Flows on Simplicial Complexes

TL;DR

This paper extends the concept of nowhere-zero flows from graphs to higher-dimensional simplicial complexes, proving the polynomial nature of the associated flow-counting function for specific cases.

Contribution

It introduces a new definition of flows on simplicial complexes and establishes polynomiality results for these flow functions in certain subclasses.

Findings

Flow functions are polynomial in certain cases.

Extension of flow concepts from graphs to simplicial complexes.

Polynomiality holds for specific subclasses and values of q.

Abstract

Given a graph $G$, the number of nowhere-zero $\ZZ_q$-flows $\phi_G(q)$ is known to be a polynomial in $q$. We extend the definition of nowhere-zero $\ZZ_q$-flows to simplicial complexes $\Delta$ of dimension greater than one, and prove the polynomiality of the corresponding function $\phi_{\Delta}(q)$ for certain $q$ and certain subclasses of simplicial complexes.