Some Inequalities for the Largest Eigenvalue of a Graph

Let λ( G ) be the largest eigenvalue of the adjacency matrix of a graph G : We show that if G is K p +1 -free then This inequality was first conjectured by Edwards and Elphick in 1983 and supersedes a series of previous results on upper bounds of λ( G ). Let T i denote the number of all i -cliques of G , λ = λ( G ) and p = cl( G ): We show Let δ be the minimal degree of G . We show This inequality supersedes inequalities of Stanley and Hong. It is sharp for regular graphs and for a class of graphs which are in some sense maximally irregular.