Graphs with few positive eigenvalues

Subject Areas : Statistics

ءMohammad reza Aboudi ¹ ( Assistant Professor, Department of Mathematics, Faculty of Science, Shiraz University, Shiraz, Iran )

Received: 2019-02-23 Accepted : 2019-11-03 Published : 2020-12-21

Keywords: ماتریس اتصال گرافها, مقادیر ویژه گرافها, گراف,

Abstract :

Let G be a simple graph with vertices v_1,..., v_n. The adjacency matrix of G denoted by A(G) is an n×n matrix whose the entry (i,j) is 1 if v_i and v_j are adjacent and is zero otherwise. By the eigenvalues of G we mean the eigenvalues of A(G). Let λ_1 (G)≥λ_2 (G)≥⋯≥λ_n (G) be the eigenvalues of G. In this paper we obtain some results related to graphs with at most three non-negative eigenvalues. We obtain all non-connected graphs with this property. In addition, we find some families of connected graphs with this property. In particular we study two following families of graphs:1. Graphs such as G with exactly two positive eigenvalues and one zero eigenvalues. In other words graphs such as G with λ_1 (G)>0 , λ_2 (G)>0 , λ_3 (G)=0 and λ_4 (G)0 , λ_2 (G)>0 , λ_3 (G)>0 and λ_4 (G)

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