# How to determine when two circulant graphs are isomorphic

I came across one problem in GTM244 about the circulant graph:

A circulant is a Cayley graph $$CG(\mathbb{Z}_n,S)$$, where $$\mathbb{Z}_n$$ is the additive group of integers modulo $$n$$. Let $$p$$ be a prime, and let $$i$$ and $$j$$ be two nonzero elements of $$\mathbb{Z}_p$$.Determine when $$CG(\mathbb{Z}_p,\{1,-1,i,-i\})\cong CG(\mathbb{Z}_p,\{1,-1,j,-j\})$$

I only can show when $$i\equiv\pm j$$ or $$ij\equiv\pm 1$$, they are isomorphic but how can I find all possibilities? Intuitively, the $$p$$-cycle $$0-1-2-\cdots-p$$ and $$0-i-2i-\cdots-pi$$ should be mapped to $$0-1-2-\cdots-p$$ and $$0-j-2j-\cdots-pj$$ under an isomorphism.Is that the right idea? How to prove it? Thanks for any help.

• Circulant graphs of prime order are isomorphic iff they have the same eigenvalues. And also iff one connection set is a multiple of the other mod $p$. Sep 2 at 6:48
• I have read the related paper and got it. Thanks very much! Sep 4 at 8:27