Further results on the join graph of a finite group

Abstract

Let G be a finite group which is not cyclic of prime power order. The join graph ∆(G) is an undirected simple whose vertices are the proper subgroups of G , which are not contained in the Frattini subgroup Φ(G) of G and two vertices H and K are joined by an edge if and only if G = ⟨H, K⟩. We classify finite groups whose join graphs have domination number ≤ 2 and independence number ≤ 3 . We show that ∆(G) ∼= ∆(A4) if and only if G ∼= A4 . We also show that if the independence number of ∆(G) is less than 15 , then G is solvable; moreover, if the equality holds and G is nonsolvable, then G/Φ(G) ∼= A5 .