On the distribution of eigenvalues of graphs

Abstract

Let G be a simple graph with n(≥ 2) vertices, and λ_i(G) be the ith largest eigenvalue of G. In this paper we obtain the following: If λ₃(G) < 0, and there exists some index $k,2 \leqslant k \leqslant [\frac{n}{2}]$,such that λ_k(G) = −1, then

$$\lambda _j (G) = - 1,j = k,k + 1, \cdots ,n - k + 1.$$

In particular, we obtain that (1) λ₂(G) = −1 implies

$$\lambda _1 (G) = n - 1,\lambda _j (G) = - 1,j = 2,3, \cdots ,n.$$

and therefore G is complete. This is a result presented in [6]; (2) λ₃(G) = −1 implies that λ_j (G) = −1, j = 3, 4,..., n −2.