Small graphs with exactly two non-negative eigenvalues

Abstract

Abstract. Let G be a graph with eigenvalues λ 1 (G) ≥ · · · ≥ λn (G). In this paper we find all simple graphs G such that G has at most twelve vertices and G has exactly two non-negative eigenvalues.In other words we find all graphs G on n vertices such that n ≤ 12 and λ 1 (G) ≥ 0, λ 2 (G) ≥ 0 and λ 3 (G) < 0. We obtain that there are exactly 1575 connected graphs G on n ≤ 12 vertices with λ 1 (G) > 0, λ 2 (G) > 0 and λ 3 (G) < 0. We find that among these 1575 graphs there are just two integral graphs.