Some lower bounds for the energy of graphs

The singular values of a matrix A are defined as the square roots of the eigenvalues of A*A, and the energy of A denoted by E(A) is the sum of its singular values. The energy of a graph G, E(G), is defined as the sum of absolute values of the eigenvalues of its adjacency matrix. In this paper, we prove that if A is a Hermitian matrix with the block form A = (B D D* C), then E(A) >= 2E(D). Also, we show that if G is a graph and H is a spanning subgraph of G such that E(H) is an edge cut of G, then E(H) <= E(G), i.e., adding any number of edges to each part of a bipartite graph does not decrease its energy. Let G be a connected graph of order n and size m with the adjacency matrix A. It is well-known that if G is a bipartite graph, then E(G) >= root 4m + n(n - 2)vertical bar det(A)vertical bar(2/n). Here, we improve this result by showing that the inequality holds for all connected graphs of order at least 7. Furthermore, we improve a lower bound for E(G) given in Oboudi (2019) [14]. (C) 2020 Elsevier Inc. All rights reserved.