## Return to Question

3 added 220 characters in body

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

2 added table of values

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 $h(q,-1)$?In particular,

Update 10/24

$f(q,-1)$ is either of them easy to compute for arbitrary graphs?

Motivation: this comes up when trying to count the number of proper $q$ coloringsin the convex dual, $h(q,-1)$ 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

1

# Derivative of Tutte polynomial at -1

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 $h(q,-1)$? In particular, is either of them easy to compute for arbitrary graphs?

Motivation: this comes up when trying to count the number of proper colorings in the convex dual