}\mbox{and}~{}V_{i}:=\\{(b,a):b\in
C_{i-1},a\in\mathbb{Z}_{m}\\},~{}\mbox{for}~{}2\leq i\leq\alpha+1,$

form an equitable partition for $\mathcal{G}_{E}(G)$. It is clear that for
each $2\leq i\leq\alpha+1$, $V_{i}$ is a complete graph on $m(p-1)$ vertices.
Thus,

$\mathcal{G}_{E}(G)\cong
K_{1,\alpha}[K_{m},\underbrace{K_{m(p-1)},\dotsc,K_{m(p-1)}}_{\alpha}].$ (15)

Suppose $T^{D}$ is the distance quotient matrix of $\mathcal{G}_{E}(G)$
corresponding to the equitable partition given in (15). Then

$T^{D}=\left[\begin{array}[]{cccc}m-1&m(p-1)\mathbf{1}_{\alpha}^{\prime}\\\
m\mathbf{1}_{\alpha}&2m(p-1)J_{\alpha}-(mp-m+1)I_{\alpha}\end{array}\right].$

By Theorem 2, we have

$\displaystyle\phi_{D}(\mathcal{G}_{E}(G),x)$
$\displaystyle=\phi(T^{D},x)(x+1)^{(mp-m-1)\alpha+(m-1)}.$

We now compute the characteristic polynomial of $T^{D}$. Since $p$ is a prime
and $n\geq 2$, $2mp^{n}-mp-m-1$ is not an eigenvalue of $T^{D}$. Thus,

$\displaystyle\phi(T^{D},x)$ $\displaystyle={\rm det}(xI_{\alpha+1}-T^{D})$
$\displaystyle=\left|\begin{array}[]{cccc}x-m+1&-(p-1)m\mathbf{1}_{\alpha}^{\prime}\\\
-m\mathbf{1}_{\alpha}&(x+mp-m+1)I_{\alpha}-2m(p-1)J_{\alpha}\end{array}\right|$
$\displaystyle=\frac{m}{x-(2mp^{n}-mp-m-1)}{\rm det}(T^{\prime}),$

where

$T^{\prime}=\left[\begin{array}[]{cccc}\displaystyle\frac{(x-m+1)(x-2mp^{n}+mp+m+1)}{m}&-m(p-1)\mathbf{1}_{\alpha}^{\prime}\\\
-(x-2mp^{n}+mp+m+1)\mathbf{1}_{\alpha}&(x+mp-m+1)I_{\alpha}-2m(p-1)J_{\alpha}\end{array}\right].$

Replacing the first column of $T^{\prime}$ by the sum of all the columns, the
determinant of $T^{\prime}$ is equal to

$\displaystyle\left|\begin{array}[]{cccc}\displaystyle\frac{(x-m+1)(x-2mp^{n}+mp+m+1)}{m}-m(p^{n}-1)&-m(p-1)\mathbf{1}_{\alpha}^{\prime}\\\
\mathbf{0}&(x+mp-m-1)I_{\alpha}-2m(p-1)J_{\alpha}\end{array}\right|.$

Hence,

$\phi(T^{D},x)=(x^{2}+(mp+2-2mp^{n})x+m^{2}(p^{n}-p)-2mp^{n}+mp+1)(x+mp-m+1)^{\alpha-1}.$

The proof is complete. ∎

Setting $m=1$ in Theorem 11, we immediately have the distance spectra of the
enhanced power graph of elementary abelian groups. This is stated below.

######  Theorem 12.

Suppose $\alpha=\frac{p^{n}-1}{p-1}$. Then

$\phi_{D}(\mathcal{P}(\mathrm{El}(p^{n})),x)=(x+p)^{\alpha-1}(x+1)^{(p-2)\alpha}(x^{2}-(2p^{n}-p-2)x-(p^{n}-1)).$

In particular, the eigenvalues of $\mathcal{P}(\mathrm{El}(p^{n}))$ are $-1$
with multiplicity $(n-2)\alpha$, $-p$ with multiplicity $\alpha-1$ and two
simple eigenvalues
$\lambda_{1,2}=\frac{(2p^{n}-p-2)\pm\sqrt{(2p^{n}-p-2)^{2}+4(p^{n}-1)}}{2}.$

### Acknowledgement

The first author thanks the Prime Minister’s Research Fund for the funding.
The second and third authors acknowledge SERB-National Post Doctoral
Fellowship (File No. PDF/2021/001899) and (File No. PDF/2022/000269),
respectively for the preparation of this work and profusely thank the Science
and Engineering Research Board, Govt. of India. The authors also acknowledge
ideal working conditions in the Department of Mathematics, Indian Institute of
Science, Bangalore.

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Authors’ address:
Anita Arora, Department of Mathematics, IISc Bangalore, Karnataka, India.
E-mail<EMAIL_ADDRESS>
Hiranya Kishore Dey, Department of Mathematics, IISc Bangalore, Karnataka,
India. E-mail<EMAIL_ADDRESS>
Shivani Goel, Department of Mathematics, IISc Bangalore, Karnataka, India.
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