A graph related to the join of subgroups of a finite group

Abstract

For a finite group $G$ different from a cyclic group of prime power order, we introduce an undirected simple graph $\Delta (G)$ whose vertices are the proper subgroups of $G$ which are not contained in the Frattini subgroup of $G$ and two vertices $H$ and $K$ are joined by an edge if and only if $G=⟨H,K⟩$ . In this paper we study $\Delta (G)$ and show that it is connected and determine the clique and chromatic number of $\Delta (G)$ and obtain bounds for its diameter and girth. We classify finite groups with complete graphs and also classify finite groups with domination number 1. Also we show that if the independence number of the graph $\Delta (G)$ is at most 7, then $G$ is solvable.