The Least Eigenvalue of the Complement of the Square Power Graph of G

Let G � (V, E) be a simple graph with vertex set V(G) � u1, u2, . . . , un 􏼈 􏼉 and edge set E(G). 'e adjacency matrix of a graph is defined to be the matrix B(G) � [bij] of order n, where bij � 1 if ui is adjacent to uj, otherwise bij � 0. As the graphs are undirected, B(G) is real and symmetric and the eigenvalues are also real and can be arranged as λ1(G)≤ λ2(G)≤ · · · ≤ λn(G). 'e eigenvalues of B(G) are referred to as the eigenvalues of G, and one can easily find that λn is the largest absolute eigenvalue among all eigenvalues 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 λmin(G) and X for the corresponding least eigenvector. A graph is said to be a square power graph denoted by Gn of order n, if V(G 2 n) � V(G) and E(Gn) � E(G)∪ uv: dG(u, v) � 2, u, v ∈ V(G) 􏼈 􏼉, where dG(u, v) denotes the distance between vertices u and v in G; also, the bipartite square power graph is denoted by Gn(p, q), where p and q are the partitions of the vertex set V(Gn). We limit our discussion to only connected graphs because 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 connected graphs whose complement is bicyclic.


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 eigenvalues 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 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 because 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 connected graphs whose complement is bicyclic.
According to Powers et al. [7], if m is the size of a graph G, then Theorem 2 (see [8]). If G is a graph whose least eigenvalue is minimal among all the connected graphs, then

either a bipartite graph, or (ii) G is a join of two nested split graphs
Theorem 3 (see [9]). Let K(n, m) denote the family of all connected bipartite graphs of order n and size m. Let G ∈ K(n, m) be the graph such that its least eigenvalue is minimal among all the connected bipartite graphs, then G must be a double nested graph.

Main Results
We begin with some definitions. Given a graph G of order n, a vector X ∈ R n is defined on G if there exists an injective function f from V(G) to the entries of X, simply written as then it is naturally defined on V(G) and every entry of X corresponds to the vertex v ∈ V(G). One can easily prove that and λ is an eigenvalue of G corresponding to the eigenvector X if and only if X ≠ 0, and Equation (4) is called the eigen equation for the graph G. Now, for any arbitrary unit vector X ∈ R n , and it will be equal iff X is a least eigenvector of G. Now, let G c denote the complement of a graph G. One can easily prove that B(G C ) � J − I − B(G), where I and J denote the identity matrix and the ones matrix of the same size as B(G). Now, for any vector X, A graph G is said to be a tree iff there is a path between any pair of vertices and has no cycle, and a tree is said to be a star iff there exists a vertex of degree n − 1 and all other vertices have degree one. e former is called the central vertex, and the later is called the pendant vertex. If a graph G is a path of order less than or equal to 4 then λ min (G c ) � −1. If a graph G contains K 1,2 as an induced subgraph, then . If a graph G is complete or its complement is a null graph or union of disjoint complete graph, then λ min (G c ) � −1. So, we are limiting our discussion to (G 2 n ) c − K c n , where K n c denotes all those graphs whose λ min is −1.
Special tree graph denoted by G(p, q) is obtained from two stars S p−1 and S q−1 of orders p and q, respectively, by joining any two vertices of degree one from the two stars. G 2 (p, q) is a square power graph of this tree and G 2 (p, q) c is its complement graph, as shown in Figure 1; the order of G 2 (p, q) c is n � p + q, and its size is m � pq − 3.

Lemma 1.
Given n ≥ 6, for any positive integers p and q such that p ≥ q ≥ 1 and p + q � n, with equality iff p � (p + q)/2 and q � (p + q)/2 .
Proof. For n ≥ 6, let λ min be the least eigenvalue of G 2 n (p, q) c and X be the corresponding least eigenvector. Let X i denote the entries of X corresponding to the vertices of (G 2 n ) c . Now, by eigen equation, all the vertices u 1 , u 2 , u 3 , . . . , u p−2 have the same values, say X 1 and X 2 denote the entry of X corresponding to the vertex u p−1 , X 3 for the vertex u p , and all the vertices v 1 , v 2 , v 3 , . . . , v q−2 have the same value X 4 . X 5 represents the entry of X corresponding to the vertex v q−1 , and for the vertex v q , the entry of X is X 6 . Now, λ min (G 2 n (p, q) c ) < −1. So, by an eigen equation, Transform (8) into a matrix equation (B − λ 1 I)X ′ � 0, where X ′ � (X 1 , X 2 , X 3 , X 4 , X 5 , X 6 ) T and the matrix B of order 6 is omitted. We get Note that λ min is a least root of f(λ; p, q). In addition, f(−1, 6180; p, q) � 6.8538 − 17.9426(p + q) < 0 as p + q � n ≥ 6, which implies λ min < − 1.6180.
If p ≥ q, by the above discussion, for λ min < − 1.6180, we have In particular, f(λ min ; p − 1, q + 1) < 0, which implies is complete the proof.

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Considering the family of square power graphs G(n, m) of order n and size m obtained from the family of graphs F(n, k) of order n and size k with m ≥ k, we have the following result.

Data Availability
e data used to support the findings of this study are cited at relevant places within the text as references.  Figure 1: (G 2 n ) c � G 2 (p, q) c graph with m � pq − 3.