This erroneous version of the wikipedia article for Cycle Graphs stated that the cycle graph of different groups could be the same. The immediate error is that it stated that the cycle graph of $\mathbb Z_2 \times \mathbb Z_8$ is the same as $\mathbb Z_{16}$.
My question is, in general, if a group $G$ and $H$ are not isomorphic, can they have isomorphic cycle graphs?
Let $p$ be an odd prime and let $G$ and $H$ be two non-isomorphic $p$-groups of order $p^a$ and of exponent $p$. (These always exists for $a\geq 3$.)
If I understood the definition correctly, $G$ and $H$ will both have the same cycle graph.