Let $G$ be a Cayley graph, and $H$ a graph cospectral with $G$. Must $H$ be a Cayley graph? Does a counterexample exist? If $G$ is a circulant graph, does a counterexample exist?



Graphs cospectral with Cayley graphs
To rectify joro's answer; he is certainly correct about the general direction, although it is not clear from his answer whether any of these 35-vertex SRGs is Cayley.


However, on 25 vertices there are a number of SRGs of degree 12, one of them is a certainly a Cayley graph (the Paley graph), and several SRGs with the same parameters (25,12,5,6) (hence the same spectrum) have intransitive automorphism group.


