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 $Gcong A_5$.
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