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 ) ≥ 2 max ( 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.