Artin-Ihara L-functions for hypergraphs

TL;DR

This paper extends the concept of Artin-Ihara L-functions from graphs to hypergraphs by leveraging associated bipartite graphs, establishing their properties and expressing hypergraph Ihara zeta functions as products of these L-functions.

Contribution

It introduces a novel framework for defining and analyzing Artin-Ihara L-functions for hypergraphs via bipartite graph associations, generalizing existing graph theory concepts.

Findings

Artin-Ihara L-functions for hypergraphs can be derived from bipartite graph L-functions.

The Ihara zeta function of a hypergraph factors into a product of Artin-Ihara L-functions.

Various properties of these L-functions for hypergraphs are established.

Abstract

We generalize Artin-Ihara L-functions for graphs to hypergraphs by exploring several analogous notions, such as (unramified) Galois coverings and Frobenius elements. To a hypergraph $H$, one can naturally associate a bipartite graph $B_H$ encoding incidence relations of $H$. We study Artin-Ihara $L$-functions of hypergraphs $H$ by using Artin-Ihara $L$-functions of associated bipartite graphs $B_H$. As a result, we prove various properties for Artin-Ihara L-functions for hypergraphs. For instance, we prove that the Ihara zeta function of a hypergraph $H$ can be written as a product of Artin-Ihara $L$-functions.