g-noncommuting graph of some finite groups

M. Nasiri
Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran.
A. Erfanian
Department of Pure Mathematics and Center of Excellence in Analysis on Algebraic Structures, Ferdowsi University of Mashhad, Mashhad, Iran.
M. Ganjali
Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran.
A. Jafarzadeh
Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran.

\(^{1}\)Corresponding Author: mahnasiri@yahoo.com

Copyright © 2016 M. Nasiri, A. Erfanian, M. Ganjali, A. Jafarzadeh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Published: December, 2016.

Abstract

Let \(G\) be a finite non-abelian group and \(g\) a fixed element of \(G\). In 2014, Tolue et al. introduced the g-noncommuting graph of \(G\), which was denoted by \(Γ^{g}_G\) with vertex set \(G\) and two distinct vertices \(x\) and \(y\) join by an edge if \([x, y] \neq g\) and \(g^{−1}\). In this paper, we consider induced subgraph of \(Γ^{g}_{G}\) on \(G /Z(G)\) and survey some graph theoretical properties like connectivity, the chromatic and independence numbers of this graph associated to symmetric, alternating and dihedral groups.

Keywords:

Non-commuting graph, g-noncommuting graph, connected graph.