Let $G$ be a finite group which is not cyclic of prime power order. The join graph $\Delta(G)$ is an undirected simple whose vertices are the proper subgroups of $G$, which are not contained in the Frattini subgroup $\Phi(G)$ of $G$ and two vertices $H$ and $K$ are joined by an edge if and only if $G=\langle H,K\rangle$. We classify finite groups whose join graphs have domination number $\leq 2$ and independence number $\leq 3$. We show that $\Delta(G)\cong \Delta(A_4)$ if and only if $G\cong A_4$. We also show that if the independence number of $\Delta(G)$ is less than $15$, then $G$ is solvable; moreover, if the equality holds and $G$ is nonsolvable, then $G/\Phi(G)\cong A_5$.
BAHRAMI, ZAHRA and TAERI, BIJAN
"Further results on the join graph of a finite group,"
Turkish Journal of Mathematics: Vol. 43:
5, Article 2.
Available at: https://journals.tubitak.gov.tr/math/vol43/iss5/2