# A study on $A_{\alpha}$-spectrum and $A_{\alpha}$-energy of unitary addition
Cayley graphs

Najiya V K, Chithra A V
Department of Mathematics
National Institute of Technology, Calicut
Kerala, India-673601
<EMAIL_ADDRESS><EMAIL_ADDRESS>
&Naveen Palanivel
Department of Mathematics,
SASTRA Deemed to be University
Thanjavur, India-613401
<EMAIL_ADDRESS>

###### Abstract

The unitary addition Cayley graph $G_{n}$, $n\in Z^{+}$ is the graph whose
vertex set is $Z_{n}$, the ring of integers modulo $n$ and two vertices $u$
and $v$ are adjacent if and only if $u+v\in\cup_{n}$ where $\cup_{n}$ is the
set of all units of the ring. The $A_{\alpha}$-matrix of a graph $G$ is
defined as $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$, $\alpha\in[0,1]$, where
$D(G)$ is the diagonal matrix of vertex degrees and $A(G)$ is the adjacency
matrix of $G$. In this paper, we investigate the $A_{\alpha}$-eigenvalues for
unitary addition Cayley graph and its complement. We determine bounds for
$A_{\alpha}$-eigenvalues of unitary addition Cayley graph when its order is
odd. Consequently, we compute the $A_{\alpha}$-energy of both $G_{n}$ and its
complement, $\overline{G_{n}}$, for $n={p}^{m}$ where ${p}$ is a prime number
and $n$ even. Moreover, we obtain some bounds for energies of $G_{n}$ and
$\overline{G}_{n}$ when $n$ is odd. We also define
$A_{\alpha}$-borderenergetic and $A_{\alpha}$-hyperenergetic graphs and
observe some classes for each.

_K_ eywords Spectrum of a graph, Energy of a graph, Unitary Addition Cayley
Graph, $A_{\alpha}$-matrix

##  1 Introduction

A graph $G=\left(V,E\right)$ consists of finite nonempty set $V$ of $n$
vertices together with a set $E$ of $m$ edges. The degree of a vertex $v$ is
the number of edges incident with it and is denoted as $deg\left(v\right)$.
The complement $\overline{G}$ of a graph $G$ has $V\left(G\right)$ as its
vertex set, but two vertices are adjacent in $\overline{G}$ if and only if
they are not adjacent in $G$. Throughout the paper we consider only simple,
undirected graphs.

Let $\mathcal{G}$ be a group and let $S$ be a subset of $\mathcal{G}$ that is
closed under taking inverses and does not contain the identity. Then the
Cayley graph $X\left(\mathcal{G},S\right)$ is the graph with vertex set
$\mathcal{G}$ and edge set $E=\\{gh:hg^{-1}\in S\\}$.

In [2] Dejter et al. introduced the concept of unitary Cayley graph. Later in
[14] Deepa et al. introduced the unitary addition Cayley graphs(UACG). Naveen
et al. in [10, 8] studied the structural properties, adjacency spectrum and
energy of UACG.

For a simple graph $G$ of order $n$, the adjacency matrix $A\left(G\right)$ of
order $n$ is a symmetric matrix with

$a_{ij}=\begin{cases}1&\text{if}\quad v_{i}\sim v_{j}\\\
0&\text{otherwise.}\end{cases}$

In [17] Nikiforov introduced a convex linear combination of the adjacency
matrix of $G$ and degree matrix of $G$ called the $A_{\alpha}$-matrix. For
$\alpha\in[0,1]$, $A_{\alpha}\left(G\right)=\alpha
D\left(G\right)+\left(1-\alpha\right)A(G),$ where $D\left(G\right)$ is the
diagonal matrix with entries as corresponding vertex degrees. Clearly,
$A_{0}(G)=A(G)$, $A_{1}(G)=D(G)$ and $Q(G)=2A_{\frac{1}{2}}(G)=A(G)+D(G)$,
where $Q(G)$ is the signless Laplacian matrix. Also
$A_{\alpha}(G)-A_{\beta}(G)=(\alpha-\beta)L(G)=(\alpha-\beta)(D(G)-A(G))$,
where $L(G)$ is the Laplacian matrix of $G$ and $\beta\in[0,1]$.
$A_{\alpha}$-matrix is a generalization of other matrices like
$A(G),Q(G),L(G)$ etc. Thus, it helps in studying spectral properties of those
matrices as well as uncountable many convex combinations of $D(G)$ and $A(G)$.

The characteristic polynomial of a real symmetric matrix $M$ of order $n$ is
defined as $|\lambda I-M|$ where $I$ is the $n$ order identity matrix. The
eigenvalues of $M$, $\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{n}$,
together with their multiplicities gives the spectrum of $M$. The spectrum of
a graph is the spectrum of the matrix considered. We represent the
$M$-spectrum of the graph $G$ as
$\operatorname{spec}_{M}\left(G\right)=\begin{pmatrix}\lambda_{1}&\lambda_{2}&\cdots&\lambda_{t}\\\
m_{1}&m_{2}&\cdots&m_{t}\end{pmatrix}$ where $\lambda_{i}$’s are the $t$
distinct eigenvalues of the matrix $M$ considered with multiplicity $m_{i}$.
The roots of the characteristic polynomial of the adjacency matrix of a graph
$G$ are called the adjacency eigenvalues of $G$. Similarly, the roots of
$A_{\alpha}$-matrix gives the $A_{\alpha}$-spectrum of $G$. If $G$ is an
r-regular graph, then its $A_{\alpha}$-eigenvalues are $\alpha
r+\left(1-\alpha\right)\lambda_{i}\left(A\left(G\right)\right),i=1,2,\cdots,n$.
The sum of absolute values of the adjacency eigenvalues of a graph gives the
adjacency energy, $\varepsilon\left(G\right)$, of the graph. The
$A_{\alpha}$-energy of a graph $G$, for $\alpha\in\left[0,1\right)$, is
defined as
$\varepsilon_{\alpha}\left(G\right)=\displaystyle\sum_{i=1}^{n}\left|\lambda_{i}\left(A_{\alpha}\left(G\right)\right)-\frac{2\alpha
m}{n}\right|,1\leq i\leq n.$ If $G$ is regular, then
$\varepsilon_{\alpha}\left(G\right)=\left(1-\alpha\right)\varepsilon\left(G\right)$.

If every vertex of a graph is connected to every other vertices, then it is
called a complete graph. Initially researchers believed that complete graphs
have the maximum energy among all the graphs on $n$ vertices. That is, if
$G\neq K_{n}$ is a graph on $n$ vertices then $\varepsilon(G)<2n-2$. But this
was disproved by Gutman and gave rise to the question whether there exist
graphs whose energy equals or exceeds the energy of complete graph. In [5],
Gutman introduced about graphs whose energy exceeds the energy of complete
graphs and called it hyperenergetic graphs. Later in 2015, Gong et al. in [4]
introduced the notion of borderenergetic graphs, that is the graphs on $n$
vertices having energy same as that of $K_{n}$. In 2017, Tura in [15] gave the
Laplacian analogous of borderenergetic graphs. Several families of
borderenergetic graphs can be found in [4, 7, 3, 16].

The tensor product $A\otimes B$ of two matrices $A$ and $B$ of order $m\times
n$ and $p\times q$, respectively, is the $mp\times nq$ block matrix
$[a_{ij}B].$ All one matrix, identity matrix and anti diagonal matrix, with
non zero entries one, of order $n$ are represented by $J_{n},I_{n}$ and
$\widetilde{D}_{n}$, respectively.

Consider a vector
${s}=\left(s_{0},s_{1},\cdots,s_{n-1}\right)\in\mathbb{R}^{n}$. The left
circulant matrix $C_{L}({s})$ associated with ${s}$ is

$C_{L}({s})=\left[\begin{array}[]{cccc}s_{0}&s_{1}&\cdots&s_{n-1}\\\
s_{1}&s_{2}&\cdots&s_{0}\\\ \vdots&\vdots&\ddots&\vdots\\\
s_{n-1}&s_{0}&\cdots&s_{n-2}\end{array}\right]$

and the right circulant matrix $C_{R}({s})$ associated with ${s}$ is

$C_{R}({s})=\left[\begin{array}[]{cccc}s_{0}&s_{1}&\cdots&s_{n-1}\\\
s_{n-1}&s_{0}&\cdots&s_{n-2}\\\ \vdots&\vdots&\ddots&\vdots\\\
s_{1}&s_{2}&\cdots&s_{0}\end{array}\right].$

From [8], the eigenvalues of right circulant matrix are
$\lambda_{j}=s_{0}+s_{1}\omega^{j}+s_{2}\omega^{2j}+\cdots+s_{n-1}\omega^{(n-1)j}$,
$j=0,1,\ldots,n-1$, where $\omega$ is a complex primitive $n$-th root of
unity.

The eigenvalues of left circulant matrix are
$\lambda_{0},\pm\left|\lambda_{1}\right|,\cdots,\pm\left|\lambda_{(n-1)/2}\right|$
if $n$ is odd and
$\lambda_{0},\lambda_{n/2},\pm\left|\lambda_{1}\right|,\cdots,$
$\pm\left|\lambda_{(n-2)/2}\right|$ if $n$ is even where $\lambda_{k}$ are the
eigenvalues of right circulant matrix. For a positive integer $n$ the Möbius
function $\mu(n)$ is defined as

$\mu(n)=\begin{cases}0,&\text{ if }n\text{ has a squared prime factor }\\\
(-1)^{k},&\text{ if }n\text{ is a product of }k\geq 0\text{ distinct primes.
}\end{cases}$

For $0\leq k\leq n-1$, the Ramanujan sum is
$c(k,n)=\mu\left(t_{k}\right)\frac{\phi(n)}{\phi\left(t_{k}\right)}$, where
$t_{k}=\frac{n}{\gcd(k,n)}$ and $\phi(n)$ is the Euler’s phi function which is
the number of positive integers $\leq n$ which are relatively prime to $n$.
Also, the eigenvalues of unitary Cayley graph $X_{n}$ are
$\lambda_{i}(X_{n})=\displaystyle\sum_{\begin{subarray}{c}1\leq j<n\\\
\gcd(j,n)=1\end{subarray}}\omega^{kj}=c(k,n)$, where $\omega$ is a complex
primitive $n$-th root of unity.

Let $\mathbb{M}_{m\times n}(\mathbb{R})$ be the set of all $m\times n$
matrices with real entries. The Frobenius norm for $M\in\mathbb{M}_{m\times
n}(\mathbb{R})$ is defined as

$\|M\|_{F}=\sqrt{\sum_{i=1}^{m}\sum_{j=1}^{n}\left|m_{ij}\right|^{2}}=\sqrt{\operatorname{trace}\left(M^{T}M\right)},$

where trace of a square matrix is defined as sum of the elements on the main
diagonal. Let $\lambda_{i}$’s be the eigenvalues of the matrix $M$. If
$MM^{T}=$ $M^{T}M$, then
$\|M\|_{F}^{2}=\displaystyle\sum_{i=1}^{n}\left|\lambda_{i}(M)\right|^{2}$.
The Zagreb index $\zeta(G)$ of a graph $G$ is defined as the sum of the
squares of the vertices degrees, that is, $\zeta(G)=\displaystyle\sum_{v\in
V(G)}deg(v)^{2}$.

In real world, modelling a problem using graph gives a regular graph or a non-
regular graph. In literature, there are more studies happened in regular
graphs compared to the non-regular graph. The study of non-regular graph thus
has an significant role in vast area of science and technologies. The UACG
$G_{n}$ is a non-regular graph for odd values of $n$. In this paper, the exact
$A_{\alpha}$-eigenvalues of UACG and its complement are found for the values
$n={p}^{m}$ and $n$ even. For the case when $n$ is odd, it’s difficult to find
an exact value, but we have found a bound for $A_{\alpha}$-eigenvalues of
$G_{n}$ and its complement. From the $A_{\alpha}$-eigenvalues, the
$A_{\alpha}$-energy for $G_{n}$ and $\overline{G_{n}}$ are also computed for
the values $n={p}^{m}$ and $n$ even. We also introduce the concept of
$A_{\alpha}$-borderenergetic and $A_{\alpha}$-hyperenergetic graphs and found
some graphs which are $A_{\alpha}$-borderenergetic and
$A_{\alpha}$-hyperenergetic.

##  2 Preliminaries

######  Definition 2.1.

[8] The unitary Cayley graph
$X_{n}=\operatorname{Cay}\left(Z_{n},\cup_{n}\right)$ is the graph with vertex
set $Z_{n}$, the integers modulo $n$ and $u,v\in Z_{n}$ are adjacent if and
only if $u-v\in\cup_{n}$, where $\cup_{n}$ represents set of all units of the
ring $Z_{n}$.
The graph $X_{n}$ is regular with degree
$\left|\cup_{n}\right|=\phi\left(n\right)$.

######  Definition 2.2.

[14] For a positive integer $n>1$, the unitary addition Cayley graph(UACG)
$G_{n}=Cay^{+}\left(Z_{n},\cup_{n}\right)$ is the graph whose vertex set is
$Z_{n}=\\{0,1,2,\cdots,n-1\\}$ and the edge set
$E\left(G_{n}\right)=\left\\{ab\mid a,b\in Z_{n},a+b\in\cup_{n}\right\\}$
where $\cup_{n}=\left\\{a\in Z_{n}\mid\right.$
$\operatorname{gcd}\left(a,n\right)=1\\}$.
If $n$ is even, then $G_{n}$ is regular and if $n$ is odd, then $G_{n}$ is
semi regular .

012345678 Figure 1: Unitary addition Cayley graph $G_{9}$

######  Theorem 2.1.

[14] The UACG $G_{n}$ is isomorphic to $X_{n}$, the unitary Cayley graph, if
and only if $n$ is even.

######  Theorem 2.2.

[14] Let $v_{i}$, $i=0,1,2,\cdots,n-1$, be any vertex of the UACG $G_{n}$.
Then

$\operatorname{deg}\left(v_{i}\right)=\begin{cases}\phi\left(n\right)&\text{
if }n\text{ is even }\\\ \phi\left(n\right)&\text{ if }n\text{ is odd and
}\operatorname{gcd}\left(i,n\right)\neq 1\\\ \phi\left(n\right)-1&\text{ if
}n\text{ is odd and }\operatorname{gcd}\left(i,n\right)=1\end{cases}$

######  Corollary 2.3.

[14] The total number of edges in the UACG $G_{n}$ is

$\left|E\left(G_{n}\right)\right|=\begin{cases}\frac{1}{2}n\phi\left(n\right)&\text{
if }n\text{ is even }\\\ \frac{1}{2}\left(n-1\right)\phi\left(n\right)&\text{
if }n\text{ is odd }\end{cases}$

######  Theorem 2.4.

[6] The energy of unitary Cayley graph $X_{n}$ equals
$2^{l}\phi\left(n\right)$, where $l$ is the total number of unique prime
factors that divides $n$.

######  Theorem 2.5.

[6] The complement of unitary Cayley graph $X_{n}$ has energy

$\varepsilon\left(\overline{X_{n}}\right)=2(n-1)+\left(2^{k}-2\right)\phi(n)-\prod_{i=1}^{k}{p}_{i}+\prod_{i=1}^{k}\left(2-{p}_{i}\right),$

where $\displaystyle\prod_{i=1}^{k}{p}_{i}$ is the largest divisor of $n$ that
is square-free.

######  Theorem 2.6.

[9] Let $M,M_{1},M_{2}$ be three real symmetric matrices of order $n$ such
that $M=M_{1}+M_{2}$. Then the eigenvalues of these matrices satisfy the
following inequalities: for $1\leq i\leq n$ and $0\leq
j\leq\min\\{i-1,n-i\\}$,

$\lambda_{i-j}\left(M_{1}\right)+\lambda_{1+j}\left(M_{2}\right)\geq\lambda_{i}(M)\geq\lambda_{i+j}\left(M_{1}\right)+\lambda_{n-j}\left(M_{2}\right)\text{.
}$

######  Theorem 2.7.

[11] Let $G$ be a connected graph with $n\geq 3$ vertices, $m$ edges, maximum
degree $\Delta$ and Zagreb index $\zeta(G)$. Let $\alpha\in[0,1)$. Then

  1. 1.

$\displaystyle\varepsilon_{\alpha}(G)\geq\sqrt{2\left(\alpha^{2}\zeta(G)+(1-\alpha)^{2}\|A(G)\|_{F}^{2}-\frac{2(\alpha
m)^{2}}{n}\right)}.$

  2. 2.

$\displaystyle\varepsilon_{\alpha}(G)\geq 4(1-\alpha)\frac{m}{n}.$
Equality occurs if and only if $G$ is a regular graph with one positive and
$n-1$ negative adjacency eigenvalues.

  3. 3.

$\displaystyle\varepsilon_{\alpha}(G)\geq
2\sqrt{\frac{\zeta(G)}{n}}-\frac{4\alpha m}{n}.$
Equality occurs if and only if $G$ is a regular graph with one positive and
$n-1$ negative adjacency eigenvalues.

  4. 4.

$\displaystyle\varepsilon_{\alpha}(G)\geq\left(\alpha(\triangle+1)+\sqrt{\alpha^{2}(\triangle+1)^{2}+4\Delta(1-2\alpha)}\right)-\frac{4\alpha
m}{n}$
with equality if and only if $G\cong K_{1,\Delta}$.

######  Theorem 2.8.

[13] Let $G$ be a graph on $n$ vertices, $m$ edges and Zagreb index
$\zeta(G)$. For $\alpha\in[0,1)$, we have

$\varepsilon_{\alpha}(G)\leq\sqrt{n(\alpha^{2}\zeta(G)+(1-\alpha)^{2}||A(G)||^{2}_{F})-4\alpha^{2}m^{2}}$

######  Lemma 2.9.

[19] Let $M_{1},M_{2},M_{3}$ and $M_{4}$ be matrices with $M_{1}$ invertible.
Let

$\widetilde{M}=\begin{pmatrix}M_{1}&M_{2}\\\ M_{3}&M_{4}\end{pmatrix}.$

Then
$\det\left(\widetilde{M}\right)=\det\left(M_{1}\right)\det\left(X-M_{3}M_{1}^{-1}M_{2}\right)$.

If $M_{4}$ is invertible, then
$\det\left(\widetilde{M}\right)=\det\left(M_{4}\right)\det\left(M_{1}-M_{2}M_{4}^{-1}M_{3}\right).$

######  Definition 2.3.

[1] The $M$-coronal $\Gamma_{M}\left(x\right)$ of a matrix $M$ of order $n$ is
defined as the sum of the entries of the matrix $\left(xI_{n}-M\right)^{-1}$
(if it exists), that is,

$\Gamma_{M}\left(x\right)=J_{n\times
1}^{T}\left(xI_{n}-M\right)^{-1}J_{n\times 1}.$

If each row sum of an $n\times n$ matrix $M$ is constant, say t, then
$\Gamma_{M}\left(x\right)=\frac{n}{x-t}$.

######  Lemma 2.10.

[18] Let $A$ be a real matrix of order $n$. Then

$\det\left(yI_{n}-A-cJ_{n}\right)=\left(1-c\Gamma_{A}\left(y\right)\right)\det\left(yI_{n}-A\right).$

######  Lemma 2.11.

[12] Let $A$ and $B$ be symmetric matrices of order $n$ and $m$, respectively.
If $\mu_{1},\cdots,\mu_{n}$ and $\lambda_{1},\cdots,\lambda_{m}$are the
eigenvalues of $A$ and $B$, respectively, then the eigenvalues of $A\otimes B$
are given by $\mu_{j}\lambda_{i};j=1,\cdots,n;i=1,\cdots,m$.

##  3 $A_{\alpha}$-spectrum and energy of unitary addition Cayley graphs

This section discusses the $A_{\alpha}$-spectrum and $A_{\alpha}$-energy of
UACG for even $n$ and $n={p}^{m}$ where ${p}$ is a prime number. Also, bound
for $A_{\alpha}$-eigenvalues and $A_{\alpha}$-energy are found for odd $n$.

######  Proposition 3.1.

Let $n={p}^{m}$, ${p}\geq 3$, $m\geq 1$. Then the $A_{\alpha}$-spectrum of
UACG $G_{n}$ is

$\begin{pmatrix}{p}^{m-1}\left({p}\alpha-1\right)-1&\frac{x-y}{2}&{p}^{m-1}\left({p}-1\right)\alpha-1&\alpha\left({p}^{m}-{p}^{m-1}\right)&{p}^{m-1}\left(\left({p}-2\right)\alpha+1\right)-1&\frac{x+y}{2}\\\
\frac{{p}-3}{2}&1&\left({p}-1\right)\left({p}^{m-1}-1\right)&{p}^{m-1}-1&\frac{{p}-1}{2}&1\end{pmatrix},$

where $x=\left(1+\alpha\right){p}^{m}-2{p}^{m-1}-1$ and
$y=\sqrt{\left(1-{p}^{m}+\alpha{p}^{m}\right)^{2}+\left(1-\alpha\right)4{p}^{m-1}}$.

###### Proof.

The $A_{\alpha}$-matrix of UACG of order ${p}^{m}$ after proper labelling is
$A_{\alpha}\left(G_{{p}^{m}}\right)=J_{{p}^{m-1}}\otimes
C-I_{{p}^{m-1}}\otimes\left(C-B\right)$,
where
$B=\begin{bmatrix}\alpha\phi\left({{p}^{m}}\right)&\left(1-\alpha\right)J_{1\times{p}-1}\\\
\left(1-\alpha\right)J_{{p}-1\times
1}&\left(1-\alpha\right)(J-\widetilde{D})-\left(1-\alpha\phi\left({{p}^{m}}\right)\right)I_{{p}-1}\end{bmatrix}$
and $C=\left(1-\alpha\right)\begin{bmatrix}0&J_{1\times{p}-1}\\\
J_{{p}-1\times 1}&J_{{p}-1}-\widetilde{D}\end{bmatrix}$.

The matrix $A_{\alpha}(G_{{p}^{m}})$ is permutationally similar to

$\hat{A}_{\alpha}\left(G_{{p}^{m}}\right)=\begin{bmatrix}\alpha\phi\left({{p}^{m}}\right)I_{{p}^{m-1}}&\left(1-\alpha\right)J_{{p}^{m-1}\times\frac{{p}^{m}-{p}^{m-1}}{2}}&\left(1-\alpha\right)J_{{p}^{m-1}\times\frac{{p}^{m}-{p}^{m-1}}{2}}\\\
\left(1-\alpha\right)J_{\frac{{p}^{m}-{p}^{m-1}}{2}\times{p}^{m-1}}&\left(1-\alpha\right)J+\left(\alpha\phi\left({{p}^{m}}\right)-1\right)I&\left(1-\alpha\right)J-\left(1-\alpha\right)\widetilde{D}\otimes
J\\\
\left(1-\alpha\right)J_{\frac{{p}^{m}-{p}^{m-1}}{2}\times{p}^{m-1}}&\left(1-\alpha\right)J-\left(1-\alpha\right)\widetilde{D}\otimes
J&\left(1-\alpha\right)J+\left(\alpha\phi\left({{p}^{m}}\right)-1\right)I\end{bmatrix},$

where $\widetilde{D}$ is the anti-diagonal matrix with non-zero entries one of
order $\frac{{p}-1}{2}.$

Then
$A_{\alpha}\left({G_{{p}^{m}}}\right)=P\hat{A_{\alpha}}\left({G_{{p}^{m}}}\right)P^{-1}$,
where
$P=\left[\begin{array}[]{lllllllllllll}A_{11}^{T}&A_{12}^{T}&\cdots&A_{1{p}^{m-2}}^{T}&A_{21}^{T}&A_{22}^{T}&\cdots&A_{2{p}^{m-2}}^{T}&\cdots&A_{{p}1}^{T}&A_{{p}2}^{T}&\cdots&A_{{p}{p}^{m-2}}^{T}\end{array}\right]$
is a permutation matrix of order ${p}^{m},m\geq 2,$

$A_{ij}=\left[a_{st}\right],a_{st}=\begin{cases}1&\text{ if
}\left(s,t\right)\in H_{ij}\\\ 0&\text{ otherwise }\end{cases}$

$s=1,2,\cdots,{p}$; $t=1,2,\cdots,{p}^{m}$ and
$H_{ij}=\\{\left(1,i+\left(j-1\right){p}\right),\left(2,i+\left(j-1\right){p}+{p}^{m-1}\right),\cdots,\left({p},i+\left(j-1\right){p}+\left({p}-1\right){p}^{m-1}\right)\\},\\\
i=1,2,\cdots,{p},j=1,2,\cdots,{p}^{m-2}$.
$P$ becomes the identity matrix when $m=1$.

$|\lambda I-\hat{A}_{\alpha}\left(G_{{p}^{m}}\right)|=$

$\begin{vmatrix}\left(\lambda-\alpha\phi\left({{p}^{m}}\right)\right)I_{{p}^{m-1}}&-\left(1-\alpha\right)J_{{p}^{m-1}\times\frac{{p}^{m}-{p}^{m-1}}{2}}&-\left(1-\alpha\right)J_{{p}^{m-1}\times\frac{{p}^{m}-{p}^{m-1}}{2}}\\\
-\left(1-\alpha\right)J_{\frac{{p}^{m}-{p}^{m-1}}{2}\times{p}^{m-1}}&\left(\lambda-\alpha\phi\left({{p}^{m}}\right)+1\right)I-\left(1-\alpha\right)J&-\left(1-\alpha\right)J+\left(1-\alpha\right)D\otimes
J\\\
-\left(1-\alpha\right)J_{\frac{{p}^{m}-{p}^{m-1}}{2}\times{p}^{m-1}}&-\left(1-\alpha\right)J+\left(1-\alpha\right)D\otimes
J&\left(\lambda-\alpha\phi\left({{p}^{m}}\right)+1\right)I-\left(1-\alpha\right)J\end{vmatrix}.$

By Lemma 2.9,

$|\lambda
I-\hat{A}_{\alpha}\left(G_{{p}^{m}}\right)|=\left(\lambda-\alpha\phi\left({{p}^{m}}\right)\right)^{{p}^{m-1}}\times\det{S},$

$\text{where
}S=\begin{bmatrix}\left(\lambda-\alpha\phi\left({{p}^{m}}\right)+1\right)I-\left(1-\alpha\right)J&-\left(1-\alpha\right)J+\left(1-\alpha\right)\widetilde{D}\otimes
J\\\ -\left(1-\alpha\right)J+\left(1-\alpha\right)\widetilde{D}\otimes
J&\left(\lambda-\alpha\phi\left({{p}^{m}}\right)+1\right)I-\left(1-\alpha\right)J\end{bmatrix}-\frac{{p}^{m-1}\left(1-\alpha\right)^{2}}{\lambda-\alpha\phi\left({{p}^{m}}\right)}J_{{p}^{m}-{p}^{m-1}}$

$\displaystyle\det{S}$
$\displaystyle=\bigg{|}\left(\lambda-\alpha\phi\left({{p}^{m}}\right)+1\right)I-\left(1-\alpha\right)J-\left(1-\alpha\right)J+\left(1-\alpha\right)\widetilde{D}\otimes
J-\frac{2{p}^{m-1}\left(1-\alpha\right)^{2}}{\lambda-\alpha\phi\left({{p}^{m}}\right)}J\bigg{|}$
$\displaystyle\hskip
113.81102pt\times\bigg{|}\left(\lambda-\alpha\phi\left({{p}^{m}}\right)+1\right)I-\left(1-\alpha\right)J+\left(1-\alpha\right)J-\left(1-\alpha\right)\widetilde{D}\otimes
J\bigg{|}.$ Using Lemma 2.10 and Definition 2.3, we get $\displaystyle\det{S}$
$\displaystyle=\left(1-2\left(1-\alpha+\frac{{p}^{m-1}\left(1-\alpha\right)^{2}}{\lambda-\alpha\phi\left({{p}^{m}}\right)}\right)\Gamma_{-\left(1-\alpha\right)\widetilde{D}\otimes
J}\left(\lambda-\alpha\phi\left({{p}^{m}}\right)+1\right)\right)$
$\displaystyle\hskip
71.13188pt\times\bigg{|}\left(\lambda-\alpha\phi\left({{p}^{m}}\right)+1\right)I+\left(1-\alpha\right)\widetilde{D}\otimes
J\bigg{|}\bigg{|}\left(\lambda-\alpha\phi\left({{p}^{m}}\right)+1\right)I-\left(1-\alpha\right)\widetilde{D}\otimes
J\bigg{|}$
$\displaystyle=\left(1-\left(1-\alpha+\frac{{p}^{m-1}\left(1-\alpha\right)^{2}}{\lambda-\alpha\phi\left({{p}^{m}}\right)}\right)\frac{{p}^{m}-{p}^{m-1}}{\lambda-\alpha\phi\left({{p}^{m}}\right)+1+\left(1-\alpha\right){p}^{m-1}}\right)$
$\displaystyle\hskip
227.62204pt\bigg{|}\left(\lambda-\alpha\phi\left({{p}^{m}}\right)+1\right)^{2}I-\left(1-\alpha\right)^{2}\left(\widetilde{D}\otimes
J\right)^{2}\bigg{|}.$ Therefore, $\displaystyle|\lambda I-$
$\displaystyle\hat{A}_{\alpha}\left(G_{{p}^{m}}\right)|=\left(\lambda-\alpha\phi\left({{p}^{m}}\right)\right)^{{p}^{m-1}}\left(1-\left(1-\alpha+\frac{{p}^{m-1}\left(1-\alpha\right)^{2}}{\lambda-\alpha\phi\left({{p}^{m}}\right)}\right)\frac{{p}^{m}-{p}^{m-1}}{\lambda-\alpha\phi\left({{p}^{m}}\right)+1+\left(1-\alpha\right){p}^{m-1}}\right)$
$\displaystyle\hskip
170.71652pt\bigg{|}\left(\lambda-\alpha\phi\left({{p}^{m}}\right)+1\right)^{2}I-\left(1-\alpha\right)^{2}\left(\widetilde{D}\otimes
J\right)^{2}\bigg{|}$
$\displaystyle=\left(\lambda-\alpha\phi\left({{p}^{m}}\right)\right)^{{p}^{m-1}}\left(1-\frac{\left(\left(1-\alpha\right)\left(\lambda-\alpha\phi\left({{p}^{m}}\right)\right)+{p}^{m-1}\left(1-\alpha\right)^{2}\right)\left({p}^{m}-{p}^{m-1}\right)}{\left(\lambda-\alpha\phi\left({{p}^{m}}\right)\right)\left(\lambda-\alpha\phi\left({{p}^{m}}\right)+1+\left(1-\alpha\right){p}^{m-1}\right)}\right)$
$\displaystyle\hskip
56.9055pt\displaystyle\prod_{i=1}^{\frac{{p}-1}{2}}\left(\left(\lambda-\alpha\phi\left({{p}^{m}}\right)+1\right)^{2}-\left(1-\alpha\right)^{2}\left({p}^{m-1}\mu_{i}\right)^{2}\right)\left(\lambda-\alpha\phi\left({{p}^{m}}\right)+1\right)^{{p}^{m}-{p}^{m-1}-{p}+1},$
where $\mu_{i}$’s are the eigenvalues of $\widetilde{D}$ and they are $\pm 1$.
$\displaystyle|\lambda I-$
$\displaystyle\hat{A}_{\alpha}\left(G_{{p}^{m}}\right)|=\left(\lambda-\alpha\phi\left({{p}^{m}}\right)\right)^{{p}^{m-1}}\left(1-\frac{\left(\left(1-\alpha\right)\left(\lambda-\alpha\phi\left({{p}^{m}}\right)\right)+{p}^{m-1}\left(1-\alpha\right)^{2}\right)\left({p}^{m}-{p}^{m-1}\right)}{\left(\lambda-\alpha\phi\left({{p}^{m}}\right)\right)\left(\lambda-\alpha\phi\left({{p}^{m}}\right)+1+\left(1-\alpha\right){p}^{m-1}\right)}\right)$
$\displaystyle\hskip
85.35826pt\left(\left(\lambda-\alpha\phi\left({{p}^{m}}\right)+1\right)^{2}-\left(1-\alpha\right)^{2}\left({p}^{m-1}\right)^{2}\right)^{\frac{{p}-1}{2}}\left(\lambda-\alpha\phi\left({{p}^{m}}\right)+1\right)^{{p}^{m}-{p}^{m-1}-{p}+1}$
$\displaystyle=\left(\lambda-\alpha\phi\left({{p}^{m}}\right)\right)^{{{p}^{m-1}-1}}\left(\lambda-\alpha\phi\left({{p}^{m}}\right)+1+\left(1-\alpha\right){p}^{m-1}\right)^{\frac{{p}-3}{2}}\left(\lambda-\alpha\phi\left({{p}^{m}}\right)+1-\left(1-\alpha\right){p}^{m-1}\right)^{\frac{{p}-1}{2}}$
$\displaystyle\left(\left(\lambda-\alpha\phi\left({{p}^{m}}\right)\right)\left(\lambda-\alpha\phi\left({{p}^{m}}\right)+1+\left(1-\alpha\right){p}^{m-1}\right)-\left(1-\alpha\right)\left(\left(\lambda-\alpha\phi\left({{p}^{m}}\right)\right)+{p}^{m-1}\left(1-\alpha\right)\right)\left({p}^{m}-{p}^{m-1}\right)\right)$
$\displaystyle\hskip
284.52756pt\left(\lambda-\alpha\phi\left({{p}^{m}}\right)+1\right)^{{p}^{m}-{p}^{m-1}-{p}+1}.$

Thus, the $A_{\alpha}$-spectrum of $G_{{p}^{m}}$ is
$\operatorname{spec}_{A_{\alpha}}\left(G_{{p}^{m}}\right)=$

$\begin{pmatrix}{p}^{m-1}\left({p}\alpha-1\right)-1&\frac{x-y}{2}&{p}^{m-1}\left({p}-1\right)\alpha-1&\alpha\left({p}^{m}-{p}^{m-1}\right)&{p}^{m-1}\left(\left({p}-2\right)\alpha+1\right)-1&\frac{x+y}{2}\\\
\frac{{p}-3}{2}&1&\left({p}-1\right)\left({p}^{m-1}-1\right)&{p}^{m-1}-1&\frac{{p}-1}{2}&1\end{pmatrix},$

where $x=\left(1+\alpha\right){p}^{m}-2{p}^{m-1}-1$ and
$y=\sqrt{\left(1-{p}^{m}+\alpha{p}^{m}\right)^{2}+\left(1-\alpha\right)4{p}^{m-1}}$.

∎

Next we calculate the $A_{\alpha}$-energy of UACG $G_{n}$ for $n={p}^{m}$.

######  Corollary 3.1.

The $A_{\alpha}$-energy of UACG $G_{{p}^{m}}$, ${p}\geq 3$ is

$\displaystyle\varepsilon_{\alpha}\left(G_{{p}^{m}}\right)=$
$\displaystyle\frac{3{p}^{m}-5{p}^{m-1}-{p}-1}{2}+\frac{\alpha}{2p}\left(-3{p}^{m+1}+9{p}^{m}-4{p}^{m-1}+{p}^{2}-2{p}+1\right)+$
$\displaystyle\frac{{p}-1}{2{p}}\left|\left({p}^{m}-{p}\right)\left(1-\alpha\right)-\alpha\right|+\left|\frac{x-y}{2}-\frac{\alpha\left({p}^{m}-1\right)\left({p}-1\right)}{{p}}\right|+\left|\frac{x+y}{2}-\frac{\alpha\left({p}^{m}-1\right)\left({p}-1\right)}{{p}}\right|,$

where $x=\left(1+\alpha\right){p}^{m}-2{p}^{m-1}-1$ and
$y=\sqrt{\left(1-{p}^{m}+\alpha{p}^{m}\right)^{2}+\left(1-\alpha\right)4{p}^{m-1}}$.

###### Proof.

The number of vertices and edges of $G_{{p}^{m}}$ is ${p}^{m}$ and
$\frac{\left({p}^{m}-1\right)\left({p}^{m}-{p}^{m-1}\right)}{2}$ respectively.
Thus,

$\displaystyle\varepsilon_{\alpha}\left(G_{{p}^{m}}\right)=$
$\displaystyle\frac{{p}-3}{2}\left|{p}^{m-1}\left({p}\alpha-1\right)-1-\frac{\alpha\left({p}^{m}-1\right)\left({p}-1\right)}{{p}}\right|+\left|\frac{x-y}{2}-\frac{\alpha\left({p}^{m}-1\right)\left({p}-1\right)}{{p}}\right|$
$\displaystyle+\left({p}-1\right)\left({p}^{m-1}-1\right)\left|{p}^{m-1}\left({p}-1\right)\alpha-1-\frac{\alpha\left({p}^{m}-1\right)\left({p}-1\right)}{{p}}\right|$
$\displaystyle+\left({p}^{m-1}-1\right)\left|\alpha\left({p}^{m}-{p}^{m-1}\right)-\frac{\alpha\left({p}^{m}-1\right)\left({p}-1\right)}{{p}}\right|$
$\displaystyle+\frac{{p}-1}{2}\left|{p}^{m-1}\left(\left({p}-2\right)\alpha+1\right)-1-\frac{\alpha\left({p}^{m}-1\right)\left({p}-1\right)}{{p}}\right|+\left|\frac{x+y}{2}-\frac{\alpha\left({p}^{m}-1\right)\left({p}-1\right)}{{p}}\right|$
$\displaystyle=$
$\displaystyle\frac{{p}-3}{2}\left(-{p}^{m-1}\left({p}\alpha-1\right)+1+\frac{\alpha\left({p}^{m}-1\right)\left({p}-1\right)}{{p}}\right)$
$\displaystyle+\left({p}-1\right)\left({p}^{m-1}-1\right)\left(-{p}^{m-1}\left({p}-1\right)\alpha+1+\frac{\alpha\left({p}^{m}-1\right)\left({p}-1\right)}{{p}}\right)$
$\displaystyle+\left({p}^{m-1}-1\right)\left(\alpha\left({p}^{m}-{p}^{m-1}\right)-\frac{\alpha\left({p}^{m}-1\right)\left({p}-1\right)}{{p}}\right)+\left|\frac{x-y}{2}-\frac{\alpha\left({p}^{m}-1\right)\left({p}-1\right)}{{p}}\right|$
$\displaystyle+\frac{{p}-1}{2}\left|{p}^{m-1}\left(\left({p}-2\right)\alpha+1\right)-1-\frac{\alpha\left({p}^{m}-1\right)\left({p}-1\right)}{{p}}\right|+\left|\frac{x+y}{2}-\frac{\alpha\left({p}^{m}-1\right)\left({p}-1\right)}{{p}}\right|$
$\displaystyle=$
$\displaystyle\frac{3{p}^{m}-5{p}^{m-1}-{p}-1}{2}+\frac{\alpha}{2p}\left(-3{p}^{m+1}+9{p}^{m}-4{p}^{m-1}+{p}^{2}-2{p}+1\right)$
$\displaystyle\hskip
56.9055pt+\frac{{p}-1}{2p}\left|\left({p}^{m}-p\right)\left(1-\alpha\right)-\alpha\right|+\left|\frac{x-y}{2}-\frac{\alpha\left({p}^{m}-1\right)\left({p}-1\right)}{{p}}\right|$
$\displaystyle\hskip
113.81102pt+\left|\frac{x+y}{2}-\frac{\alpha\left({p}^{m}-1\right)\left({p}-1\right)}{{p}}\right|,$

where $x=\left(1+\alpha\right){p}^{m}-2{p}^{m-1}-1$ and
$y=\sqrt{\left(1-{p}^{m}+\alpha{p}^{m}\right)^{2}+\left(1-\alpha\right)4{p}^{m-1}}.$
∎

######  Remark 3.1.

If $n$ is even, then $G_{n}$ is regular and its $A_{\alpha}$-spectrum will be
$\alpha\phi\left(n\right)+\left(1-\alpha\right)\lambda_{i}\left(A\left(G_{n}\right)\right),i=1,2,\cdots,n$.
Its $A_{\alpha}$-energy will be
$\varepsilon_{\alpha}\left(G_{n}\right)=2^{k}\left(1-\alpha\right)\phi\left(n\right)$,
where $k$ is the number of distinct prime factors dividing $n$ and
$\alpha\in\left[0,1\right)$.

In general, computing exact eigenvalues of UACG for odd values of $n$ is a
tedious work. But, in the following proposition, we determine a bound for
$A_{\alpha}$-eigenvalues of UACG $G_{n}$ when $n$ is odd.

######  Proposition 3.2.

For an odd integer $n$, the $A_{\alpha}$-spectrum of the UACG $G_{n}$ satisfy
the following inequalities:

$(1-\alpha)\mu(t_{k})\frac{\phi(n)}{\phi(t_{k})}+\alpha\phi(n)-1\leq\lambda_{k}\left(A_{\alpha}\left(G_{n}\right)\right)\leq(1-\alpha)\mu(t_{k})\frac{\phi(n)}{\phi(t_{k})}+\alpha\phi(n)$

for $1\leq k\leq\frac{(n+1)}{2}$ and

$-(1-\alpha)\mu(t_{k})\frac{\phi(n)}{\phi(t_{k})}+\alpha\phi(n)-1\leq\lambda_{k}\left(A_{\alpha}\left(G_{n}\right)\right)\leq-(1-\alpha)\mu(t_{k})\frac{\phi(n)}{\phi(t_{k})}+\alpha\phi(n)$

for $\frac{(n+3)}{2}\leq k\leq n$.

###### Proof.

Let $A_{\alpha}=(a_{ij})$, $0\leq i,j\leq n-1$, be the $A_{\alpha}$ matrix of
$G_{n}$, where

$a_{ij}=\left\\{\begin{array}[]{l l l l}1-\alpha&\quad\text{if
}\gcd(n,i+j)=1\text{ and }i\neq j,\\\ \alpha\phi(n)&\quad\text{if
}\gcd(n,i+j)\neq 1\text{ and }i=j,\\\ \alpha(\phi(n)-1)&\quad\text{if
}\gcd(n,i+j)=1\text{ and }i=j,\\\ 0&\quad\text{otherwise}.\end{array}\right.$

Then $A_{\alpha}=B+C$, where $B=[b_{ij}],0\leq i,j\leq n-1,$

$b_{ij}=\left\\{\begin{array}[]{l l}1-\alpha&\quad\text{if }\gcd(n,i+j)=1,\\\
0&\quad\text{otherwise},\end{array}\right.$

and $C=[c_{ij}],0\leq i,j\leq n-1,$

$c_{ij}=\left\\{\begin{array}[]{l l l}\alpha\phi(n)&\quad\text{if
}\gcd(n,i+j)\neq 1\text{ and }i=j,\\\ \alpha\phi(n)-1&\quad\text{if
}\gcd(n,i+j)=1\text{ and }i=j,\\\ 0&\quad\text{otherwise}.\end{array}\right.$

According to the definition of $B$, it is a left circulant matrix with initial
row $(s_{0},s_{1},\cdots,s_{n-1})$ where

$s_{j}=\left\\{\begin{array}[]{l l}1-\alpha&\quad\text{if }\gcd(j,n)=1,\\\
0&\quad\text{otherwise}.\end{array}\right.$

Thus eigenvalues of $B$ are $(1-\alpha)\mu(t_{k})\frac{\phi(n)}{\phi(t_{k})}$,
for $0\leq k\leq\frac{(n-1)}{2}$ and
$-(1-\alpha)\mu(t_{k})\frac{\phi(n)}{\phi(t_{k})}$ for $\frac{(n+1)}{2}\leq
k\leq n-1$.
Eigenvalues of $C$ are $\alpha\phi(n)$ repeated ${n-\phi(n)}$ times and
$\alpha\phi(n)-1$ repeated ${\phi(n)}$ times.
The required result is therefore obtained from the eigenvalues of $B,C$ and
Theorem 2.6. ∎

In the following proposition, we estimate some bounds for $A_{\alpha}$-energy
of UACG $G_{n}$ when $n$ is odd.

######  Proposition 3.3.

For an odd integer $n\geq 3$ and $\alpha\in[0,1)$,

$\displaystyle\epsilon_{\alpha}(G_{n})\geq\sqrt{\frac{\phi(n)}{n}\left[\alpha^{2}\left(\phi(n)\left(3n^{2}-4n-1\right)+2n\right)+(1-2\alpha)(n-1)2n\phi(n)\right]}.$

$\displaystyle\epsilon_{\alpha}(G_{n})\geq\frac{2(1-\alpha)(n-1)\phi(n)}{n}.$

$\displaystyle\epsilon_{\alpha}(G_{n})\geq\sqrt{\frac{\phi(n)(\phi(n)(n-2)+1)}{n}}-\frac{4\alpha(n-1)\phi(n)}{n}$.

$\displaystyle\epsilon_{\alpha}(G_{n})\geq\alpha(\phi(n)+1)+\sqrt{\alpha^{2}(\phi(n)+1)^{2}+4\phi(n)(1-\alpha)}-\frac{2\alpha(n-1)\phi(n)}{n}.$

$\displaystyle\epsilon_{\alpha}(G_{n})\leq\sqrt{\phi(n)(n^{2}(1-\alpha)^{2}-n+2\alpha
n-\alpha^{2}\phi(n)}.$

###### Proof.

The Zagreb index of UACG is $\phi(n)^{2}(n-2)+\phi(n)$ and
$||A(G_{n})||^{2}_{F}=(n-1)\phi(n)$. Then from Theorems 2.7 and 2.8 the result
follows. ∎

##  4 $A_{\alpha}$-spectrum and energy of complement of unitary addition
Cayley graphs

In this section, we compute the $A_{\alpha}$-spectrum and $A_{\alpha}$-energy
of complement of UACG for even $n$ and $n={p}^{m}$. Also we find bounds for
$A_{\alpha}$-eigenvalues and $A_{\alpha}$-energy for odd $n$.

######  Proposition 4.1.

Let $n={p}^{m}$, ${p}\geq 3$, $m\geq 1$. Then the $A_{\alpha}$-spectrum of
complement of UACG $G_{n}$ is

$\begin{pmatrix}-{p}^{m-1}&\alpha{p}^{m-1}-1&\alpha{p}^{m-1}&{p}^{m-1}-1&{p}^{m-1}\\\
\frac{{p}-1}{2}&{p}^{m-1}-1&\left({p}-1\right)\left({p}^{m-1}-1\right)&1&\frac{{p}-1}{2}\end{pmatrix}.$

###### Proof.

The $A_{\alpha}$-matrix of complement of UACG $G_{{p}^{m}}$ after proper
labelling is
$A_{\alpha}\left(\overline{G_{{p}^{m}}}\right)=J_{{p}^{m-1}}\otimes
C-I_{{p}^{m-1}}\otimes\left(C-B\right)$,

where $B=\begin{bmatrix}\alpha\left(1-{p}^{m-1}\right)&0_{1\times{p}-1}\\\
0_{{p}-1\times
1}&\left(1-\alpha\right)\widetilde{D}_{{p}-1}-\alpha{p}^{m-1}I\end{bmatrix}$
and $C=\left(1-\alpha\right)\begin{bmatrix}1&0_{1\times{p}-1}\\\
0_{{p}-1\times 1}&\widetilde{D}_{{p}-1}\end{bmatrix}$.

The matrix $A_{\alpha}(\overline{G_{{p}^{m}}})$ is permutationally similar to

$\displaystyle\hat{A}_{\alpha}\left(\overline{G_{{p}^{m}}}\right)=$
$\displaystyle\begin{bmatrix}\left(1-\alpha\right)J_{{p}^{m-1}}-\left(1-\alpha{p}^{m-1}\right)I_{{p}^{m-1}}&0_{{p}^{m-1}\times\frac{{p}^{m}-{p}^{m-1}}{2}}&0_{{p}^{m-1}\times\frac{{p}^{m}-{p}^{m-1}}{2}}\\\
0_{\frac{{p}^{m}-{p}^{m-1}}{2}\times{p}^{m-1}}&\alpha{p}^{m-1}I&\left(1-\alpha\right)\widetilde{D}\otimes
J\\\
0_{\frac{{p}^{m}-{p}^{m-1}}{2}\times{p}^{m-1}}&\left(1-\alpha\right)\widetilde{D}\otimes
J&\alpha{p}^{m-1}I\end{bmatrix},$ where $\widetilde{D}$ is the anti-diagonal
matrix of order $\frac{{p}-1}{2}$ with all non-zero entries one. Then
$A_{\alpha}\left(\overline{G_{{p}^{m}}}\right)=P\hat{A_{\alpha}}\left(\overline{G_{{p}^{m}}}\right)P^{-1}$,
where $P$ is the same matrix as used in proof of Proposition 3.1.
$\displaystyle|\lambda
I-\hat{A}_{\alpha}\left(\overline{G_{{p}^{m}}}\right)|=$
$\displaystyle\begin{vmatrix}\left(\lambda+1-\alpha{p}^{m-1}\right)I_{{p}^{m-1}}-\left(1-\alpha\right)J_{{p}^{m-1}}&0_{{p}^{m-1}\times\frac{{p}^{m}-{p}^{m-1}}{2}}&0_{{p}^{m-1}\times\frac{{p}^{m}-{p}^{m-1}}{2}}\\\
0_{\frac{{p}^{m}-{p}^{m-1}}{2}\times{p}^{m-1}}&\left(\lambda-\alpha{p}^{m-1}\right)I&-\left(1-\alpha\right)\widetilde{D}\otimes
J\\\
0_{\frac{{p}^{m}-{p}^{m-1}}{2}\times{p}^{m-1}}&-\left(1-\alpha\right)\widetilde{D}\otimes
J&\left(\lambda-\alpha{p}^{m-1}\right)I\end{vmatrix}.$ By Lemma 2.9,
$\displaystyle|\lambda
I-\hat{A}_{\alpha}\left(\overline{G_{{p}^{m}}}\right)|=$
$\displaystyle\left|\left(\lambda+1-\alpha{p}^{m-1}\right)I-\left(1-\alpha\right)J\right|\begin{vmatrix}\left(\lambda-\alpha{p}^{m-1}\right)I&-\left(1-\alpha\right)\widetilde{D}\otimes
J\\\ -\left(1-\alpha\right)\widetilde{D}\otimes
J&\left(\lambda-\alpha{p}^{m-1}\right)I\end{vmatrix}$ $\displaystyle=$
$\displaystyle\left(\lambda+1-{p}^{m-1}\right)\left(\lambda+1-\alpha{p}^{m-1}\right)^{{p}^{m-1}-1}\left(\lambda-\alpha{p}^{m-1}\right)^{{p}^{m}-{p}^{m-1}-{p}+1}$
$\displaystyle\hskip
85.35826pt\left(\left(\lambda-\alpha{p}^{m-1}\right)^{2}-\left(1-\alpha\right)^{2}{p}^{2m-2}\right)^{\frac{{p}-1}{2}}.$

Thus, the $A_{\alpha}$-spectrum of $\overline{G_{{p}^{m}}}$ is

$\operatorname{spec}_{A_{\alpha}}\left(\overline{G_{{p}^{m}}}\right)=\begin{pmatrix}-{p}^{m-1}&\alpha{p}^{m-1}-1&\alpha{p}^{m-1}&{p}^{m-1}-1&{p}^{m-1}\\\
\frac{{p}-1}{2}&{p}^{m-1}-1&\left({p}-1\right)\left({p}^{m-1}-1\right)&1&\frac{{p}-1}{2}\end{pmatrix}.$

∎

Using Proposition 4.1 we now calculate the $A_{\alpha}$-energy of complement
of UACG $G_{n}$ for $n={p}^{m}$.

######  Corollary 4.1.

The $A_{\alpha}$-energy of complement of UACG $G_{{p}^{m}}$, ${p}\geq 3$ is

$\varepsilon_{\alpha}\left(\overline{G_{{p}^{m}}}\right)=\frac{{p}^{m+1}-{p}+\alpha\left(2-{p}-2{p}^{m-1}+{p}^{m}\right)}{{p}}+\left|\frac{{p}^{m}-{p}-\alpha{p}^{m}+\alpha}{{p}}\right|$

.

###### Proof.

The number of vertices and edges of $\overline{G_{{p}^{m}}}$ is ${p}^{m}$ and
$\frac{{p}^{m-1}\left({p}^{m}-1\right)}{2}$ respectively. Thus,

$\displaystyle\varepsilon_{\alpha}\left(\overline{G_{{p}^{m}}}\right)=$
$\displaystyle\left(\frac{{p}-1}{2}\right)\left|-{p}^{m-1}-\frac{\alpha\left({p}^{m}-1\right)}{{p}}\right|+\left({p}^{m-1}-1\right)\left|\alpha{p}^{m-1}-1-\frac{\alpha\left({p}^{m}-1\right)}{{p}}\right|$
$\displaystyle+\left({p}-1\right)\left({p}^{m-1}-1\right)\left|\alpha{p}^{m-1}-\frac{\alpha\left({p}^{m}-1\right)}{{p}}\right|+\left|{p}^{m-1}-1-\frac{\alpha\left({p}^{m}-1\right)}{{p}}\right|$
$\displaystyle+\left(\frac{{p}-1}{2}\right)\left|{p}^{m-1}-\frac{\alpha\left({p}^{m}-1\right)}{{p}}\right|$
$\displaystyle=$
$\displaystyle\left(\frac{{p}-1}{2}\right)\left({p}^{m-1}+\frac{\alpha\left({p}^{m}-1\right)}{{p}}\right)+\left({p}^{m-1}-1\right)\left(-\alpha{p}^{m-1}+1+\frac{\alpha\left({p}^{m}-1\right)}{{p}}\right)$
$\displaystyle+\left({p}-1\right)\left({p}^{m-1}-1\right)\alpha+\left|{p}^{m-1}-1-\frac{\alpha\left({p}^{m}-1\right)}{{p}}\right|+\left(\frac{{p}-1}{2}\right)\left(\frac{{p}^{m}-\alpha{p}^{m}+\alpha}{{p}}\right)$
$\displaystyle=$
$\displaystyle\frac{{p}^{m+1}-{p}+\alpha\left(2-{p}-2{p}^{m-1}+{p}^{m}\right)}{{p}}+\left|\frac{{p}^{m}-{p}-\alpha{p}^{m}+\alpha}{{p}}\right|$
$\displaystyle=$
$\displaystyle\begin{cases}\frac{{p}^{m+1}+{p}^{m}-2{p}+\alpha\left(3-{p}-2{p}^{m-1}\right)}{{p}}&\text{if
}\alpha\leq\frac{{p}^{m}-{p}}{{p}^{m}-1}\\\
\frac{{p}^{m+1}-{p}^{m}+\alpha\left(1-{p}-2{p}^{m-1}+2{p}^{m}\right)}{{p}}&\text{if
}\alpha>\frac{{p}^{m}-{p}}{{p}^{m}-1}\end{cases}.$

∎

######  Remark 4.1.

If $n$ is even, then $\overline{G_{n}}$ is regular and its
$A_{\alpha}$-spectrum will be $n-1-\phi\left(n\right)$ and
$\alpha\left(n-\phi\left(n\right)\right)-\left(1-\alpha\right)\lambda_{i}\left(A\left(G_{n}\right)\right)-1,i=2,3,\cdots,n$.
Its $A_{\alpha}$-energy will be
$\left(1-\alpha\right)\left(2n-2+\left(2^{k}-2\right)\phi\left(n\right)-\displaystyle\prod_{i=1}^{k}{p}_{i}+\prod_{i=1}^{k}\left(2-{p}_{i}\right)\right)$,
where ${p}_{1}{p}_{2}\cdots{p}_{k}$ is the largest square-free number that
divides $n$.

In general, computing exact eigenvalues of complement of UACG for odd values
of $n$ is difficult. But, we estimate a bound for $A_{\alpha}$-eigenvalues of
complement of UACG $G_{n}$ when $n$ is odd in the following proposition.

######  Proposition 4.2.

The $A_{\alpha}$-spectrum of the complement of UACG $G_{n}$, for odd $n$,
satisfy the ensuing inequalities:

$(1-\alpha)\mu(t_{k})\frac{\phi(n)}{\phi(t_{k})}+\alpha
n-\alpha\phi(n)-1\leq\lambda_{k}\left(A\left(\overline{G_{n}}\right)\right)\leq(1-\alpha)\mu(t_{k})\frac{\phi(n)}{\phi(t_{k})}+\alpha
n-\alpha\phi(n)$

for $2\leq k\leq\frac{(n+1)}{2}$ and

$-(1-\alpha)\mu(t_{k})\frac{\phi(n)}{\phi(t_{k})}+\alpha
n-\alpha\phi(n)-1\leq\lambda_{k}\left(A\left(\overline{G_{n}}\right)\right)\leq-(1-\alpha)\mu(t_{k})\frac{\phi(n)}{\phi(t_{k})}+\alpha
n-\alpha\phi(n)$

for $\frac{(n+3)}{2}\leq k\leq n$ and

$n-\phi(n)-1\leq\lambda_{1}\left(A\left(\overline{G_{n}}\right)\right)\leq
n-\phi(n).$

###### Proof.

Let $\bar{A}_{\alpha}=(\bar{a}_{ij})$, $0\leq i,j\leq n-1$, be the
$A_{\alpha}$ matrix of $\overline{G_{n}}$, where

$\bar{a}_{ij}=\left\\{\begin{array}[]{l l l l}1-\alpha&\quad\text{if
}\gcd(n,i+j)\neq 1\text{ and }i\neq j,\\\ \alpha(n-1-\phi(n))&\quad\text{if
}\gcd(i+j,n\neq 1\text{ and }i=j,\\\ \alpha(n-\phi(n))&\quad\text{if
}\gcd(n,i+j)=1\text{ and }i=j,\\\ 0&\quad\text{otherwise}.\end{array}\right.$

Then $\bar{A}_{\alpha}=\tilde{B}+\tilde{C}$, where $\tilde{B}=[b_{ij}],0\leq
i,j\leq n-1,$

$b_{ij}=\left\\{\begin{array}[]{l l}1-\alpha&\quad\text{if }\gcd(n,i+j)\neq
1,\\\ 0&\quad\text{otherwise},\end{array}\right.$

and $\tilde{C}=[c_{ij}],0\leq i,j\leq n-1,$

$c_{ij}=\left\\{\begin{array}[]{l l l}\alpha n-\alpha\phi(n)-1&\quad\text{if
}\gcd(n,i+j)\neq 1\text{ and }i=j,\\\ \alpha(n-\phi(n))&\quad\text{if
}\gcd(n,i+j)=1\text{ and }i=j,\\\ 0&\quad\text{otherwise}.\end{array}\right.$

According to the definition of $\tilde{B}$, it is a left circulant matrix with
initial row $(s_{0},s_{1},\cdots,s_{n-1})$ where

$s_{j}=\left\\{\begin{array}[]{l l}1-\alpha&\quad\text{if }\gcd(j,n)=1,\\\
0&\quad\text{otherwise}.\end{array}\right.$

So eigenvalues of $\tilde{B}$ are
$(1-\alpha)(n-\phi(n)),(1-\alpha)\mu(t_{k})\frac{\phi(n)}{\phi(t_{k})}$ for
$1\leq k\leq\frac{(n-1)}{2}$ and
$-(1-\alpha)\mu(t_{k})\frac{\phi(n)}{\phi(t_{k})}$ for $\frac{(n+1)}{2}\leq
k\leq n-1$.
Eigenvalues of $\tilde{C}$ are $\alpha n-\alpha\phi(n)-1$ repeated
${n-\phi(n)}$ times and $\alpha(n-\phi(n))]$ repeated ${\phi(n)}$ times.
Then from the eigenvalues of $\tilde{B},\tilde{C}$ and Theorem 2.6 we get the
desired result.

∎

In the following proposition, we estimate some bounds for $A_{\alpha}$-energy
of complement of UACG $G_{n}$ when $n$ is odd.

######  Proposition 4.3.

For an odd integer $n\geq 3$ and $\alpha\in[0,1)$,

$\displaystyle\epsilon_{\alpha}(\overline{G}_{n})\geq$

$\sqrt{(n-\phi(n))\left(\alpha^{2}(n-\phi(n)(n^{2}+2n-2)+2n\alpha^{2}(2\phi(n)+n-1)+2n(n-1)(1-2\alpha)\right)+2n\alpha^{2}\phi(n)}.$

$\displaystyle\epsilon_{\alpha}(\overline{G}_{n})\geq\frac{2(1-\alpha)(n-1)(n-\phi(n))}{n}.$

$\displaystyle\epsilon_{\alpha}(\overline{G}_{n})\geq
2\sqrt{(n-\phi(n))^{2}+\frac{\phi(n)(2(n-\phi(n))+1)}{n}}-\frac{2\alpha(n-1)(n-\phi(n))}{n}.$

$\displaystyle\epsilon_{\alpha}(\overline{G}_{n})\geq\alpha(\phi(n)+4)-n\alpha-\frac{2\alpha\phi(n)}{n}+\sqrt{(n-\phi(n)+2)(\alpha^{2}(n-\phi(n)+2)+4(1-2\alpha)}.$

$\displaystyle\epsilon_{\alpha}(\overline{G}_{n})\leq$

$\sqrt{\alpha^{2}n^{4}+n^{3}(-\alpha-2\alpha+2\phi(n)+3)-n^{2}(1+p)(2\alpha^{2}+(1-\alpha)^{2})+np(-2\alpha^{2}{p}-2\alpha+1)+{p}^{2}}.$

###### Proof.

The Zagreb index of complement of UACG is
$(n-\phi(n))^{2}n+\phi(n)(2(n-\phi(n))+1)$ and
$||A(\overline{G}_{n})||^{2}_{F}=(n-1)(n-\phi(n))$. Then by using Theorems 2.7
and 2.8 we get the desired result. ∎

##  5 $A_{\alpha}$-borderenergetic and $A_{\alpha}$-hyperenergetic graphs

In this section we define $A_{\alpha}$ analogous of borderenergetic and
hyperenergetic graphs.

On studying about the $A_{\alpha}$-energy we are interested to know whether
there is a graph on $n$ vertices whose $A_{\alpha}$-energy equals the
$A_{\alpha}$-energy of complete graph on $n$ vertices, that is
$2(1-\alpha)(n-1)$. We refer them as $A_{\alpha}$-borderenergetic graphs.
Formally, a graph $G$ is $A_{\alpha}$-borderenergetic if
$\varepsilon_{\alpha}(G)=\varepsilon_{\alpha}(K_{n})$, for some
$\alpha\in[0,1)$. In general borderenergetic graphs are not
$A_{\alpha}$-borderenergetic, but regular borderenergetic graphs are
$A_{\alpha}$-borderenergetic for every value of $\alpha$.

The graph whose $A_{\alpha}$-energy exceeding the $A_{\alpha}$-energy of
complete graph is called $A_{\alpha}$-hyperenergetic. That is, a graph $G$ is
$A_{\alpha}$-hyperenergetic if
$\varepsilon_{\alpha}(G)\geq\varepsilon_{\alpha}(K_{n})$, for some
$\alpha\in[0,1)$.

In the following section we observed some graphs which are
$A_{\alpha}$-borderenergetic and $A_{\alpha}$-hyperenergetic.

## Observations

In this section with the help of Matlab software we study about
$A_{\alpha}$-energy of UACG and its complement as $\alpha$ varies.

Observation 1 : From Table 1 we observe the following:

  1. 1.

$A_{\alpha}$-energy of UACG $G_{n}$ decreases as $\alpha$ increases for
$n={p}^{m}$.

  2. 2.

$G_{{p}^{m}}$ is $A_{\alpha}$-hyperenergetic for ${p}\geq 5,m\geq 2$.

  3. 3.

For all $m$, $\overline{G}_{{p}^{m}}$ is $A_{\alpha}$-hyperenergetic for some
$\alpha$ greater than $0.4$.

Eg: $\overline{G}_{9}$ is hyperenergetic for $\alpha>0.428571428571$.
$\overline{G}_{25}$ is hyperenergetic for $\alpha>0.438596491228$.

| 0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 0.9999
---|---|---|---|---|---|---|---|---|---|---|---
$\varepsilon_{\alpha}(G_{9})$ | 14.717 | 13.457 | 12.197 | 10.940 | 9.686 | 8.438 | 7.2 | 5.980 | 4.810 | 4 | 3.9996
$\varepsilon_{\alpha}(\overline{G_{9}})$ | 10 | 9.8 | 9.6 | 9.4 | 9.2 | 9 | 8.8 | 8.6 | 8.666 | 9 | 9.333
$\varepsilon_{\alpha}(K_{9})$ | 16 | 14.4 | 12.8 | 11.2 | 9.6 | 8 | 6.4 | 4.8 | 3.2 | 1.6 | 0.0016
$\varepsilon_{\alpha}(G_{27})$ | 50.683 | 46.618 | 42.554 | 38.490 | 34.427 | 30.367 | 26.309 | 22.257 | 18.220 | 14.247 | 11.998
$\varepsilon_{\alpha}(\overline{G_{27}})$ | 34 | 33.4 | 32.8 | 32.2 | 31.6 | 31 | 30.4 | 29.8 | 29.2 | 28.6 | 29.332
$\varepsilon_{\alpha}(K_{27})$ | 52 | 46.8 | 41.6 | 36.4 | 31.2 | 26 | 20.8 | 15.6 | 10.4 | 5.2 | 0.005
$\varepsilon_{\alpha}(G_{81})$ | 158.672 | 146.206 | 133.740 | 121.274 | 108.809 | 96.344 | 83.880 | 71.418 | 58.961 | 46.523 | 35.996
$\varepsilon_{\alpha}(\overline{G_{81}})$ | 106 | 104.2 | 102.4 | 100.6 | 98.8 | 97 | 95.2 | 93.4 | 91.6 | 89.8 | 89.329
$\varepsilon_{\alpha}(K_{81})$ | 160 | 144 | 128 | 112 | 96 | 80 | 64 | 48 | 32 | 16 | 0.016
$\varepsilon_{\alpha}(G_{5})$ | 6.472 | 5.841 | 5.212 | 4.588 | 3.969 | 3.361 | 2.772 | 2.224 | 1.774 | 1.546 | 1.599
$\varepsilon_{\alpha}(\overline{G_{5}})$ | 4 | 4.08 | 4.16 | 4.24 | 4.32 | 4.4 | 4.48 | 4.56 | 4.64 | 4.72 | 4.799
$\varepsilon_{\alpha}(K_{5})$ | 8 | 7.2 | 6.4 | 5.6 | 4.8 | 4 | 3.2 | 2.4 | 1.6 | 0.8 | 0.001
$\varepsilon_{\alpha}(G_{25})$ | 54.413 | 49.533 | 44.656 | 39.778 | 34.902 | 30.026 | 25.153 | 20.286 | 15.432 | 10.641 | 7.999
$\varepsilon_{\alpha}(\overline{G_{25}})$ | 28 | 27.76 | 27.52 | 27.28 | 27.04 | 26.8 | 26.56 | 26.32 | 26.08 | 26.48 | 27.199
$\varepsilon_{\alpha}(K_{25})$ | 48 | 43.2 | 38.4 | 33.6 | 28.8 | 24 | 19.2 | 14.4 | 9.6 | 4.8 | 0.004
$\varepsilon_{\alpha}(G_{125})$ | 294.402 | 268.722 | 243.043 | 217.363 | 191.684 | 166.005 | 140.326 | 114.648 | 88.973 | 63.306 | 39.996
$\varepsilon_{\alpha}(\overline{G_{125}})$ | 148 | 146.96 | 145.92 | 144.88 | 143.84 | 142.8 | 141.76 | 140.72 | 139.68 | 138.64 | 139.196
$\varepsilon_{\alpha}(K_{125})$ | 248 | 223.2 | 198.4 | 173.6 | 148.8 | 124 | 99.2 | 74.4 | 49.6 | 24.8 | 0.0248
$\varepsilon_{\alpha}(G_{625})$ | 1494.401 | 1364.721 | 1235.041 | 1105.361 | 975.681 | 846.001 | 716.321 | 586.641 | 456.962 | 327.285 | 199.981
$\varepsilon_{\alpha}(\overline{G_{625}})$ | 748 | 742.96 | 737.92 | 732.88 | 727.84 | 722.8 | 717.76 | 712.72 | 707.68 | 702.64 | 699.180
$\varepsilon_{\alpha}(K_{625})$ | 1248 | 1123.2 | 998.4 | 873.6 | 748.8 | 624 | 499.2 | 374.4 | 249.6 | 124.8 | 0.124
$\varepsilon_{\alpha}(G_{49})$ | 118.291 | 107.405 | 96.520 | 85.635 | 74.751 | 63.867 | 52.984 | 42.103 | 31.226 | 20.372 | 11.998
$\varepsilon_{\alpha}(\overline{G_{49}})$ | 54 | 53.742 | 53.485 | 53.228 | 52.971 | 52.714 | 52.457 | 52.2 | 51.942 | 52.028 | 53.141
$\varepsilon_{\alpha}(K_{49})$ | 96 | 86.4 | 76.8 | 67.2 | 57.6 | 48 | 38.4 | 28.8 | 19.2 | 9.6 | 0.009
$\varepsilon_{\alpha}(G_{121})$ | 318.183 | 288.092 | 258.001 | 227.911 | 197.820 | 167.73 | 137.639 | 107.550 | 77.461 | 47.378 | 19.998
$\varepsilon_{\alpha}(\overline{G_{121}})$ | 130 | 129.727 | 129.454 | 129.181 | 128.909 | 128.636 | 128.363 | 128.091 | 127.818 | 127.545 | 129.089
$\varepsilon_{\alpha}(K_{121})$ | 240 | 216 | 192 | 168 | 144 | 120 | 96 | 72 | 48 | 24 | 0.024
Table 1: $A_{\alpha}$-energy of $G_{{p}^{m}}$,
$\left(\overline{G_{{p}^{m}}}\right)$ and $K_{{p}^{m}}$ for different values
of $\alpha$.

Observation 2 : From Table 1 we also observe the following:

  1. 1.

For all $m$ and some $\alpha$, $G_{3^{m}}$ is $A_{\alpha}$-borderenergetic and
as $m$ increases, $\alpha\rightarrow 0$.

Eg: $G_{9}$ is borderenergetic for $\alpha=0.375$, (see Table 2). However
$G_{{p}^{m}}$ is not $A_{\alpha}$-borderenergetic for ${p}\geq 5$ and $m\geq
2$.

  2. 2.

$G_{{p}^{m}}$ is not adjacency borderenergetic for any ${p}$ and $m$.

  3. 3.

For all $m$ and ${p}\geq 5$, $\overline{G}_{{p}^{m}}$ is
$A_{\alpha}$-borderenergetic for some $\alpha\in(0.4,0.5)$.

Eg: $\overline{G}_{5}$ is borderenergetic for $\alpha=0.454545454545$.
$\overline{G}_{49}$ is borderenergetic for $\alpha=0.449541284404$.

$n$ | $\alpha$ | $\varepsilon_{\alpha}(G_{n})$ | $\varepsilon_{\alpha}(K_{n})$
---|---|---|---
9 | 0.375 | 10 | 10
27 | 0.115981467677 | 45.968963680773 | 45.968963680773
81 | 0.037573807884 | 153.988190738623 | 153.988190738623
243 | 0.012405082000 | 477.995940311853 | 477.995940311853
5 | 0.755511381250 | 1.955908950000 | 1.955908950000
Table 2: Values of $\alpha$ for which $G_{{p}^{m}}$ is $A_{\alpha}$-borderenergetic. $n$ | $\alpha$ | $\varepsilon_{\alpha}(\overline{G}_{n})$ | $\varepsilon_{\alpha}(\overline{K}_{n})$
---|---|---|---
9 | 0.428571428571 | 9.142857142857 | 9.142857142857
27 | 0.391304347826 | 31.652173913043 | 31.652173913043
81 | 0.380281690141 | 99.154929577465 | 99.154929577465
5 | 0.454545454545 | 4.363636363636 | 4.363636363636
25 | 0.438596491228 | 26.947368421053 | 26.947368421053
125 | 0.420875420875 | 143.622895622896 | 143.622895622896
625 | 0.417501670007 | 726.957915831663 | 726.957915831663
49 | 0.449541284404 | 52.844036697248 | 52.844036697248
121 | 0.463601532567 | 128.735632183908 | 128.735632183908
Table 3: Values of $\alpha$ for which $\overline{G}_{{p}^{m}}$ is
$A_{\alpha}$-borderenergetic.

##  6 Conclusion

In this article the $A_{\alpha}$-eigenvalues of unitary addition Cayley graph
$G_{n}$ when its order is ${p}^{m}$ or even is calculated. When its order is
odd, a bound for the $A_{\alpha}$-eigenvalues is found. In addition, the
$A_{\alpha}$-energy of $G_{n}$ when $n$ is ${p}^{m}$ or even is computed. For
odd $n$, bounds for $A_{\alpha}$-energy of $G_{n}$ and $\overline{G}_{n}$ are
estimated. Also the concept of $A_{\alpha}$-borderenergetic and
$A_{\alpha}$-hyperenergetic graphs is introduced and found some families of
each.

## Acknowledgement

The authors would like to thank the DST, Government of India, for providing
support to carry out this work under the scheme
’FIST’(No.SR/FST/MS-I/2019/40). The first author gratefully acknowledges the
financial support of University Grants Commission(UGC), India.

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