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 τ, it is proved that 2τ−2c≤E(G)≤2τΔ, where c is the number of odd cycles in G and Δ 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 τ copies of K1,Δ together with some isolated vertices.