Nowhere-Zero $\vec k$-Flows on Graphs

TL;DR

This paper introduces a multivariate counting function for nowhere-zero flows on graphs with edge-specific capacities, showing it is a piecewise polynomial satisfying a reciprocity law involving cyclic orientations.

Contribution

It establishes that the counting function is a piecewise polynomial and reveals a reciprocity law connecting capacities and cyclic orientations.

Findings

Counting function is a piecewise polynomial.

Reciprocity law links capacities with cyclic orientations.

Provides new combinatorial insights into graph flows.

Abstract

We introduce and study a multivariate function that counts nowhere-zero flows on a graph G, in which each edge of G has an individual capacity. We prove that the associated counting function is a piecewise-defined polynomial in these capacities, which satisfy a combinatorial reciprocity law that incorporates totally cyclic orientations of G.