Bounded Treewidth and Complete Monotonicity for Scott-Sokal Spanning-Tree Polynomials

Scott and Sokal asked for a structural description of the finite graphs \(G\) for which inverse powers \(T_G^{-\beta}\) of the spanning-tree polynomial are completely monotone.

We prove the following bounded-treewidth criterion: if \(G\) is a finite connected simple graph with \(\operatorname{tw}(G)\le k\), then \(T_G^{-\beta}\) is completely monotone for every \(\beta>(k-1)/2\).

Consequently, every partial \(3\)-tree is covered throughout the first Scott--Sokal open interval \(1<\beta<3/2\), including finite Apollonian networks, \(K_5-e\), and the four-spoke wheel \(W_4\).

The proof combines the real Riesz/Wishart integral for determinants, star--mesh elimination of simplicial vertices, and a Gaussian Laplace kernel for the degree-\(d\) star.

General bounded-treewidth graphs follow by chordal completion and monotone deletion of completion edges.