New type bounds for energy of graphs and spread of matrices

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), . . . ,lambda(n) are the eigenvalues of M, then the spread of M is max(1 <= i,j <= 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(G). 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(G) >= root 2m(n(+)+n(-))/n(+)n(-) and the equality holds if and only if G=r K-s boolean OR tK(1) or G=rK(p), . . . ,p(q) boolean OR tK(1) or G=r(1)K(a1),(b1)boolean OR center dot center dot center dot boolean OR r(h) K-ah, (bh), for some non-negative integers r, s, t, p, q and r(1),a(1),b(1), . . . ,r(h),a(h),b(h) such that a(1)b(1)=center dot center dot center dot=a(h)b(h), p >= 2 and q >= 3. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.