Some Inequalities For The Largest Eigenvalue Of A Graph

Let (G) be the largest eigenvalue of the adjacency matrix of a graph G: We show that if G is Kp+1-free then (G) r 2 p 1 p e (G): This inequality was ...rst conjectured by Edwards and Elphick in 1983 and supersedes a series of previous results on upper bounds of (G) : Let Ti denote the count of all i-cliques of G; = (G) and p = cl (G) : We show p T2 p 2 + ::: + (i 1) Ti p i + ::: + (p 1) Tp: Let be the minimal degree of G: We show (G) 1 2 + s 2e (G) n + ( + 1)2 4 : 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. 1 

• Type: Journal
• Year: 2002
• Where: CPC
• Authors: Vladimir Nikiforov
• Keywords: Adjacency Matrix, CPC 2002, Largest Eigenvalue, Upper Bounds