On the distribution of eigenvalues of graphs

Let G a simple undirected graph with n ⩾ 2 vertices and let α0(G) ⩾ …, αn−1(G) be the eigenvalues of the adjacency matrix of G. It is shown by Cao and Yuen (1995) that if α1(G) = − 1 then G is a complete graph, and therefore α0(G) = n − 1 and αi(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 αk(G)=−1 then αi(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.