Research Article
Some Algebraic Properties of a Class of Integral Graphs
Determined by Their Spectrum
Jia-Bao Liu ,
1
S. Morteza Mirafzal ,
2
and Ali Zafari
3
1
School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, China
2
Lorestan University, Department of Mathematics, Faculty of Science, Khorramabad, Iran
3
Department of Mathematics, Faculty of Science, Payame Noor University, P.O. Box 19395-4697, Tehran, Iran
Correspondence should be addressed to Ali Zafari; zafari.math.pu@gmail.com
Received 28 December 2020; Revised 27 January 2021; Accepted 5 February 2021; Published 17 February 2021
Academic Editor: Ghulam Mustafa
Copyright © 2021 Jia-Bao Liu et al. is is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Let Γ �(V, E) be a graph. If all the eigenvalues of the adjacency matrix of the graph Γ are integers, then we say that Γ is an integral
graph. A graph Γ isdeterminedbyitsspectrumifeverygraphcospectraltoitisinfactisomorphictoit.Inthispaper,weinvestigate
some algebraic properties of the Cayley graph Γ � Cay(Z
n
,S), where n � p
m
(p is a prime integer and m ∈ N) and
S � a ∈ Z
n
|(a, n)� 1 . First, we show that Γ is an integral graph. Also, we determine the automorphism group of Γ. Moreover, we
show that Γ and K
v
▽Γ are determined by their spectrum.
1. Introduction
e graphs in this paper are simple, undirected, and con-
nected. We always assume that Γ denotes the complement
graph of Γ. e eigenvalues of a graph Γ are the eigenvalues
of the adjacency matrix of Γ. e spectrum of Γ is the list of
the eigenvalues of the adjacency matrix of Γ together with
their multiplicities, and it is denoted by Spec(Γ);see[1].Ifall
the eigenvalues of the adjacency matrix of the graph Γ are
integers, then we say that Γ is an integral graph. e notion
of integral graphs was first introduced by Harary and
Schwenk in 1974; see [2]. In general, the problem of
characterizing integral graphs seems to be very difficult.
ere are good surveys in this area; see [3]. For more results
depending on the integral graphs and their applications in
engineering networks, see [4–6]. For any vertex v of a
connected graph Γ, we denote the set of vertices of Γ at
distance r from Γ by Γ
r
(v). en, we have
Γ
r
(v)� u ∈ V(Γ)| d(u, v)� r { }, (1)
where d(u, v) denotes the distance in Γ between the vertices
u and v and r is a nonnegative integer not exceeding d, the
diameter of Γ. It is clear that Γ
0
(v)� v {}, and V(Γ) is
partitioned into the disjoint subsets Γ
0
(v), ... , Γ
d
(v), for
each v in V(Γ). e graph Γ is called distance regular with
diameter d and intersection array b
0
, ... ,b
d− 1
; c
1
, ... ,c
d
if
it is regular of valency k and, for any two vertices u and v in Γ
at distance r, we have |Γ
r+1
(v) ∩Γ
1
(u)| � b
r
, (0 ≤ r ≤ d − 1),
and |Γ
r− 1
(v) ∩Γ
1
(u)| � c
r
(1 ≤ r ≤ d). e intersection
numbers c
r
,b
r
, and a
r
satisfy a
r
� k − b
r
− c
r
(0 ≤ r ≤ d),
where a
r
is the number of neighbours of u in Γ
r
(v). Let G be
a finite group and let H be a subset of G such that it is closed
under taking inverses and does not contain the identity. A
Cayley graph Γ � Cay(G, H) is the graph whose vertex set
and edge set are defined as follows:
V(Γ)� G;
E(Γ)� x, y | x
− 1
y ∈ H .
(2)
It is well known that if Γ is a distance regular graph with
valency k, diameter d, adjacency matrix A, and intersection
array
b
0
,b
1
, ... ,b
d− 1
; c
1
,c
2
, ... ,c
d
, (3)
then the tridiagonal (d + 1)×(d + 1) matrix,
Hindawi
Journal of Mathematics
Volume 2021, Article ID 6632206, 5 pages
https://doi.org/10.1155/2021/6632206