On graphs with at most two non-negative eigenvalues

‎Let $G$ be a graph with eigenvalues $\lambda_1(G)\geq\cdots\geq\lambda_n(G)$‎. ‎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 $\lambda_2(G)\geq 0$ and $\lambda_3(G) 0$‎. ‎In particular we characterize all graphs $G$ such that $\lambda_3(G) 0$