Research Article
The Least Eigenvalue of the Complement of the Square Power
Graph of G
Lubna Gul,
1
Gohar Ali ,
1
Usama Waheed,
1
and Sumiya Nasir
2
1
Department of Mathematics, Islamia College, Peshawar, Pakistan
2
College of Science and Human Studies, Prince Mohammad Bin Fahd University, Khobar Dhahran, Saudi Arabia
Correspondence should be addressed to Gohar Ali; gohar.ali@icp.edu.pk
Received 28 July 2021; Accepted 2 September 2021; Published 16 September 2021
Academic Editor: Muhammad Javaid
Copyright © 2021 Lubna Gul 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 G(n, m) represent the family of square power graphs of order n and size m, obtained from the family of graphs F(n, k) of order
n and size k, with m ≥ k. In this paper, we discussed the least eigenvalue of graph G in the family G(n, m)
c
. All graphs considered
here are undirected, simple, connected, and not a complete K
n
for positive integer n.
1. Introduction
Let G �(V, E) be a simple graph with vertex set V(G)�
u
1
,u
2
, ... ,u
n
and edge set E(G). e adjacency matrix of a
graph is defined to be the matrix B(G)�[b
ij
] of order n,
where b
ij
� 1 if u
i
is adjacent to u
j
, otherwise b
ij
� 0. As the
graphs are undirected, B(G) is real and symmetric and the
eigenvalues are also real and can be arranged as
λ
1
(G) ≤ λ
2
(G) ≤ ··· ≤ λ
n
(G). e eigenvalues of B(G) are
referred to as the eigenvalues of G, and one can easily find
that λ
n
is the largest absolute eigenvalue among all eigen-
values of G.
A graph G is called minimizing in a certain graph class if
its least eigenvalue attains the minimum among all the least
eigenvalues of the graphs of that class. We will denote the
least eigenvalue of a graph G by λ
min
(G) and X for the
corresponding least eigenvector. A graph is said to be a
square power graph denoted by G
2
n
of order n, if V(G
2
n
)�
V(G) and E(G
2
n
)� E(G) ∪ uv: d
G
(u, v)� 2, u, v ∈ V(G) ,
where d
G
(u, v) denotes the distance between vertices u and v
in G; also, the bipartite square power graph is denoted by
G
2
n
(p, q), where p and q are the partitions of the vertex set
V(G
2
n
).
We limit our discussion to only connected graphs be-
cause we know that the spectrum (the set of eigenvalues of a
graph) of a disconnected graph is the union of spectra of its
connected components and so the least eigenvalue of a
disconnected graph will be the minimum eigenvalue of its
component among all the minimum eigenvalues of its
components.
ere are many results in the literature regarding the
least eigenvalue of a graph. Fan et al. [1] discussed the least
eigenvalue of complement of trees. Hoffman [2] discussed
the limiting point of the least eigenvalue of connected
graphs. Hong and Shu [3] discussed the sharp lower bounds
of the least eigenvalue of planar graphs. Javid [4, 5] discussed
the minimizing graphs of the connected graphs whose
complements are bicyclic (with two cycles) and explained
the characterization of the minimizing graph of the con-
nected graphs whose complement is bicyclic.
Theorem 1 (see [6]). For a graph G of order n,
λ
min
(G) ≥ −
������
n
2
n
2
, (1)
with equality iff G � K
⌈n/2⌉,⌊n/2⌋
.
According to Powers et al. [7], if m is the size of a graph
G, then
λ
min
(G) ≥ −
��
m
√
. (2)
Hindawi
Complexity
Volume 2021, Article ID 5885834, 4 pages
https://doi.org/10.1155/2021/5885834