A lower bound for the energy of graphs in terms of the vertex cover number

Abstract

The energy of the graph G, denoted by E ( G ) , is the sum of the absolute values of its eigenvalues. Wang and Ma proved that if G has c odd cycles, then E ( G ) ≥ 2 ( β ( G ) − c ) , where β ( G ) is the vertex cover number of G. In this paper we strengthen this result by showing that if G and G ‾ have c o ( G ) and c o ( G ‾ ) numbers of induced odd cycles, respectively, then E ( G ) ≥ 2 ( β ( G ) − min ⁡ { c o ( G ) , c o ( G ‾ ) } ) and we conjecture that for every graph G, E ( G ) ≥ 2 β ( G ) . We prove the conjecture for some families of graphs, namely, bipartite graphs, C 4 -free regular graphs, perfect graphs, and for all graphs with β ( G ) ≤ | V ( G ) | 2 . It is shown that for every graph G, 2 ( β ( G ) − λ 1 ( G ‾ ) − λ n ( G ‾ ) ) ≤ E ( G ) , where G ‾ is the complement of G, λ 1 ( G ‾ ) and λ n ( G ‾ ) denote the largest and the smallest eigenvalues of the adjacency matrix of G ‾ , respectively. Using this we also prove that the conjecture holds for regular graphs with large degree.