Small graphs with exactly two non-negative eigenvalues

Abstract

Let $G$ be a graph with eigenvalues $lambda_1(G)geqcdotsgeqlambda_n(G)$. 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 $nleq12$ and $lambda_1(G)geq0$, $lambda_2(G)geq0$ and $lambda_3(G) 0$, $lambda_2(G)>0$ and $lambda_3(G)<0$. We find that among these $1575$ graphs there are just two integral graphs.