On graphs with at most two non-negative eigenvalues

Let G be a graph with eigenvalues λ1(G) ≥ …. ≥ λn(G). In this paper we study the graphs that have at most two non-negative eigenvalues. We obtain a new characterization of graphs that have exactly two non-negative eigenvalues. In other words we find all graphs G with λ2(G) ≥ 0 and λ 3(G) < 0. In particular we characterize all graphs G such that λ 3(G) < 0.