Small graphs with exactly two non-negative eigenvalues

Let G be a graph with eigenvalues \lambda_۱(G)\geq\cdots\geq\lambda_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 n\leq۱۲ and \lambda_۱(G)\geq۰, \lambda_۲(G)\geq۰ and \lambda_۳(G)<۰. We obtain that there are exactly ۱۵۷۵ connected graphs G on n\leq۱۲ vertices with \lambda_۱(G)>۰, \lambda_۲(G)>۰ and \lambda_۳(G)<۰. We find that among these ۱۵۷۵ graphs there are just two integral graphs.

Keywords:

Spectrum of graphs , Eigenvalues of graphs , Graphs with exactly two non-negative eigenvalues