Title

Some inequalities for the largest eigenvalue of a graph

Abstract

Let λ (G) be the largest eigenvalue of the adjacency matrix of a graph G. We show that if G is Kp+1-free then λ (G) ≤ √2 p-1/p e (G). 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 Ti denote the number of all i-cliques G, λ = λ (G) and p = cl (G). We show λp ≤ T2λp-2 + ... + (i-1) Tiλp-i + ... + (p-1) Tp. Let δ be the minimal degree of G. We show λ (G) ≤ δ-1/2 + √2e (G) - δn + (δ+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.

Publication Title

Combinatorics Probability and Computing

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