On the distribution of eigenvalues of graphs

Abstract

Let G be a simple graph with n(≥ 2) vertices, and λi(G) be the ith largest eigenvalue of G. In this paper we obtain the following: If λ3(G) < 0, and there exists some index k,2⩽k⩽[n/2],such that λk(G) = −1, then λj(G)=−1,j=k,k+1,...,n−k+1. In particular, we obtain that (1) λ2(G) = −1 implies λ1(G)=n−1,λj(G)=−1,j=2,3,...,n. and therefore G is complete. This is a result presented in [6]; (2) λ3(G) = −1 implies that λj (G) = −1, j = 3, 4,..., n −2.