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 Δ(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 Δ(G)≅Δ(H), then H is cyclic. Also we prove that Δ(G)≅Δ(A5) if and only if G≅A5. Keywords