ON THE GENUS OF THE COMMUTING GRAPHS OF FINITE NON-ABELIAN GROUPS

Year 2016, Volume: 19 Issue: 19, 91 - 109, 01.06.2016

Ashish Kumar Das Deiborlang Nongsiang

https://doi.org/10.24330/ieja.266195

Cited By: 14

Abstract

The commuting graph of a non-abelian group is a simple graph in
which the vertices are the non-central elements of the group, and two distinct
vertices are adjacent if and only if they commute. In this paper, we determine
(up to isomorphism) all finite non-abelian groups whose commuting graphs are
acyclic, planar or toroidal. We also derive explicit formulas for the genus of
the commuting graphs of some well-known class of finite non-abelian groups,
and show that, every collection of isomorphism classes of finite non-abelian
groups whose commuting graphs have the same genus is finite.

Keywords

Commuting graph, finite group, AC-group, genus of the commuting graphs