@article { author = {Derikvand, Tajedin and Oboudi, Mohammad Reza}, title = {Small graphs with exactly two non-negative eigenvalues}, journal = {Algebraic Structures and Their Applications}, volume = {4}, number = {1}, pages = {1-18}, year = {2017}, publisher = {Yazd University}, issn = {2382-9761}, eissn = {2423-3447}, doi = {10.22034/as.2017.994}, abstract = {Let $G$ be a graph with eigenvalues $\lambda_1(G)\geq\cdots\geq\lambda_n(G)$. 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 $n\leq12$ and $\lambda_1(G)\geq0$, $\lambda_2(G)\geq0$ and $\lambda_3(G)<0$. We obtain that there are exactly $1575$ connected graphs $G$ on $n\leq12$ vertices with $\lambda_1(G)>0$, $\lambda_2(G)>0$ and $\lambda_3(G)<0$. We find that among these $1575$ graphs there are just two integral graphs.}, keywords = {Spectrum of graphs,Eigenvalues of graphs,Graphs with exactly two non-negative eigenvalues}, url = {https://as.yazd.ac.ir/article_994.html}, eprint = {https://as.yazd.ac.ir/article_994_708f0d4c89ce19056a0c89be6c5bc68f.pdf} }