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 Oct 26 2010:** 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 Oct 24 2010**

$f(q,-1)$ 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:

```
computeH[graphName_, q_] := (
g = Normal[GraphData[graphName, "AdjacencyMatrix"]];
edges = Select[Position[g, 1], #[[1]] < #[[2]] &];
kr[x_, edge_] := Boole[x[[First[edge]]] == x[[Last[edge]]]];
term[x_] := Times @@ (1 + v kr[x, #] & /@ edges);
xs = Tuples[Range[q], Length[g]];
f = Total[term /@ xs];
D[f, v] /. v -> -1
);
TableForm[
Table[computeH[{#, n}, q], {q, 2, 8}, {n, 3, 6}]] & /@ {"Path",
"Cycle", "Complete"}
```