Graphs with few positive eigenvalues
Subject Areas : Statistics
1 - Assistant Professor, Department of Mathematics, Faculty of Science, Shiraz University, Shiraz, Iran
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)
[1] A.E. Brouwer, W.H. Haemers, Spectra of graphs, Springer, New York, 2012.
[2] D. M. Cvetkovi'c, M. Doob and H. Sachs, Spectra of Graphs, Theory and Application, Academic Press, Inc., New York, 1979.
[3] M.R.Oboudi, Bipartite graphs with at most six non-zero eigenvalues, Ars Mathematica Contemporanea 11 (2016) 315-325.
[4] M.R. Oboudi, Characterization of graphs with exactly two non-negative eigenvalues, Ars Mathematica Contemporanea 12 (2017) 271-286 .
[5] M.R. Oboudi, On the third largest eigenvalue of graphs, Linear Algebra and its Applications: 503 (2016) 164-179.
[6] J. H. Smith, Symmetry and multiple eigenvalues of graphs, Glas. Mat., Ser. III 12 (1977) 3-8.
