Chromatic Polynomials of Circulant Graph With Two Parameters

Question

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

Answer 1

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 $\mathbb{Z}/p\mathbb{Z}$.)