The complement of subgroup graph of a group

Abstract

Let GG be a finite group and HH a subgroup of GG. In 2012, David F. Anderson et al. introduced the subgroup graph of HH in GG as a simple graph with vertex set consisting all elements of G and two distinct vertices xx and yy are adjacent if and only if xy∈Hxy∈H. They denoted this graphby ΓH(G)ΓH(G). In this paper, we consider the complement of ΓH(G)ΓH(G), denoted by ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ΓH(G)ΓH(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.