The Zeta Functions of Dihypergraphs and Dihypergraph Coverings

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