DefinePK

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.