On Co-Maximal Subgroup Graph of a Group

Abstract

The co-maximal subgroup graph $Γ(G)$ of a group $G$ is a graph whose vertices are non-trivial proper subgroups of $G$ and two vertices $H$ and $K$ are adjacent if $HK=G$. In this paper, we continue the study of $Γ(G)$, especially when $Γ(G)$ has isolated vertices. We define a new graph $Γ^*(G)$, which is obtained by removing isolated vertices from $Γ(G)$. We characterize when $Γ^*(G)$ is connected, a complete graph, star graph, has an universal vertex etc. We also find various graph parameters like diameter, girth, bipartiteness etc. in terms of properties of $G$.