Artin-Ihara L-functions for hypergraphs

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.