Spectrum and energy of non-commuting graphs of finite groups

2020/02/24 by Fasfous, Walaa Nabil Taha, Nath, Rajat Kanti
#05C25 #05C50 #15A18 #20D60 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2002.10146

Abstract

Let G be a finite non-abelian group and Γ_nc(G) be its non-commuting graph. In this paper, we compute spectrum and energy of Γ_nc(G) for certain classes of finite groups. As a consequence of our results we construct infinite families of integral complete r-partite graphs. We compare energy and Laplacian energy (denoted by E(Γ_nc(G)) and LE(Γ_nc(G)) respectively) of Γ_nc(G) and conclude that E(Γ_nc(G)) ≤ LE(Γ_nc(G)) for those groups except for some non-abelian groups of order pq. This shows that the conjecture posed in [Gutman, I., Abreu, N. M. M., Vinagre, C. T.M., Bonifacioa, A. S and Radenkovic, S. Relation between energy and Laplacian energy, MATCH Commun. Math. Comput. Chem., 59: 343--354, (2008)] does not hold for non-commuting graphs of certain finite groups, which also produces new families of counter examples to the above mentioned conjecture.

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