It's the first question I post here :) I'm sorry if the question is too specific or if it's somehow repeating others.

In other words, my question is the following. Consider a Cayley graph $\Gamma$ of a non-abelian group. Consider also the family $\mathcal{F}$ of Cayley graphs of abelian groups. Is there $\Gamma$ such that, for all $\Gamma~' \in \mathcal{F}$, $\Gamma$ is not isomorphic to $\Gamma~'$?

I've read some interesting posts such as:

- Non-isomorphic groups with the same oriented Cayley graph;
- Does a Cayley graph on a minimal symmetric set of generators determine a finite group up to isomorphism?.

However, I haven't made any progress towards the answer.

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