On Connectivity of Comaximal Subgroup Graph

The co-maximal subgroup graph $\mathrm{\Gamma }(G)$ of a finite group $G$ is defined to be a graph with the set of all non-trivial proper subgroups of $G$ as the set of vertices and two distinct vertices $H$ and $K$ are adjacent if and only if $HK=G$. The deleted co-maximal subgroup graph of $G$, denoted by ${\mathrm{\Gamma }}^{\ast }(G)$, is defined as the graph obtained by removing the isolated vertices from $\mathrm{\Gamma }(G)$. In this paper, we prove that for any finite group $G$, ${\mathrm{\Gamma }}^{\ast }(G)$ is connected. Furthermore, we show that ${\mathrm{\Gamma }}^{\ast }(G)$ either contains a cycle or is a star. When ${\mathrm{\Gamma }}^{\ast }(G)$ contains a cycle, its girth is either $3$ or $4$. Finally, we classify all finite groups $G$ for which ${\mathrm{\Gamma }}^{\ast }(G)$ is a star.