Are These Graphs Circulant?

Question

We will say a circulant graph is a graph whose adjacency matrix is circulant (even if the graph is disconnected). Let $R$ be a Dedekind domain, and let $I$ be an ideal of $R$ such that $R/I$ is finite and nontrivial. Let $G_{R/I}$ be the unitary Cayley graph whose vertex set is $R/I$ and whose edge set is $E=\{\{X,Y\}\colon X-Y\in U(R/I)\}$, where $U(R/I)$ denotes the group of units of $R/I$. That is, two vertices are adjacent if and only if their difference is a unit in $R/I$. Is $G_{R/I}$ necessarily circulant? It seems as though this is the case, but I can't seem to be able to prove it. I have tried using the fact that we may write $I=P_1^{a_1}\cdots P_r^{a_r}$ for some prime ideals $P_1,\ldots,P_r$. By the Chinese Remainder Theorem, this shows that $G_{R/I}$ is a direct product (or tensor product) of complete $k_i$-partite graphs, where $k_i=\vert R/P_i\vert$ for $1\leq i\leq r$.

Answer 1

Check this paper: Energy of unitary Cayley graphs and gcd-graphs http://www.sciencedirect.com/science/article/pii/S0024379511002187