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 $\mathrm{tw}(G)\le k$, then $T_G^{-\beta }$ is completely monotone for every $\beta >(k-1)\mathrm{/}2$. Consequently, every partial $3$-tree is covered throughout the first Scott–Sokal open interval $1<\beta <3\mathrm{/}2$, 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.