Extensions of Real-Weighted Fractional Arboricity

We study a conductance-weighted arboricity $A_c(G)$ for a finite simple undirected graph $G=(V,E,c)$ with a conductance assignment $c:E\to (0,\mathrm{\infty })$. This functional reduces to the fractional arboricity $\gamma (G)$ when $c\equiv 1$, is isomorphism invariant, monotone under taking subgraphs and adding edges, positively homogeneous, and convex. We also prove sharp global bounds with attainment at a connected subgraph. On the analytic side, we introduce a local variant and derive conductance–resistance inequalities using effective resistances in the ambient network, which in turn yields an upper bound 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 unions of graphs, $A_c(G)$ 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.