Small graphs with exactly two non-negative eigenvalues

Document Type : Research Paper

Authors

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 λ1(G)≥⋯≥λn(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≤12 and λ1(G)≥0, λ2(G)≥0 and λ3(G)<0. We obtain that there are exactly 1575 connected graphs G on n≤12 vertices with λ1(G)>0, λ2(G)>0 and λ3(G)<0. We find that among these 1575 graphs there are just two integral graphs.

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References

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Algebraic Structures and Their Applications
Volume 4, Issue 1
February 2017
Pages 1-18

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