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arxiv: 2606.18061 · v2 · pith:GFPYQXUDnew · submitted 2026-06-16 · 🧮 math.CO

Principal minors of effective-resistance matrices and local resistance radii

Guangfu Wang This is my paper

Pith reviewed 2026-06-26 23:45 UTC · model grok-4.3

classification 🧮 math.CO
keywords effective resistance matrixprincipal minorsspanning forestsKron reductioncofactorsweighted graphsset functionsresistance radii
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The pith

The cofactor of any principal submatrix R[S] of the effective-resistance matrix equals a signed ratio of S-rooted forest to spanning-tree enumerators, while the normalized determinant reduces to a Kron-reduced trace and quadratic form.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper derives factorization formulas for the principal minors of the effective-resistance matrix on a finite connected weighted graph. For a vertex subset S the cofactor of R[S] is expressed as (-2) to the power |S|-1 times the S-rooted forest enumerator divided by the spanning tree enumerator. The ratio of determinant to cofactor is then given by two times the trace of the pseudoinverse of the Kron-reduced Laplacian plus half the quadratic form of its diagonal vector with the reduced Laplacian. Equivalently this ratio equals the maximum quadratic form value u transpose R[S] u over all vectors u that sum to one. The factorization separates the global counting contribution from the local boundary term obtained after Kron reduction.

Core claim

For every nonempty vertex set S the cofactor of the principal resistance submatrix R[S] factors as cof R[S] = (-2)^{|S|-1} κ_G(S) / τ(G) where τ(G) is the weighted spanning tree enumerator and κ_G(S) the weighted enumerator of S-rooted spanning forests. After Kron reduction to S producing the reduced Laplacian K with pseudoinverse Q and q equal to the diagonal of Q the normalized factor det R[S] / cof R[S] equals 2 over |S| times the trace of Q plus one half q transpose K q. This quantity is equivalently the maximum of u transpose R[S] u over all real vectors u on S with sum of entries equal to one. The identities follow from a principal specialization of known resistance-minor relations tog

What carries the argument

The Kron reduction of the graph Laplacian to the vertex set S, which produces a reduced Laplacian K whose pseudoinverse Q yields the local term in the normalized determinant via its trace and the quadratic form involving its diagonal.

If this is right

  • The normalized factor is monotone nondecreasing when the set S is enlarged.
  • An exact update formula exists for adding one vertex to S.
  • A support criterion determines when equality holds in the maximization characterization.
  • The resulting set function fails to be submodular or supermodular in general, as verified on small star graphs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The equivalence to a maximum over probability vectors suggests an interpretation of the normalized factor as an effective local resistance radius averaged over distributions on S.
  • The non-modularity result indicates that the set function does not satisfy the diminishing-returns property common in many combinatorial optimization settings.

Load-bearing premise

The formulas depend on the correctness of previously established resistance-minor identities that express cofactors in terms of forest enumerators.

What would settle it

Compute the effective-resistance matrix for the complete graph on three vertices with unit weights, take S as all three vertices, and check whether the ratio det R[S]/cof R[S] matches the Kron-reduced expression involving the reduced Laplacian on those vertices.

read the original 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 \[ \det R[S]/\cof R[S] =\frac{2}{|S|}\tr Q+\frac12 q^{\mathsf T}Kq. \] 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The paper derives explicit factorizations for the cofactor and the ratio det/cof of principal submatrices R[S] of the effective-resistance matrix of a finite connected weighted graph G. It shows cof R[S] = (-2)^{|S|-1} κ_G(S)/τ(G), where τ(G) is the spanning-tree enumerator and κ_G(S) the S-rooted forest enumerator. After Kron reduction to the reduced Laplacian K with pseudoinverse Q and q=diag(Q), the normalized factor is det R[S]/cof R[S] = (2/|S|) tr Q + (1/2) q^T K q. This quantity is equivalently the maximum of u^T R[S] u over probability vectors u. The paper derives monotonicity under enlargement of S, a one-point update formula, a support criterion, and shows via small star graphs that the induced set function is neither submodular nor supermodular.

Significance. If the derivations hold, the work supplies exact, parameter-free identities that cleanly separate combinatorial enumeration from boundary potential theory via Kron reduction, together with an optimization characterization of the normalized principal minor. The optimization perspective immediately yields monotonicity and an exact update rule. These results rest on standard Kron reduction (Schur complement) and known resistance-minor identities without circularity or fitted parameters, and the explicit counterexamples to sub/super-modularity clarify the analytic character of the set function.

minor comments (1)
  1. The abstract states the cofactor formula as a 'principal specialization of known resistance-minor identities'; a brief pointer to the precise prior identity (e.g., a cited theorem number) would help readers locate the specialization step.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation to accept the manuscript. No major comments were raised.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained on external identities

full rationale

The paper states that the cofactor formula is a principal specialization of known resistance-minor identities, with the new contribution being the Kron-reduction factorization of the normalized factor det R[S]/cof R[S] = (2/|S|) tr Q + (1/2) q^T K q after reducing to the Schur complement K. This uses the standard preservation property of Kron reduction (R = 1q^T + q1^T - 2Q) and the general matrix identity equating det(M)/(1^T adj(M)1) to max{u^T M u : 1^T u =1}. No step reduces by construction to a fitted input, self-definition, or load-bearing self-citation chain; the argument is independent of the target result and externally grounded.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

No free parameters are introduced; the work relies on standard domain assumptions from graph theory and prior results on resistance minors without adding new entities.

axioms (2)
  • domain assumption Properties of the graph Laplacian and its pseudoinverse define the effective resistance matrix R.
    Invoked in the definition of R and the Kron reduction to K = L^S.
  • standard math The weighted spanning tree enumerator τ(G) and S-rooted forest enumerator κ_G(S) are well-defined for finite connected weighted graphs.
    Used directly in the cofactor formula.

pith-pipeline@v0.9.1-grok · 5793 in / 1470 out tokens · 61148 ms · 2026-06-26T23:45:05.986119+00:00 · methodology

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Reference graph

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