Journal of Prime Research in Mathematics

# The complement of subgroup graph of a group

F. Kakeri
$$^{1}$$Corresponding Author: fereshte.kakeri@yahoo.com
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.