Small Graphs With Exactly Two Nonnegative Eigenvalues

Let g be a graph with eigenvalues λ1(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 λ1(g) ≥ 0, λ2(g) ≥ 0 and λ3(g) < 0. we obtain that there are exactly 1575 connected graphs g on n ≤ 12 vertices with λ1(g) > 0, λ2(g) > 0 and λ3(g) < 0. we find that among these 1575 graphs there are just two integral graphs.