Some relations between rank, chromatic number and energy of graphs

The energy of a graph GG, denoted by E(G)E(G), is defined as the sum of the absolute values of all eigenvalues of GG. Let GG be a graph of order nn and rank(G) be the rank of the adjacency matrix of GG. 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 nn is the number of vertices of GG, G¯ and χ(G)χ(G) are the complement and the chromatic number of GG, respectively. Moreover some new lower bounds for E(G)E(G) in terms of rank(G) are given.