the operators $B_p:
\Undir_{n,k} \to \Undir_{n,k}$, $p = 1 {, \dots ,} k$, and the Laplace operator $\Delta: \Undir_{n,k} \to \Undir_{n,k}$ for undirected graphs exactly as in Definition \[Df:Laplace\].

For any $G \in \Undir_{n,k}$ consider the two-variable polynomial: $$\label{Eq:DefPotts}
Z_G(q,v) = \SumSub(q^{\beta_0(H)} v^{\#\text{of edges of $H$}};G).$$ called [*Potts partition function*]{}. It is related [@Sokal Eq.(2.26)] to the Tutte polynomial $T_G$ of the graph $G$ as $$T_G(x,y) = (x-1)^{-\beta_0(G)} (y-1)^{-n} Z((x-1)(y-1), y-1;G).$$ Values of $Z_G$ in some points have a special combinatorial interpretation, in particular

\[Pp:SpecVal\]

$$\begin{aligned}
Z_G(-1,1) &= (-1)^{\beta_0(G)} 2^{\#\text{\upshape of loops of $G$}} \#
\{\Phi \in \SSC_{n,k} \subset \Gamma_{n,k} \mid \lmod \Phi\rmod = G\}.\\
Z_G(-1,-1) &= (-1)^n \# \{\Phi \in \AC_{n,k} \subset \Gamma_{n,k} \mid
\lmod \Phi\rmod = G\}.
 \end{aligned}$$

(Recall that by $\SSC_{n,k}$ and $\AC_{n,k}$ we denote the sets of all strongly semiconnected and acyclic graphs in $\Gamma_{n,k}$, respectively.)

\[Cr:NumSSC\] For any graph $G \in \Upsilon_{n,k}$ one has $$\# \{\Phi \in \SSC_{n,k} \subset \Gamma_{n,k} \mid \lmod \Phi\rmod = G\} =
(-1)^{\beta_0(G)} Z_{\widehat G}(-1,1)$$ where $\widehat G$ is the graph $G$ with all the loops deleted.

The definition of the Potts partition function implies immediately that $Z_G(q,v) = (v+1)^{\#\text{of loops of $G$}} Z_{\widehat
G}(q,v)$.

Consider now the [*universal Potts partition 