I have been working with the chromatic polynomials of *circulant graphs of prime order* $p$ with two distinct parameters, i.e. 


$P\_{p,i,j}(x):= P(C\_{p}(i,j),x)$ with $1 \leq i \neq j \leq \ n/2.$ 


In this regard, I have a couple of questions. Let $p$ be a prime number as above, and $1 \leq i' \neq j' \leq \ n/2$ and $1 \leq i \neq j \leq \ n/2$. Are the following conjectural identities true?


1. $P\_{p,i,j}(x) \stackrel{?}{=} P\_{p,i',j'}(x)$; (No. Counter example by Jeremy Martin: $C\_{11}(1,2)$ and $C\_{11}(1,3)$)
2. $P\_{p,i,j}(p) \stackrel{?}{\leq} e^2 \* p^p$.


These questions seem to answer in positive in experiments, but I have not found literature that helps me to prove (or disprove) these statements. Any literature suggestions or ideas on how to approach them would be very helpful. 


Best regards 



Chromatic Polynomials of Circulant Graph With Two Parameters
I assume that $C\_p(i,j)$ means the graph with vertices $0,\dots,p-1$ and edges between each pair of vertices with difference $i$ or $j$ mod $p$. If that is the case, then identity 1 does not appear to be true in general; Sage gives different chromatic polynomials for $C\_{11}(1,2)$ and $C\_{11}(1,3)$. (By the way, you may as well assume $i=1$, because $C\_p(i,j)\cong C\_p(1,j/i)$, with the quotient taken in $\mathbb{Z}/p\mathbb{Z}$.)


