Principal minors of effective-resistance matrices and local resistance radii

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 $τ(G)$ is the weighted spanning tree enumerator and $κ_G(S)$ is the weighted enumerator of $S$-rooted spanning forests, then \[ \cof R[S]=(-2)^{|S|-1}κ_G(S)/τ(G). \] After Kron reduction to $S$, with reduced Laplacian $K=L^S$, $Q=K^+$, and $q=\diag(Q)$, the remaining normalized factor is \[ \det R[S]/\cof R[S] =\frac{2}{|S|}\tr Q+\frac12 q^{\mathsf T}Kq. \] The cofactor factor is a principal specialization of known resistance-minor identities; the contribution here is the boundary/Kron-reduction factorization and local radius calculus. Equivalently, the normalized 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.