A new lower bound for the energy of graphs

Let G be a simple graph with n vertices v(1), ..., v(n) and m edges. Let A(G) = [a(ij)](nxn) be the adjacency matrix of G, that is a(ij) = 1 if v(i) and vj are adjacent, and a(ij) = 0, otherwise. Let lambda(1), ..., lambda(n) be the eigenvalues of A(G). The energy of G, denoted by epsilon(G), is defined as vertical bar lambda(1)vertical bar + ... + vertical bar lambda(n)vertical bar. It is well known that 2 root m <= epsilon(G) <= root 2mn. In this paper we improve the latter lower bound and prove that if all eigenvalues of G are non-zero, then epsilon(G) >= 2 root vertical bar lambda(1)lambda(n)vertical bar/vertical bar lambda(1)vertical bar + vertical bar lambda(n)vertical bar root 2mn, where vertical bar lambda(1)vertical bar >= ... >= vertical bar lambda(n)vertical bar We show that the equality holds if and only if G = n/2 K-2 or G = pK(beta+1) boolean OR T-q(beta+1), where p and q are some non-negative integers and beta >= 2 and T beta+1 is the graph K beta+1,(beta+1) \ M, where M is a perfect matching of K beta+1,(beta+1). Finally by studying the root vertical bar lambda(1)lambda(n)vertical bar/vertical bar lambda(1)vertical bar + vertical bar lambda(n)vertical bar of graphs, we find some lower bounds for the energy of graphs. In particular, we obtain that if G has no eigenvalue in the interval (-1, 1), then epsilon(G) >= 2 root m root 2n - 2/n, and the equality holds if and only if G is the complete graph K-n. (C) 2019 Elsevier Inc. All rights reserved.