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 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.
Combinatorics Probability and Computing
Nikiforov, V. (2002). Some inequalities for the largest eigenvalue of a graph. Combinatorics Probability and Computing, 11 (2), 179-189. https://doi.org/10.1017/S0963548301004928