The Zeta Functions of Dihypergraphs and Dihypergraph Coverings

As the generalization of the digraph, the directed hypergraph (dihypergraph), denoted by (H) over right arrow, is considered in this paper. First, we introduce the notions of the zeta function of (H) over right arrow and dihypergraph coverings over (H) over right arrow. Next, several expressions for the zeta function of (H) over right arrow are given, and all dihypergraph coverings over (H) over right arrow are generated by permutation voltage assignments on the incidence bipartite digraph or the edge-colored digraph of (H) over right arrow. Final, we derive two explicit decomposition formulae for the zeta function of any dihypergraph covering <(<(H)over right arrow>)over bar> over (H) over right arrow, which turn out that the zeta function of (H) over right arrow divides the zeta function of <(<(H)over right arrow>)over bar>.