some results on the join graph of finite groups

Let g be a finite group which is not cyclic of prime power order. the join graph ∆(g) of g is a graph whose vertex set is the set of all proper subgroups of g, which are not contained in the frattini subgroup g and two distinct vertices h and k are adjacent if and only if g = ⟨h, k⟩. among other results, we show that if g is a finite cyclic group and h is a finite group such that ∆(g) ∼= ∆(h), then h is cyclic. also we prove that ∆(g) ∼= ∆(a5) if and only if g ∼= a5.