Question

Let Tutte polynomial on graph with edge-set $E$ be defined as follows

$$f(q,v)=\sum_{A\subseteq E} q^{\kappa(A)} v^{|A|}$$

Here the sum is over all subgraphs $A$, $\kappa(A)$ is the number of connected components in $A$, $|A|$ is number of edges in $A$.

Let $$g(q,v)=\frac{\partial}{\partial v} \log f(q,v)$$ $$h(q,v)=\frac{\partial}{\partial v} f(q,v)$$

Is anything known about $g(q,-1)$ or $h(q,-1)$?

(Edit 10/26: actually it's the multivariate Tutte polynomial restricted to all $v$'s being equal, relation to regular Tutte polynomial in Jeremy Martin's answer)

Update 10/24

$f(q,-1)$ is the number of proper $q$ colorings, $h(q,-1)$ is ???, $g(q,-1)$ is their ratio. Below are tables of values of $h(q,-1)$ for paths, cycles, complete graphs. Columns give graph size, $n=3..6$, rows give $q=2..8$

Mathematica code to generate this

Answer 1

This polynomial isn't the usual Tutte polynomial, but it's equivalent. Provided that $G$ is connected (which is probably the case you're interested in, and I'll assume), it looks like $f_G(q,v)=qv^{n-1}T_G(q/v+1,v+1)$, where $n$ is the number of vertices of $G$.

One thing that comes to mind is the Crapo beta invariant, which is $(-1)^n\chi_G'(1)$, where $\chi_G(k)=(-1)^{n-1}kT_G(1-k,0)$ is the chromatic polynomial. (E.g., see exercise 22 of lecture 4 of Stanley's notes on hyperplane arrangements: http://www-math.mit.edu/~rstan/arrangements/arr.html.) Your polynomials, especially $g$, might be related to this, but I haven't thought about the details.