On the Distribution of Eigenvalues of Graphs

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 ≤ [n2 ],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. 1.Introduction. All graphs considered here are undirected and simple. Let G denote a graph with vertex set {v1, v2, · · · , vn}. Its adjacency matrix A(G) is the n × n one-zero matrix (aij), where aij=1 iff vi is adjacent to vj , and aij=0 otherwise. It is seen that A(G) is a symmetric (0,1) matrix with every diagonal entry equal to zero. We shall denote the characteristic polynomial of G by P (x,G) = det(xI −A(G)) = n ∑ i=0 aix n−i. Since A(G) is a real symmetric matrix, its eigenvalues, say λi(A(G))(i = 1, 2, · · · , n), are real numbers, and may be ordered as λ1(A(G)) ≥ λ2(A(G)) ≥ · · · ≥ λn(A(G)). Denote λi(A(G)) simply by λi(G). The sequence of n eigenvalues of G is known as the spectrum of G. Spectra of graphs appear frequently in the mathematical sciences. A good survey on this field can be found in [1]. The problem how to characterize a graph by the second eigenvalue has been considered by several authors([2∼ 5]). Dasong Cao and Hong Yuan showed that for a simple graph λ2(G) = −1 iff G is complete ([6]), they also established in [7]