Bounds of graph energy in terms of vertex cover number

The energy E(G) of a graph G is the sum of the absolute values of all eigenvalues of G. In this paper, we give a lower bound and an upper bound for graph energy in terms of vertex cover number. For a graph G with vertex cover number tau, it is proved that 2 tau-2c <= E(G) <= 2 tau root Delta, where c is the number of odd cycles in G and is the maximum vertex degree of G. The lower bound is attained if and only if G is the disjoint union of some complete bipartite graphs with perfect matchings and some isolated vertices, the upper bound is attained if and only if G is the disjoint union of tau copies of K-1,(Delta) together with some isolated vertices. (C) 2016 Elsevier Inc. All rights reserved.