Extensions of Real-Weighted Fractional Arboricity: Conductance-Resistance Bounds and Monoid Structure

TL;DR

This paper extends the concept of fractional arboricity to conductance-weighted graphs, establishing bounds, inequalities, and algebraic structures, with applications to network analysis and graph theory.

Contribution

It introduces a conductance-weighted arboricity, derives sharp bounds, and explores its algebraic and resistance-based properties, extending classical fractional arboricity concepts.

Findings

Established sharp global bounds for conductance-weighted arboricity.

Derived conductance-resistance inequalities using effective resistances.

Showed that the measure forms a max-invariant monoid under disjoint union.

Abstract

We study a conductance-weighted arboricity for a finite simple undirected graph $G=(V,E,c)$ with a conductance assignment $c:E\to[0,\infty)$: \[ A_c(G):=\max\bigl\{ D_c(H): H\subseteq G\text{ connected}, |V(H)|\ge 2 \bigr\},\qquad D_c(H):=\frac{\sum_{e\in E(H)}c(e)}{|V(H)|-1}. \] This functional reduces to fractional arboricity when $c\equiv 1$, is isomorphism invariant, monotone under subgraphs and edge additions, positively homogeneous, and convex. We prove sharp global bounds \[ \max_{e\in E}c(e)\le A_c(G)\le\sum_{e\in E}c(e), \] 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 $R^{G}_{\mathrm{eff}}(e)$ denotes the effective resistance between the endpoints of $e$ in $G$, we show that every connected $H\subseteq G$ satisfies \[ \sum_{e\in E(H)}c(e)\,R^{G}_{\mathrm{eff}}(e)\le |V(H)|-1, \] which in turn yields the upper bound \[ D_c(H)\le\sqrt{\frac{\sum_{e\in E(H)}c(e)/R^{G}_{\mathrm{eff}}(e)}{|V(H)|-1}} \] and hence an explicit effective resistance-based upper bound on $A_c(G)$. On the structural side, we describe the algebraic behavior of $A_c(G)$. We show that under edge-disjoint union, $A_c(G)$ behaves as a max invariant: for a finite disjoint union of weighted graphs one has $A_c(G)= \max_i A_{c_i}(G_i)$. In particular, disjoint union induces a commutative idempotent monoid structure at the level of isomorphism classes, with $A_c(G)$ idempotent with respect to this operation. We also provide a computational exhibit on the hypercube family $Q_d$, including random conductance sampling, illustrating numerical evaluation of the resulting resistance-based bound.