Let $p(x)$ be the chromatic polynomial of a special graph. Performing a certain type of operation on the graph changes $p$ by shifting it and adding a constant, say to: $q(x)=p(x+a) + b$.

I have noticed that this operation always results in $q$ having the same discriminant and splitting field as $p$. This would of course be obvious if we were just shifting $p$ by $a$, but it seems unusual given that we are also adding $b$.

I realise it is difficult to comment without knowing more details, but all I really want to know is whether or not this might be significant. The polynomials in question are irreducible cubics. How common is it that two cubics - which are not just shifted versions of each other - have the same discriminant? Does anybody have any idea what this might signify?

(edit: since posting this I have noticed that the chromatic polynomial in question is of the form $p(u,v,x)$, where $u$ and $v$ are given constants, and the discriminant of $p$ is a symmetric polynomial in $u$ and $v$. The graph operation switches $u$ and $v$...this is why the discriminant remains the same. So my question could be formulated as: what is the significance - if any - of a discriminant which is a symmetric polynomial in 2 variables?)