Dutta, J., Nath, R. (2018). Laplacian and Signless Laplacian Spectrum of Commuting Graphs of Finite Groups. Khayyam Journal of Mathematics, 4(1), 77-87. doi: 10.22034/kjm.2018.57490

Jutirekha Dutta; Rajat Kanti Nath. "Laplacian and Signless Laplacian Spectrum of Commuting Graphs of Finite Groups". Khayyam Journal of Mathematics, 4, 1, 2018, 77-87. doi: 10.22034/kjm.2018.57490

Dutta, J., Nath, R. (2018). 'Laplacian and Signless Laplacian Spectrum of Commuting Graphs of Finite Groups', Khayyam Journal of Mathematics, 4(1), pp. 77-87. doi: 10.22034/kjm.2018.57490

Dutta, J., Nath, R. Laplacian and Signless Laplacian Spectrum of Commuting Graphs of Finite Groups. Khayyam Journal of Mathematics, 2018; 4(1): 77-87. doi: 10.22034/kjm.2018.57490

Laplacian and Signless Laplacian Spectrum of Commuting Graphs of Finite Groups

^{}Department of Mathematical Sciences, Tezpur University, Napaam-784028, Sonitpur, Assam, India.

Abstract

The commuting graph of a finite non-abelian group $G$ with center $Z(G)$, denoted by $\Gamma_G$, is a simple undirected graph whose vertex set is $G\setminus Z(G)$, and two distinct vertices $x$ and $y$ are adjacent if and only if $xy = yx$. A finite non-abelian group $G$ is called super integral if the spectrum, Laplacian spectrum and signless Laplacian spectrum of its commuting graph contain only integers. In this paper, we first compute Laplacian spectrum and signless Laplacian spectrum of several families of finite non-abelian groups and conclude that those groups are super integral. As an application of our results we obtain some positive rational numbers $r$ such that $G$ is super integral if commutativity degree of $G$ is $r$. In the last section, we show that $G$ is super integral if $G$ is not isomorphic to $S_4$ and its commuting graph is planar. We conclude the paper showing that $G$ is super integral if its commuting graph is toroidal.