On Co-Maximal Subgroup Graph of a Group

Abstract

The co-maximal subgroup graph $\Gamma (G)$ 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 $\Gamma (G)$, especially when $\Gamma (G)$ has isolated vertices. We define a new graph ${\Gamma }^{\ast }(G)$, which is obtained by removing isolated vertices from $\Gamma (G)$. We characterize when ${\Gamma }^{\ast }(G)$ 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 $G$.

14 pages, 3 figures

Cited by in corpus (2)

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