The complement of subgroup graph of a group

F. Kakeri
Ferdowsi University of Mashhad, International Campus, Mashhad, Iran.
A. Erfanian
Department of Pure Mathematics and Center of Excellence in Analysis on Algebraic Structures, Ferdowsi University of Mashhad, Mashhad, Iran.

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

Copyright © 2015 F. Kakeri, A. Erfanian. 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, 2015.

Abstract

Let \(G\) be a finite group and \(H\) a subgroup of \(G\). In 2012, David F. Anderson et al. introduced the subgroup graph of \(H\) in \(G\) as a simple graph with vertex set consisting all elements of G and two distinct vertices \(x\) and \(y\) are adjacent if and only if \(xy ∈ H\). They denoted this graphby \(Γ_H(G)\). In this paper, we consider the complement of \(Γ_H(G)\), denoted by \(\overline{Γ_H(G)}\) and will give some graph properties of this graph, for instance diameter, girth, independent and dominating sets, regularity. Moreover, the structure of this graph, planerity and 1-planerity are also investigated in the paper.