A graph related to the join of subgroups of a finite group. (Q396507)

For a finite group \(G\) different from a cyclic group of prime power order, the authors 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 \(G=\langle H,K\rangle\). In this paper they 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. The authors classify finite groups with complete graphs and also classify finite groups with domination number 1. Also it is proved that if the independence number of the graph \(\Delta(G)\) is at most 7, then \(G\) is solvable.