Some results on the join graph of finite groups

Abstract

‎Let $G$ be a finite group which is not cyclic of prime power order‎. ‎The join graph $Delta(G)$ 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 $G=langle H‎, ‎Krangle$‎. ‎Among other results‎, ‎we show that if $G$ is a finite cyclic group and $H$ is a finite group such that $Delta(G)congDelta(H)$‎, ‎then $H$ is cyclic‎. ‎Also we prove that $Delta(G)congDelta(A_5)$ if and only if $Gcong A_5$‎.