New type bounds for energy of graphs and spread of matrices

Abstract

The energy of a simple graph G is defined as the sum of the absolute values of eigenvalues of the adjacency matrix of G. For a complex matrix M the spread of M is the maximum absolute value of the differences between any two eigenvalues of M. Thus if ${\lambda }_1,\dots ,{\lambda }_n$ are the eigenvalues of M, then the spread of M is ${\max }_{1\le i,j\le n}|{\lambda }_i-{\lambda }_j|$. The spread of a graph G is defined as the spread of its adjacency matrix and is denoted by $s\left(G\right)$. The inertia of G is an integer triple $(n^{+},n^{-},n^0)$ specifying the numbers of positive, negative and zero eigenvalues of the adjacency matrix of G. In this paper we find some bounds for energy of graphs in terms of some parameters of graphs such as rank, inertia and spread of graphs. We find some bounds for spread of graphs and matrices that improve the previous bounds. In particular, we show that if G is a graph with m edges and inertia $(n^{+},n^{-},n^0)$, then $s\left(G\right)\ge \sqrt{\frac{2m(n^{+}+n^{-})}{n^{+}{n}^{-}}}$ and the equality holds if and only if $G=r{K}_s\cup t{K}_1$ or $G=r{K}_{\underset{q}{\underbrace{p,\dots ,p}}}\cup t{K}_1$ or $G=r_1{K}_{a_1,b_1}\cup \cdots \cup r_h{K}_{a_h,b_h}$, for some non-negative integers $r,s,t,p,q$ and $r_1,a_1,b_1,\dots ,r_h,a_h,b_h$ such that $a_1{b}_1=\cdots =a_h{b}_h$, $p\ge 2$ and $q\ge 3$.