A NEW CHARACTERIZATION OF GRAPHS THAT HAVE EXACTLY TWO NON-NEGATIVE EIGENVALUES

Let G be a graph with eigenvalues λ 1 (G) ≥ · · · ≥ λ n (G). In this paper we study the graphs G such that λ 3 (G) 0. We find a new characterization of graphs that have exactly two non-negative eigenvalues. In other words we characterize all graphs G such that λ 1 (G) ≥ 0, λ 2 (G) ≥ 0 and λ 3 (G) 0.