Abstract:Let $G$ be a finite connected weighted graph and let $R$ be its effective-resistance matrix. For every nonempty vertex set $S$, we factor the cofactor sum and determinant of the principal resistance submatrix $R[S]$ into an enumerative term and a boundary potential-theoretic term. If $\tau(G)$ is the weighted spanning tree enumerator and $\kappa_G(S)$ is the weighted enumerator of $S$-rooted spanning forests, then \[
\cof R[S]=(-2)^{|S|-1}\kappa_G(S)/\tau(G). \] After Kron reduction to $S$, with reduced Laplacian $K=L^S$, $Q=K^+$, and $q=\diag(Q)$, the remaining normalized factor is \[
\det R[S]/\cof R[S]
=\frac{2}{|S|}\tr Q+\frac12 q^{\mathsf T}Kq. \] Equivalently, this factor is the maximum of $u^{\mathsf T}R[S]u$ over all $u\in\R^S$ satisfying $\one^{\mathsf T}u=1$. This optimization viewpoint yields monotonicity under enlargement of $S$, an exact one-point update formula, and a support criterion for equality. Small star examples show that the resulting set function is neither submodular nor supermodular in general.
From: Guangfu Wang [view email]
[v1]
Tue, 16 Jun 2026 15:37:09 UTC (13 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。