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Paper IPM / M / 8726  


Abstract:  
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(X(G),n −X(―G), where n is the number of vertices of G,
―G and X(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.
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