TY - JOUR
ID - 57490
TI - Laplacian and Signless Laplacian Spectrum of Commuting Graphs of Finite Groups
JO - Khayyam Journal of Mathematics
JA - KJM
LA - en
SN -
AU - Dutta, Jutirekha
AU - Nath, Rajat Kanti
AD - Department of Mathematical Sciences, Tezpur University, Napaam-784028,
Sonitpur, Assam, India.
Y1 - 2018
PY - 2018
VL - 4
IS - 1
SP - 77
EP - 87
KW - Commuting graph
KW - spectrum
KW - integral graph
KW - finite group
DO - 10.22034/kjm.2018.57490
N2 - 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 $Gsetminus Z(G)$, 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 obtainsome 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.
UR - http://www.kjm-math.org/article_57490.html
L1 - http://www.kjm-math.org/article_57490_293c3a034fe521dab3aecbbd7b850f8f.pdf
ER -