Let $p$ be a prime number greater than or equal to 11. Are there any cospectral non-isomorphic graphs with circulant graphs on $p$ vertices.
Which circulant graphs over prime number of vertices greater than or equal to 11 are determined by the spectrum?



Non-DS circulant graphs
No, and you don't need $p\ge 11$ either.


B Elspas, J Turner
Graphs with circulant adjacency matrices,
J. Combinatorial Theory, 9 (1970), pp. 297–307.


This paper shows that no two non-isomorphic circulant graphs on a prime number of vertices have the same spectrum. It doesn't say anything about circulant graphs cospectral with non-circulant graphs.


