On the distribution of eigenvalues of graphs

Abstract Let G a simple undirected graph with n ⩾ 2 vertices and let α 0 ( G ) ⩾ …, α n −1 ( G ) be the eigenvalues of the adjacency matrix of G . It is shown by Cao and Yuen (1995) that if α 1 ( G ) = − 1 then G is a complete graph, and therefore α 0 ( G ) = n − 1 and α i ( G ) = − 1 for 1 ⩽ i ⩽ n − 1. We obtain similar results for graphs whose complement is bipartite. We show in particular, that if the complement of G is bipartite and there exists an integer k such that 1⩽ k n − 1)/2 and α k ( G )=−1 then α i ( G )=−1 for k ⩽ i ⩽ n − k + 1. We also compare and discuss the relation between some properties of the Laplacian and the adjacency spectra of graphs.