Ramsey numbers, graph eigenvalues, and a conjecture of Cao and Yuan

Abstract

Let N be a positive integer and R(N,N) denote the Ramsey number (see [15] or [11]) such that any graph with at least R(N,N) vertices contains a clique with N vertices or an independent set with N vertices. We show that any graph G with order at least R(N,N) must have its Nth largest eigenvalue λN(G)≥−1, and that any graph G with order at least R(N+1,N+1) must have its Nth smallest eigenvalue λN′(G)≤0, where the bounds −1 and 0 are best possible. This reveals some connection between graph spectra and Ramsey numbers, and enables us to give bounds of some eigenvalues for graphs with order greater than certain numbers. Moreover, it leads to a disproof of a conjecture on limit points of graph eigenvalues posted by Cao and Yuan [3] in 1995.Finally we post some problems for further research.