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 it 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(chi (G). n - chi ((G) over bar)), where n is the number of vertices of G, (G) over bar and chi(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. (c) 2008 Elsevier B.V. All rights reserved.