On Weighted Arboricity: Conductance-Resistance Bounds and Monoid Structure

Rowan Moxley

Paper

On Weighted Arboricity: Conductance-Resistance Bounds and Monoid Structure

Abstract

We study a conductance-weighted arboricity for a finite simple undirected graph

with a conductance assignment

This functional reduces to the classical Nash--Williams identity when

, is isomorphism invariant, monotone under subgraphs and edge additions, positively homogeneous, and convex. We prove sharp global bounds

with attainment by some connected subgraph. On the analytic side, we introduce a local variant and derive conductance--resistance inequalities using effective resistances in the ambient network. If

denotes the effective resistance between the endpoints of

, we show that every connected

satisfies

which in turn yields the upper bound

and hence an explicit effective resistance-based upper bound on

. On the structural side, we describe the algebraic behavior of

. We show that under edge-disjoint union,

behaves as a max invariant: for a finite disjoint union of weighted graphs one has

. In particular, disjoint union induces a commutative idempotent monoid structure at the level of isomorphism classes, with

idempotent with respect to this operation. We also provide a computational exhibit on the hypercube family

, including random conductance sampling, illustrating numerical evaluation of the resulting resistance-based bound.