Cayley graphs on $Z_{11}$ and $Z_p$

Question

I want to find all cayley graphs on $Z_{11}$. I know how many connected cayley graphs exist but i want to find all of them, connected or not, to find their eigenvalues. I found some of them and a theorem about isomorphism of caykey graphs on $Z_p$, p is a prime number. Also I trid to work with GAP to construct these cayley graphs but I can't. I know cayley graphs on $Z_p$ are circulant. Is there any special property about the cayley graph on $Z_p$? Or is there any category for them? Thanks for your helping.

Answer 1

If you are interested in graphs (not digraphs), then the elements of the connection set must come in pairs, so you are only looking at subsets $$ C \subseteq \{\pm1, \pm2, \pm3, \pm4, \pm5\}. $$

Moreover, we know that the graph with connection set $C$ is isomorphic to the graph with connection set $kC$ ($k \ne 0$ and all calculations mod 11) so if the graph is not empty, we can assume that $\pm 1 \in C$.

So if $|C| = 1$, we get the $11$-cycle.

If $|C| = 2$, then either $C = \{\pm 1, \pm 2\}$ or $C = \{\pm 1, \pm 3\}$ (the choices $C = \{\pm 1, \pm4\}$ and $C = \{ \pm 1, \pm 5\}$ are each isomorphic to one of the previous ones.)

If $|C| > 2$ then the graph is the complement of one already found.

So you only have three graphs and their complements to check