Department of Pure Mathematics, Ferdowsi University of Mashhad (in cooperation with the Center of Excellence in Analysis on Algebraic Structures and Tusi Mathematical Research Group)Khayyam Journal of Mathematics2423-47884120180101Laplacian and Signless Laplacian Spectrum of Commuting Graphs of Finite Groups77875749010.22034/kjm.2018.57490ENJutirekhaDuttaDepartment of Mathematical Sciences, Tezpur University, Napaam-784028,
Sonitpur, Assam, India.Rajat KantiNathDepartment of Mathematical Sciences, Tezpur University, Napaam-784028,
Sonitpur, Assam, India.Journal Article20171214The 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 $Gsetminus Z(G)$, and two distinct vertices $x$ and $y$ are adjacent if and only if $xy = yx$.<br />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.<br />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<br />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.http://www.kjm-math.org/article_57490_293c3a034fe521dab3aecbbd7b850f8f.pdf