Some relations between rank, chromatic number and energy of graphs

The energy of a graph G, denoted by E(G), is defined as the sum of the absolute values of all eigenvalues of G. Let G be a graph of order n and rank(G) be the rank of the adjacency matrix of G. In this paper we characterize all graphs with E(G) = rank(G). Among other results we show that apart from a few families of graphs, E(G) ≥ 2max(χ(G), n − χ(G)), where n is the number of vertices of G, G and χ(G) are the complement and the chromatic number of G, respectively. Moreover some new lower bounds for E(G) in terms of rank(G) are given.