Derikvand, T., Oboudi, M. (2017). Small graphs with exactly two non-negative eigenvalues. Algebraic Structures and Their Applications, 4(1), 1-18. doi: 10.29252/asta.4.1.1

Tajedin Derikvand; Mohammad Reza Oboudi. "Small graphs with exactly two non-negative eigenvalues". Algebraic Structures and Their Applications, 4, 1, 2017, 1-18. doi: 10.29252/asta.4.1.1

Derikvand, T., Oboudi, M. (2017). 'Small graphs with exactly two non-negative eigenvalues', Algebraic Structures and Their Applications, 4(1), pp. 1-18. doi: 10.29252/asta.4.1.1

Derikvand, T., Oboudi, M. Small graphs with exactly two non-negative eigenvalues. Algebraic Structures and Their Applications, 2017; 4(1): 1-18. doi: 10.29252/asta.4.1.1

Small graphs with exactly two non-negative eigenvalues

^{1}Department of Mathematics, Marvdasht Branch, Islamic Azad University, Marvdasht, Iran

^{2}Department of Mathematics, College of Sciences, Shiraz University, Shiraz, 71457-44776, Iran

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.

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