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 \(\Gamma ^*(G)\), which is obtained by removing isolated vertices from \(\Gamma (G)\). We characterize when \(\Gamma ^*(G)\) 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.