The Zeta Functions of Dihypergraphs and Dihypergraph Coverings

Abstract

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

1 Introduction

Let $G=(V_G, E_G)$ be a finite graph or digraph. $\overleftrightarrow {G}$ is the symmetric digraph corresponding to G which is obtained from G by changing every undirected edge of G to a pair of oppositely directed edges (i.e., arcs), whereas $\overleftrightarrow {G}=G$ when G is a digraph. We denote the arc from the vertex u to the vertex v as ordered pair (u, v). If $e=(u, v)\in E_G$ is an arc, set $u=o_G(e)$ and $v=t_G(e)$, and $e^{-1}=(v, u)$ means the reverse arc to e. A path $P=(v_1, v_2, \ldots , v_t, v_{t+1})$ in G is a sequence such that consecutive vertices $v_i, v_{i+1}$ form an arc $e_i=(v_i, v_{i+1})$ in $\overleftrightarrow {G}$ for each $i=1, \ldots , t$. The length of P is $|P|=t$, the number of arcs in P. A path P is said to have a backtrack if $e_{i+1} = e_{i}^{-1}$ for some $i = 1, 2, \ldots , t-1$. Note that a path in G and the same path traversed in the opposite direction are different path in G. A path P is called to have a tail if $e_t = e_1^{-1}$. A path is called a cycle (or a closed path) if $v_1 = v_{t+1}$. It is noted that when G is a digraph, there are no path in G with a backtrack, and there are no path in G with a tail. A cycle is said to be reduced if it has no backtrack or tail. Let $D^r$ be the cycle obtained by going r times around a cycle D. A cycle C is prime if $C \ne D^r$ for any cycle D and any integer $r > 1$. Two cycles $C_1=(v_1, v_2, \ldots , v_t, v_{1})$ and $C_2=(v_{i_1}, v_{i_2}, \ldots , v_{i_t}, v_{i_1})$ are called equivalent if there exists an integer k such that $e_{i_j} = e_{j+k}$ for all j, i.e., $C_1$ is a cyclic permutation of $C_2$. For the cycle $C=(v_1, v_2, \ldots , v_t, v_{1})$, the equivalence class [C] of the cycle C means $[C]=\{(v_1, v_2, \ldots , v_t, v_{1})$, $(v_2, \ldots , v_t, v_{1}, v_{2}), \ldots , (v_{t}, v_1,$ $\ldots , v_{t-1}, v_{t})\}$.

The zeta function [1,2,3,4] of a finite graph(or digraph) G is defined to be the function of a complex number u with |u| sufficiently small, given by

$$\begin{aligned} \varsigma _G(u)=\prod _{[C]}(1-u^{|C|})^{-1}, \end{aligned}$$

where [C] runs over all equivalence classes of prime, reduced cycles of G.

Both the zeta function of a graph and the zeta function of a digraph have determinant formulae.

Theorem 1

( [5]) Let G be a graph with m edges and n vertices. The degree of any vertex in G is at least 2. Let A(G) and D(G) denote the adjacency matrix and the degree diagonal matrix of G, respectively. Then,

$$\begin{aligned} \varsigma _G(u)^{-1}=(1-u^2)^{m-n}\det (I_n-uA(G)+u^2Q(G)), \end{aligned}$$

where $I_n$ is the identity matrix with order n and $Q(G)=D(G)-I_n$.

Theorem 2

( [1, 6]) Let D be a connected digraph with n vertices. Then, the zeta function of D is given by

$$\begin{aligned} \varsigma _G(u)^{-1}=\det (I_n-uA(D)), \end{aligned}$$

where A(D) is the adjacency matrix of the digraph D.

Storm [7] introduced the following definition of the zeta function of a hypergraph, which can be viewed as the generalization of the zeta function of a graph: For $u\in {\mathbb {C}}$ (where ${\mathbb {C}}$ is the complex field) with |u| sufficiently small, the zeta function of a finite hypergraph H is defined by

$$\begin{aligned} \varsigma _H(u)=\prod _{[C]}(1-u^{|C|})^{-1}, \end{aligned}$$

where [C] runs over all equivalence classes of prime, reduced hypercycles of H. Storm also gave two explicit determinant expressions for the zeta function of a hypergraph utilizing Theorem 1 and Theorem 2:

Theorem 3

([7]) Let H be a finite connected hypergraph such that each vertex is in at least two hyperedges. Then, we have

$$\begin{aligned} \varsigma _H(u)^{-1}= & {} \det (I-u\mathbf {T}(H))\\= & {} \varsigma _{B_H}(\sqrt{u})^{-1}\\= & {} (1-u)^{|E_{B_H}|-|V_{B_H}|}\det (I_{|V_{B_H}|+|E_{B_H}|}-\sqrt{u}A(B_H)+uQ(B_H)), \end{aligned}$$

where $\mathbf {T}(H)$ is the square matrix which is referred to as the Perron–Frobenius operator of H and $B_H$ is the incidence graph of H.

We refer the reader to [7] for an in-depth treatment of the zeta functions of hypergraphs. In addition, Sato provided another determinant expression of the zeta function of a hypergraph in [8].

Let G be a finite graph. A homomorphism map $\pi$ from a graph $\bar{G}$ to G is called covering projection if it is a surjection $\pi : V_{\bar{G}}\rightarrow V_{G}$ such that $\pi |_{N({\widetilde{v}})}:N({\widetilde{v}})\rightarrow N(v)$ is a bijection for all vertices $v\in V_G$ and ${\widetilde{v}}\in \pi ^{-1}(v)$, where N(v), the neighborhood of the vertex v, is the set of vertices adjacent to v. The graph $\bar{G}$ is called a graph covering over G, and $\bar{G}$ is a k-fold covering if $\pi$ is k-to-one. Covering graphs have been studied in many literatures, such as [3, 9,10,11,12,13]. In particular, Terras showed the zeta function of a finite graph divides the zeta function of any covering over this graph in [4].

Let $S_k$ be the symmetric group on $[k]=\{1, 2, \ldots , k\}$. The function $\phi : E_{\overleftrightarrow {G}} \rightarrow S_k$ is called a permutation voltage assignment on G if $\phi (e)^{-1}=\phi (e^{-1})$ for each $e\in E_{\overleftrightarrow {G}}$. $\phi$ is called trivial if $\phi (e)^{-1}=\phi (e^{-1})=1$ for each $e\in E_{\overleftrightarrow {G}}$, where 1 is the identity of $S_k$. Otherwise, $\phi$ is nontrivial.

The derived graph $G^\phi$ associated with $\phi$ is a graph with the vertex set $V_{G^\phi }=V_G \times [k]=\{(v_i, s)|v_i\in V_G, s\in [k]\}$, and $(v_i, s)$ and $(v_j, t)$ are adjacent if and only if $e=(v_i, v_j)\in E_{\overleftrightarrow {G}}$ and $t=\phi (e)s$. Gross and Tucker showed the following corresponding relationship:

Lemma 4

[14] There is a one-to-one correspondence between each k-fold covering $\overline{G}$ over G and some derived graph $G^\phi$.

With this corresponding relationship, Mizuno and Sato gave an explicit composition formula for the zeta function of any graph covering over a finite graph in [15].

Digraph coverings over a finite digraph are also researched in recent years [16,17,18,19,20]. For a digraph $D=(V_D, E_D)$, $N_D^+(u)=\{e\in E_D|o_D(e)=u\}$ denotes the out-arc set of the vertex u, and $N_D^-(u)=\{e\in E_D|t_D(e)=u\}$ is the in-arc set of the vertex u. Let $\bar{D}$ and D be two digraphs. A digraph homomorphism from $\bar{D}$ to D is a pair of map $\pi =(\pi _0, \pi _1):(V_{\bar{D}},E_{\bar{D}})\rightarrow (V_{D},E_{D})$ such that $o_{D}(\pi _1(e))=\pi _0(o_{\bar{D}}(e))$ and $t_{D}(\pi _1(e))=\pi _0(t_{\bar{D}}(e))$ for each $e\in E_{\bar{D}}$. A digraph homomorphism $\pi =(\pi _0, \pi _1): \bar{D}\rightarrow D$ is a covering projection if $\pi _0:V_{\bar{D}}\rightarrow V_D$ and $\pi _1:E_{\bar{D}}\rightarrow E_D$ are both surjections and $\pi _1\mid _{N_{\bar{D}}^+({\widetilde{v}})}:N_{\bar{D}}^+({\widetilde{v}})\rightarrow N_{D}^+(v)$ and $\pi _1\mid _{N_{\bar{D}}^-({\widetilde{v}})}:N_{\bar{D}}^-({\widetilde{v}})\rightarrow N_{D}^-(v)$ are bijections for all vertices $v\in V_D$ and ${\widetilde{v}}\in \pi _0^{-1}(v)$. The digraph $\bar{D}$ is called a digraph covering (covering for short) over D. If $\pi$ is k-to-one, $\bar{D}$ is a k-fold covering. The function $\phi : E_{D} \rightarrow S_k$ is called a permutation voltage assignment on D. $\phi$ is called trivial if $\phi (e)=1$ for each arc $e\in E_{D}$. Otherwise, $\phi$ is nontrivial. The pair $(D, \phi )$ is called a permutation voltage graph. A derived digraph $D^\phi$ of $(D, \phi )$ is a digraph with the vertex set $V_{D^\phi }=V_D \times [k]=\{(v_i, s)|v_i\in V_D, s\in [k]\}$, and there is an arc from $(v_i, s)$ to $(v_j, t)$ if and only if $e=(v_i, v_j)\in E_{D}$ and $t=\phi (e)s$. The corresponding relationship between a covering over D and a derived graph $D^\phi$ of $(D, \phi )$ is obtained in [17].

Lemma 5

[17] Let $\pi :\bar{D}\rightarrow D$ be a k-fold covering over a digraph D. Then, there exists a permutation voltage assignment $\phi :E_D\rightarrow S_k$ such that the digraph covering $(\bar{D},\pi )$ is just $(D^{\phi }, \pi _{\phi })$ up to a digraph isomorphism.

Using Lemma 5, the zeta function of any digraph covering is formulated in [21].

Recently, in [22] the definition of hypergraph coverings over a hypergraph is introduced and decomposition formulae for the zeta function of any hypergraph covering are obtained, which is considered the generalization of graph coverings.

Motivated by above results, in this paper, we start from the zeta function of a directed hypergraph (dihypergraph), which can be viewed as the generalization of a digraph. In Sect. 2 we introduce the definition of the zeta function of a finite dihypergraph, and several expressions for the zeta function of a dihypergraph are obtained. In Sect. 3, we generalize the definition of digraph coverings over a finite digraph to dihypergraphs. And by permutation voltage assignments on the incidence bipartite digraph and the edge-colored digraph of a finite dihypergraph, all dihypergraph coverings are generated. Final, based on the representation theory of symmetric group we find the zeta function of any dihypergraph covering can be expressed by the zeta function of this dihypergraph in Sect. 4.

2 The Zeta Function of a Dihypergraph

A finite dihypergraph [23] is an ordered pair: ${H}=(V_{{H}}, E_{{H}}=\{{e_i}:i\in I\}),$ where $V_{{H}}$ is a finite set of vertices and $E_{{H}}$ is a set of hyperarcs with finite index set I. The repeated hyperarcs are possible. Each hyperarc ${e_i}$ is an ordered pair ${e_i}=(e_i^{+}, e_i^{-}),$ where $e_i^{+}\subseteq V_{{H}}$ is called the tail of the hyperarc ${e_i}$ and $e_i^{-}\subseteq V_{{H}}$ is the head of ${e_i}$. The set of tails is denoted by $E_{{H}}^+$ and the set of heads is denoted by $E_{{H}}^-$. The vertices of ${e_i}$ are denoted by $e_i=e_i^+\cup e_i^-$ and $E_{H}=(e_i)_{i\in I}$. The hypergraph $H=(V_{{H}}, E_{H})$ is called the underlying hypergraph of ${H}=(V_{{H}}, E_{{H}})$. We suppose that, for all ${e}_i=(e_i^{+}, e_i^{-})\in E_{{H}}$, $e_i^{+}\cap e_i^{-}=\emptyset$, $e_i^{+}\ne \emptyset$ and $e_i^{-}\ne \emptyset$.

To define the zeta function of a dihypergraph, we begin by extending what we mean by an equivalence class [C] of a prime hypercycle. A hyperpath in ${H}$ from the vertex $v_1$ to the vertex $v_{t+1}$ in ${H}$ is a sequence $P=(v_1, {e_1}, v_2, {e_2}, v_3,$ $\ldots , v_t, {e_t}, v_{t+1})$ such that $v_1\in e_1^+, v_{t+1}\in e_t^-$ and $v_i\in e_{i-1}^-\cap e_i^+$ for any $i=2, \ldots , t.$ The integer t is the length of hyperpath P which is denoted by $|P|=t$. If $v_1=v_{t+1}$ the hyperpath P is said to be a hypercycle (or closed hyperpath) in ${H}$. Let $C^m$ denote the hypercycle obtained by going m times around a hypercycle C. A hypercycle C is said to be prime if $C\ne B^m$ for any hypercycle B and any integer $m\ge 2.$ Two hypercycles $C=(v_1, {e_1}, v_2, {e_2}, v_3,$ $\ldots , v_t, {e_t}, v_{1})$ and ${\widetilde{C}}=(v_{i_1}, {e_{i_1}}, v_{i_2}, {e_{i_2}}, v_{i_3},\ldots , v_{i_t},$ ${e_{i_t}}, v_{i_1})$ are called equivalent if there exists some integer k such that $v_{i_j}=v_{j+k}$ and ${e}_{i_j}={e}_{j+k}$ for all j. $[C]=\{(v_1, {e_1}, v_2, {e_2}, v_3,\ldots , v_t, {e_t}, v_{1}), (v_2, {e_2}, v_3, {e_3},$ $\ldots , {e_t}, v_{1}, {e}_1, v_2),\ldots , (v_t, {e_t}, v_1, {e_1},\ldots , v_{t-1}, {e_{t-1}}, v_t)\}$ is called the equivalence class containing the hypercycle C.

We define the generalized Ihara–Selberg zeta function (zeta function for short) of a dihypergraph as follows:

Definition 6

Let ${H}$ be a finite dihypergraph. For a complex variable u with |u| sufficiently small, the generalized Ihara–Selberg zeta function of ${H}$ is defined by

$$\begin{aligned} \varsigma _{{H}}(u)=\prod _{[C]}(1-u^{|C|})^{-1}, \end{aligned}$$

where [C] runs all over equivalence classes of prime hypercycles of ${H}.$

Next, we will present several expressions for the zeta function of a dihypergraph, in which the determinants of explicit matrices may be involved.

Theorem 7

Let ${H}$ be a finite dihypergraph, and $N_m$ the number of all hypercycles with length m in ${H}$. Then, the zeta function of ${H}$ is given by

$$\begin{aligned} \varsigma _{{H}}(u)=\exp \left( \sum _{m\ge 1}\frac{N_m}{m}u^m\right) . \end{aligned}$$

Proof

Since $|[C]|=|C|$ and $-\log (1-u^{|C|})=\sum _{m\ge 1}\frac{1}{m}u^{|C|m}$, we have

$$\begin{aligned} \varsigma _{{H}}(u)= & {} \prod _{[C]}(1-u^{|C|})^{-1}\\= & {} \exp (-\sum _{[C]}\log (1-u^{|C|}))\\= & {} \exp \left( \sum _{[C]}\sum _{m\ge 1}\frac{1}{m}u^{|C|m}\right) \\= & {} \exp \left( \sum _{m\ge 1}\sum _{C}\frac{1}{|C|m}u^{|C|m}\right) \\= & {} \exp \left( \sum _{m\ge 1}\frac{N_m}{m}u^m\right) . \end{aligned}$$

$\square$

A dihypergraph can be converted into its incidence bipartite digraph, edge-colored digraph and oriented line graph in order to represent this dihypergraph.

The incidence bipartite digraph $B_{{H}}$ of the dihypergraph ${H}$ is a directed bipartite graph (where a digraph is bipartite if the underlying graph is bipartite) with the vertex set $V_{B_{{H}}}=V_{{H}}\cup E_{{H}}$ and the arc set $E_{B_{{H}}}= \{(v_i, {e_j}): v_i\in e_j^+\}\cup \{({e_j}, v_i): v_i\in e_j^-\}$. An example of the incidence bipartite digraph $B_{{H}}$ of ${H}$ is shown in Figure 1. The relationship between a dihypergraph and its incidence bipartite digraph is the following:

Lemma 8

Let ${H}$ be a finite dihypergraph with the incidence bipartite digraph $B_{{H}}$. Then, there exists a one-to-one correspondence between prime hypercycles of length t in ${H}$ and prime cycles of length 2t in $B_{{H}}.$

Fig. 1

The dihypergraph ${H}$ and its incidence bipartite digraph $B_{{H}}$

Proof

Let $C_{{H}}=\{v_1, {e_1}, v_2, {e_2}, v_3,$ $\ldots , v_t, {e_t}, v_{1} \}$ be a hypercycle in ${H}$. Then we can see that $C_{B_{{H}}}=\{v_1, {e_1}, v_2, {e_2}, v_3,$ $\ldots , v_t, {e_t}, v_{1} \}$ is a cycle in $B_{{H}}$ with length 2t. Also, it is clear that $C_{B_{{H}}}$ in $B_{{H}}$ is prime if $C_{{H}}$ in ${H}$ is prime. Note that $|C_{B_{{H}}}|=2|C_{{H}}|=2t.$ Thus, each prime hypercycle of length t in ${H}$ corresponds to a prime cycle of length 2t in $B_{{H}}$. On the other hand, with the similar method, we can see that for a given prime cycle in $B_{{H}}$, it corresponds to a prime hypercycle of half the length in ${H}$. $\square$

Theorem 9

Let ${H}$ be a finite dihypergraph. Then, the zeta function of ${H}$ is given by

$$\begin{aligned} \varsigma _{{H}}(u)=\varsigma _{B_{{H}}}(\sqrt{u})=\det (I-\sqrt{u}A(B_{{H}}))^{-1}. \end{aligned}$$

Proof

Let $[C_{{H}}]$ be an equivalence class of prime hypercycles in ${H}$ and $[C_{B_{{H}}}]$ an equivalence class of prime cycles in $B_{{H}}$, respectively. By using Lemma 8 and Theorem 2, we obtain

$$\begin{aligned} \varsigma _{{H}}(u)= & {} \prod _{[C_{{H}}]}(1-u^{|C_{{H}}|})^{-1}\\= & {} \prod _{[C_{{H}}]}(1-u^{|\frac{2C_{{H}}}{2}|})^{-1} \\= & {} \prod _{[C_{B_{{H}}}]}(1-\sqrt{u}^{|C_{B_{{H}}}|})^{-1}\\= & {} \varsigma _{B_{{H}}}(\sqrt{u})\\= & {} \det (I-\sqrt{u}A(B_{{H}}))^{-1}. \end{aligned}$$

$\square$

The edge-colored digraph $G{H}_c=(V_{G{H}_c}, E_{G{H}_c})$ of ${H}$ is constructed as follows: We label the hyperarcs of ${H}$: $E_{{H}}=\{{e_1}, {e_2}, \ldots ,$ ${e_m}\}$ and fix m colors $\{ c_1, c_2, \ldots , c_m\}$ for each hyperarc. The vertex set $V_{G{H}_c}=V_{{H}}.$ For each hyperarc ${e_i}\in E_{{H}}$, we establish a complete bipartite digraph (where a digraph is complete bipartite if the underlying graph is) ${C}_i=(e_i^+, e_i^-)$ with partition $e_i^+\cup e_i^-$ by joining the arc from u to v for each pair of vertices $u\in e_i^+$ and $v\in e_i^-$. Then, color the arcs of ${C}_i$ with color $c_i$. Obviously, $G{H}_c$ is composed of m complete bipartite digraph ${C}_i$, $i\in [m]$, and we have colored arcs depend upon which hyperarc they are induced from. Once $G{H}_c$ is established, the oriented line graph ${H}_L=(V_{{H}_L}, E_{{H}_L})$ of ${H}$ is constructed by $V_{{H}_L}=E_{G{H}_c}$ and $E_{{H}_L}=\{(\alpha _i, \alpha _j)\in E_{G{H}_c}\times E_{G{H}_c}: t(\alpha _i)=o(\alpha _j)\},$

The edge-colored digraph $G{H}_c$ and the oriented line graph ${H}_L$ of ${H}$ being given in Fig. 1 are exhibited in Fig. 2, respectively. Note that the construction of ${H}_L$ has nothing to do with the color of arcs in $G{H}_c$.

Fig. 2

The edge-colored digraph $G{H}_c$ and the oriented line graph ${H}_L$ of ${H}$ depicted in Fig. 1

There is the following correspondence between a dihypergraph and its oriented line graph:

Lemma 10

Let ${H}$ be a finite dihypergraph with the oriented line graph ${H}_L$. Then, there is a one-to-one correspondence between equivalence classes of prime hypercycles of length t in ${H}$ and equivalence classes of prime cycles of length t in ${H}_L.$

Proof

Suppose that $u\in e^+$ and $v\in e^-$ are two vertices contained in the hyperarc ${e}\in E_{{H}}.$ Then, (u, v) forms an arc in $G{H}_c$. Let $C_{{H}}=(v_1, {e_1}, v_2, {e_2}, \ldots , v_t, {e_t}, v_1)$ be a prime hypercycle in ${H}$. Then, there exists a prime cycle $C_{G{H}_c}=((v_1, v_2), (v_2, v_3), \ldots , (v_t, v_1))$ in $G{H}_c$ corresponding to the hypercycle $C_{{H}}$ of ${H}$. From the definition of the edge-colored digraph, consequently, the corresponding cycle $C_{{H}_L}=(((v_1, v_2), (v_2, v_3)), ((v_2, v_3), (v_3, v_4)),$ $\ldots ,((v_{t-1}, v_t), (v_t, v_1)))$ in ${H}_L$ is a prime cycle with length t.

Similarly, for any prime cycle $C_{{H}_L}$ in ${H}_L$, there exists a prime hypercycle $C_{{H}}$ in ${H}$ corresponding to $C_{{H}_L}$ and $C_{{H}}$ is the same length as $C_{{H}_L}$. $\square$

By using Lemma 10 and Theorem 2, a determinant expression of the zeta function of a dihypergraph is obtained immediately as follows:

Theorem 11

Let ${H}$ be a finite dihypergraph and ${H}_L$ its oriented line graph. Then, the zeta function of ${H}$ is given by

$$\begin{aligned} \varsigma _{{H}}(u)=\det (I-uA({H}_L))^{-1}. \end{aligned}$$

If ${H}$ degenerates into a digraph, it is apparent that Theorem 11 turns into the case of the oriented line graph of a graph which is characterized in [1].

3 Dihypergraph Coverings Over a Dihypergraph

In this section, we will give the concept of dihypergraph coverings over a finite dihypergraph, and generate all dihypergraph coverings.

Let ${H}=(V_{{H}}, E_{{H}})$ be a finite dihypergraph(we allow hyperarcs to repeat). For each vertex $v\in V_{{H}}$, we assume that $E_{{H}}^+(v)=\{{e}=(e^+, e^-)\in E_{{H}}:v\in e^+\}$ and $E_{{H}}^-(v)=\{{e}=(e^+, e^-)\in E_{{H}}:v\in e^-\}$.

Fig. 3

The dihypergraph ${H}$ and its two 2-fold dihypergraph coverings are showed on the left, middle and right of this figure, respectively, where $v_i^j=(v_i, j)$

A homomorphism from a finite dihypergraph $\overline{{H}}$ to ${H}$ is a pair of maps $\pi =(\pi _0, \pi _1): (V_{\overline{{H}}}, E_{\overline{{H}}})\rightarrow (V_{{H}}, E_{{H}})$ which preserves vertices, hyperarcs, and direction, i.e., for all ${e}=(e^+, e^-)\in E_{\overline{{H}}}$, we have $\pi _1({e}~)=(\pi _0(e^+), \pi _0(e^-))\in E_{{H}}$. A homomorphism map $\pi =(\pi _0, \pi _1): \overline{{H}}\rightarrow {H}$ is called a covering projection if $\pi _0: V_{\overline{{H}}}\rightarrow V_{{H}}$ and $\pi _1: E_{\overline{{H}}}\rightarrow E_{{H}}$ are surjections such that $\pi _1|_{E_{\overline{{H}}}^+({\widetilde{v}})}: E_{\overline{{H}}}^+({\widetilde{v}})\rightarrow E_{{H}}^+(v)$ and $\pi _1|_{E_{\overline{{H}}}^-({\widetilde{v}})}: E_{\overline{{H}}}^-({\widetilde{v}})\rightarrow E_{{H}}^-(v)$ are bijections for all vertices $v\in V_{{H}}$ and ${\widetilde{v}}\in \pi _0^{-1}(v)$. The dihypergraph $\overline{{H}}$ is called a dihypergraph covering over H, or a covering for short. If $\pi$ is k-to-one, we call $\overline{{H}}$ a k-fold dihypergraph covering. From the definition of the dihypergraph covering, it is obviously that the covering over ${H}$ is not unique, and each vertex v of ${H}$ and each hyperarc ${e}$ of ${H}$ have k vertices and k hyperarcs in their respective preimages $\pi _0^{-1}(v)$ and $\pi _1^{-1}({e}~)$ when $\overline{{H}}$ is a k-fold dihypergraph covering. For an illustration example of a dihypergraph ${H}$ and two 2-fold dihypergraph coverings over ${H}$, see Fig. 3.

All dihypergraph coverings can be generated by the representation graphs of dihypergraphs. First, we establish a representation to all k-fold dihypergraph coverings over a finite dihypergraph utilizing the incidence bipartite digraph. The vertex set and the hyperarc set of ${H}$ are denoted by $V_{{H}}=\{v_1, v_2, \ldots , v_n\}$ and $E_{{H}}=\{{e}_1, {e}_2, \ldots , {e}_m\}$, respectively. Suppose that $\phi : E_{B_{{H}}}\rightarrow S_k$ is a permutation voltage assignment on the incidence bipartite digraph $B_{{H}}$ of ${H}$. From Lemma 5, the derived digraph $B_{{H}}^\phi$ of $(B_{{H}}, \phi )$ brings into a correspondence with some k-fold covering over $B_{{H}}$. The derived dihypergraph ${H}^{B_{{H}}^{\phi }}$ associated with $B_{{H}}^\phi$ is constructed as follows: The vertex set $V_{{H}^{B_{{H}}^{\phi }}}=\{(v_i, t)|v_i\in V_{{H}}, t\in [k]\}$. There is a hyperarc ${e}=\{(e^+, e^-)|(v_i, t)\in e^+$ and $(v_j, s)\in e^- \}$ in ${H}^{B_{{H}}^{\phi }}$ if and only if there exists some vertex $({e}_l, h)\in V_{B_{{H}}^\phi }$ such that any vertex $(v_i, t)\in e^+$ satisfies $\phi ((v_i, {e}_l))t=h$ and each vertex $(v_j, s)\in e^-$ satisfies $\phi (({e}_l, v_j))h=s$. Intuitively speaking, for each vertex $({e}_l, h)\in V_{B_{{H}}^\phi }$, all vertices flowing into $({e}_l, h)$ in $B_{{H}}^\phi$ form the tail of a hyperarc of ${H}^{B_{{H}}^{\phi }}$ and all vertices flowing out $({e}_l, h)\in V_{B_{{H}}^\phi }$ consist of the head of this hyperarc. Accordingly, the digraph $B_{{H}}^{\phi }$ is the incidence bipartite digraph of ${H}^{B_{{H}}^{\phi }}$ if the hyperarc of ${H}^{B_{{H}}^{\phi }}$ with the tail $(v_i, t)$ and the head $(v_j, s)$ is labeled to $({e}_l, h)$. On the other hand, it is easy to check that ${H}^{B_{{H}}^{\phi }}$ is a k-fold dihypergraph covering over ${H}$. Thus, for each permutation voltage assignment $\phi$ on $B_{{H}}$, there must exist a k-fold dihypergraph covering $\overline{{H}}$ over ${H}$ corresponding to ${H}^{B_{{H}}^{\phi }}$. However, different permutation voltage assignments on $B_{{H}}$ may generate the same derived dihypergraph ${H}^{B_{{H}}^{\phi }}$ because the vertex $({e}_l, h)$ in $B_{{H}}^{\phi }$ transforms into a hyperarc when $B_{{H}}^{\phi }$ acts as a graph representation of ${H}^{B_{{H}}^{\phi }}$. Therefore, each dihypergraph covering over ${H}$ can be generated by at least one permutation voltage assignment $\phi$ on $B_{{H}}$.

Fig. 4

The dihypergraph ${H}$, the incidence bipartite digraph $B_{{H}}$ of ${H}$, and a 2-fold dihypergraph covering $\overline{{H}}$ over ${H}$. $B^{\phi _1}_{{H}}$ and $B^{\phi _2}_{{H}}$ can be regarded as the incidence bipartite digraph of $\overline{{H}}$, where $v_i^j=(v_i, j)$ and ${e}_i^{~j}=({e}_i, j)$

Example 12

In Fig. 4, two distinct permutation voltage assignments $\phi _1$ and $\phi _2$ on $B_{{H}}$ induce to the same 2-fold dihypergraph covering $\overline{{H}}$ over ${H}$, where $\phi _1(({e}_1, v_2))=\phi _1(({e}_2, v_4))$ $=\phi _1((v_4, {e}_3))=\phi _1(({e}_3, v_3))=1$, $\phi _1(({e}_1, v_1))=\phi _1((v_3, {e}_1 ))=\phi _1((v_1, {e}_2))=\phi _1(({e}_3, v_2))=(12),$ and $\phi _2(({e}_1, v_1))=\phi _2((v_3, {e}_1 ))=\phi _2((v_1, {e}_2))=\phi _2(({e}_3, v_2))=1,$ $\phi _2(({e}_3, v_3))=\phi _2(({e}_2, v_4))$ $=\phi _2(({e}_1, v_2))=\phi _2((v_4, {e}_3))=(12)$.

Note that each dihypergraph is one-to-one corresponding to its incidence bipartite digraph after labeling the vertices and the hyperarcs. Therefore, by Lemma 5 we obtain the following theorem:

Theorem 13

Let $\overline{{H}}$ be any k-fold dihypergraph covering over a finite dihypergraph ${H}$. Then, there is at least a permutation voltage assignment $\phi : E_{B_{{H}}}\rightarrow S_k$ on the incidence bipartite digraph $B_{{H}}$ of ${H}$ such that the derived dihypergraph ${H}^{B_{{H}}^{\phi }}$ associated with $B_{{H}}^\phi$ is isomorphic to $\overline{{H}}$.

Next, we establish another correspondence to all k-fold dihypergraph coverings with the edge-colored digraph. Assume that the hyperarc set of ${H}$ is denoted by $E_{{H}}=\{{e_1}=(e_1^+, e_1^-), {e_2}=(e_2^+, e_2^-), \ldots ,$ ${e_m}=(e_m^+, e_m^-)\}$ and giving each hyperarc ${e_i}$ a color $c_i$, $i\in [m]$.

Lemma 14

( [22]) Let $K_n$ be a complete graph with $n (n\ge 3)$ vertices. Then, there exists a nontrivial permutation voltage assignment $\phi :E_{\overleftrightarrow {K_n}}\rightarrow S_k(k>1)$ on $K_n$ such that $\phi ((v_i, v_j))\phi ((v_j, v_t))\phi ((v_t, v_i))$ $=1$ for every three distinct vertices $v_i, v_j$, and $v_t$.

Lemma 15

Let ${K}_n=(V_1, V_2)$ be a complete bipartite digraph with $n (n\ge 3)$ vertices. Then, there exists a nontrivial permutation voltage assignment $\phi :E_{{K}_n}\rightarrow S_k(k>1)$ on ${K}_n$ such that the derived digraph ${K}_n^{\phi }$ being k complete bipartite digraphs isomorphic to ${K}_n$.

Proof

We add the directed edges to ${K}_n$ such that ${K}_n$ turns into the symmetric digraph $\overleftrightarrow {K_n}$ corresponding to the complete graph $K_n$. Then, we expand the permutation voltage assignment $\phi$ to the digraph $\overleftrightarrow {K_n}$ such that $\phi (e^{-1})=\phi (e)^{-1}$ for each $e\in E_{\overleftrightarrow {K_n}}$. By Lemma 5, there exists a digraph covering over $\overleftrightarrow {K_n}$ corresponding to the derived digraph $\overleftrightarrow {K_n}^{\phi }$ associated with $\phi$. By Lemma 14, it is obtained that the derived digraph $\overleftrightarrow {K_n}^{\phi }$ is k symmetric digraphs isomorphic to $\overleftrightarrow {K_n}$ after attaching the condition $\phi ((v_i, v_j))\phi ((v_j, v_t))\phi ((v_t, v_i))$ $=1$ for every three distinct vertices $v_i, v_j$, and $v_t$ in $\overleftrightarrow {K_n}$ to $\phi$. The proof is completed after deleting the arcs added in ${K}_n$ and the arcs (in $\overleftrightarrow {K_n}^{\phi }$) corresponding to the added arcs in ${K}_n$. $\square$

Remark 16

Let ${K}_n=(V_1, V_2)$ be a complete bipartite digraph with $n (n\ge 3)$ vertices, whose arcs are from u to v for each pair of vertices $u\in V_1$ and $v\in V_2$. Suppose that $\phi :E_{{K}_n}\rightarrow S_k$ is a nontrivial permutation voltage assignment on ${K}_n$. If $\phi$ satisfies Lemma 15, then it should be held that $\phi ((u, s))\phi ((t, s)^{-1})\phi ((t, u))=1$ for every three distinct vertices $u, t\in V_1$ and $s\in V_2$ or $\phi ((s, u))\phi ((u, t))\phi ((s, t)^{-1})=1$ for every three distinct vertices $s\in V_1$ and $u, t\in V_2$ after $\phi$ is extended to $\overleftrightarrow {K_n}$ with $\phi (e^{-1})=\phi (e)^{-1}$ for each $e\in E_{\overleftrightarrow {K_n}}$.

The permutation voltage assignment $\phi : E_{G{H}_c}\rightarrow S_k$ on $G{H}_c$ satisfying Remark 16 for any three distinct arcs with the same color is called the strong permutation voltage assignment on $G{H}_c$, which is denoted by ${\widetilde{\phi }}$. Note that this restriction is invalid if all the arcs in $G{H}_c$ have different color. Let $C({\widetilde{\phi }}; k)$ denote the set of all strong permutation voltage assignments ${\widetilde{\phi }}: E_{G{H}_c}\rightarrow S_k$ on $G{H}_c$. Let $\pi =(\pi _0, \pi _1): G{H}_c^{{\widetilde{\phi }}} \rightarrow G{H}_c$ be a covering projection. From Lemma 15, it is obtained that any strong permutation voltage assignment ${\widetilde{\phi }}\in C({\widetilde{\phi }}; k)$ turns each complete bipartite digraph ${C}_i=(e_i^{+}, e_i^{-})$ in $G{H}_c$ with the color $c_i$ into k complete bipartite digraphs ${C}_i^j=((e_i^{+}, j), (e_i^{-}, j))$ in $G{H}_c^{{\widetilde{\phi }}}$, where $\pi _0((e_i^{+}, j))=e_i^{+}$ and $\pi _0((e_i^{-}, j))=e_i^{-}$, $j\in [k]$. We still color these k complete bipartite digraphs with $c_i$. In the following, the derived dihypergraph ${H}^{{\widetilde{\phi }}}$ associated with ${\widetilde{\phi }}$ is established: The vertex set ${H}^{{\widetilde{\phi }}}$ is the vertex set of the derived digraph $G{H}_c^{{\widetilde{\phi }}}$ of $(GH_c, {\widetilde{\phi }})$. For each complete bipartite digraph ${C}_i^j=((e_i^{+}, j), (e_i^{-}, j))$ in $G{H}_c^{{\widetilde{\phi }}}$, a hyperarc is built with the tail $(e_i^{+}, j)$ and the head $(e_i^{-}, j)$. From the construction of ${H}^{{\widetilde{\phi }}}$, it is found that ${H}^{{\widetilde{\phi }}}$ must be a k-fold dihypergraph covering over ${H}$ and $G{H}_c^{{\widetilde{\phi }}}$ can be considered as the edge-colored digraph of ${H}^{{\widetilde{\phi }}}$.

Fig. 5

The edge-colored digraph $G{H}_c$ of the dihypergraph depicted in Fig. 1, the derived graph $G{H}_c^{{\widetilde{\phi }}}$ and the dihypergraph covering corresponding to $G{H}_c^{{\widetilde{\phi }}}$, where $v_i^j=(v_i, j)$

Theorem 17

Let $\overline{{H}}$ be any k-fold dihypergraph covering over the dihypergraph ${H}$. Then, we can establish a one-to-one correspondence between $\overline{{H}}$ and some derived dihypergraph ${H}^{{\widetilde{\phi }}}$ associated with the strong permutation voltage assignments ${\widetilde{\phi }}\in C({\widetilde{\phi }}; k)$.

Proof

Note that the edge-colored digraph $G\overline{{H}}_c$ of $\overline{{H}}$ is a k-fold covering over $G{H}_c$. Then, by Lemma 5 and Lemma 15, there exists a strong permutation voltage assignment ${\widetilde{\phi }}$ on $G{H}_c$ such that the derived digraph $G{H}_c^{{\widetilde{\phi }}}$ associated with ${\widetilde{\phi }}$ is corresponding to $G\overline{{H}}_c$. The proof is completed by noting that every dihypergraph is one-to-one corresponding to its edge-colored digraph. $\square$

Example 18

In Fig. 1, we assign the strong permutation voltage ${\widetilde{\phi }}$ on $\overleftrightarrow {G{H}_c}$ as follows: ${\widetilde{\phi }}((v_1, v_2))={\widetilde{\phi }}((v_1, v_3))={\widetilde{\phi }}((v_3, v_4))={\widetilde{\phi }}((v_3, v_5))={\widetilde{\phi }}((v_4, v_6))={\widetilde{\phi }}((v_5, v_6))={\widetilde{\phi }}((v_6, v_7))={\widetilde{\phi }}((v_7, v_1))=(12),$ ${\widetilde{\phi }}((v_2, v_3))={\widetilde{\phi }}((v_2, v_4))={\widetilde{\phi }}((v_4, v_5))={\widetilde{\phi }}((v_4, v_7))={\widetilde{\phi }}((v_5, v_7))={\widetilde{\phi }}((v_6, v_1))=1.$ Then, the derived digraph $G{H}_c^{{\widetilde{\phi }}}$ and the corresponding 2-fold dihypergraph covering $\overline{{H}}$ over ${H}$ are obtained, see Fig. 5.

4 The Zeta Function of a Dihypergraph Covering

In this section, we derive two explicit decomposition formulae for the zeta function of any covering over a finite dihypergraph.

The Kronecker product $A\otimes B$ of matrices $A=(a_{ij})_{m\times n}$ and $B=(b_{ij})_{p\times q}$ is $mp\times nq$ matrix obtained from A by replacing each element $a_{ij}$ with the block $a_{ij}B$. This is an associative operation with the property $(A\otimes B)(C\otimes D)=AC\otimes BD$ whenever the products AC and BD exist. The latter implies that $(A\otimes B)^{-1}=A^{-1}\otimes B^{-1}$ for nonsingular matrices A and B. We use the block diagonal sum $A_1\oplus \cdots \oplus A_n$ or $\oplus _{i=1}^n A_i$ to denote the square matrix $diag (A_1, A_2, \ldots , A_n)$ for square matrices $A_1, \ldots , A_n$, and we write as $n\circ A$ for short when $A_1= \cdots = A_n=A$.

The complex general linear group of degree k, denoted by $GL(k, {\mathbb {C}})$, is the group of all $k\times k$ invertible matrices over the complex field ${\mathbb {C}}$ with respect to multiplication. Let $\mathcal {G}$ be a finite group. A representation $\rho$ of the group $\mathcal {G}$ over ${\mathbb {C}}$ is a homomorphism from $\mathcal {G}$ to $GL(k, {\mathbb {C}})$, and k is also called the degree of the representation $\rho$. If the representation $\rho : \mathcal {G}\rightarrow GL(k, {\mathbb {C}})$ sends each $g\in \mathcal {G}$ to a permutation matrix, then $\rho$ is called the permutation representation of $\mathcal {G}$. The permutation representation $\mathbf{P}$ of the symmetric group $S_k$ sends each $g\in S_k$ to the $k\times k$ permutation matrix $\mathbf{P} _g=(p_{ij}^{~g})$, where

$$\begin{aligned} p_{ij}^{~g}= {\left\{ \begin{array}{ll} 1~~~~~~~{\hbox {if}}~~~ i=g(j),\\ 0~~~~~~~{\hbox {otherwise}}. \end{array}\right. } \end{aligned}$$

(1)

Lemma 19

([24]) Let $\rho _1=I_1, \rho _2, \ldots , \rho _s$ be a complete set of inequivalent irreducible representations of $S_k$ and $f_i$ be the degree of $\rho _i$ for each $i\in [s]$, where $f_1=1$. Moreover, for each $i\in [s]$ suppose that $m_i$ is the multiplicity of the inequivalent irreducible representation $\rho _i$ in the permutation representation $\mathbf {P}$ which is defined by Eq. (1). Then, there exists a nonsingular matrix T such that for all $g\in S_k$

$$\begin{aligned} T^{-1}\mathbf {P}_gT=m_1\circ I_1\oplus m_2\circ \rho _2(g)\oplus \cdots \oplus m_s\circ \rho _s(g). \end{aligned}$$

Let ${H}$ be a finite dihypergraph with n vertices and m hyperarcs. Suppose that $\phi$ and ${\widetilde{\phi }}$ are a permutation voltage assignment on $B_{{H}}$ and a strong permutation voltage assignment on $G{H}_c$, respectively. Let $\Gamma _1=\langle \phi (e)|e\in E_{B_{{H}}}\rangle$ and $\Gamma _2=\langle {\widetilde{\phi }}(e)|e\in E_{\overleftrightarrow {G{H}_c}}\rangle$ be the subgroups of $S_{k}$, respectively, whose elements act as the permutation on the set [k]. $\mathbf {P}_1$ and $\mathbf {P}_2$ are permutation representations of $\Gamma _1$ and $\Gamma _2$ associated with the set [k], respectively, which are defined as Eq. (1). For simplicity, all inequivalent irreducible representations of $\Gamma _1$ and $\Gamma _2$ are uniformly denoted by $\rho _{1}=I_1, \rho _{2}, \ldots , \rho _{s}$. Moreover, the multiplicity and the degree of $\rho _{i}$ in $\mathbf {P}_1$ (or $\mathbf {P}_2$) are uniformly denoted by $m_{i}$ and $f_{i}$ for $i\in [s]$, respectively. Two decomposition formulae for the zeta function of any dihypergraph covering are obtained as follows.

Theorem 20

Let $\phi : E_{B_{{H}}}\rightarrow S_{k}$ be a permutation voltage assignment on $B_{{H}}$ such that $\overline{{H}}$ is a k-fold dihypergraph covering over ${H}$ corresponding to $\phi$. Then, the zeta function of $\overline{{H}}$ can be decomposed to the follows:

$$\begin{aligned} \varsigma _{\overline{{H}}}(u)= & {} \varsigma _{{H}}(u)^{m_1} \prod _{i=2}^s\det \big (I_{f_i(m+n)} -\sqrt{u}\Sigma _{g\in \Gamma _1}(\mathbf {A}_{g} \otimes \rho _i(g))\big )^{-m_i}, \end{aligned}$$

where $\mathbf {A}_{g}$ is defined by Eq. (2).

Proof

We utilize Theorem 9 to prove this theorem. The proof is an analogue of the method in [21].

Suppose that the vertex set and the hyperarc set of ${H}$ are denoted by $V_{{H}}=\{v_1, v_2, \ldots ,$ $v_n\}$ and $E_{{H}}=\{{e}_1, {e}_2, \ldots , {e}_m\}$, respectively. Then, the vertices of $B_{{H}}$ are arranged into 2 blocks:

$$\begin{aligned} v_1, v_2, \ldots , v_n; {e}_1, {e}_2, \ldots , {e}_m, \end{aligned}$$

and we arrange the vertices of the incidence bipartite digraph $B_{\overline{{H}}}$ into 2k blocks:

$$\begin{aligned} v_1^{1},\ldots , v_n^{1}; {e}_1^{~1}, \ldots , {e}_m^{~1}; \ldots ; v_1^{k},\ldots , v_n^{k}; {e}_1^{~k}, \ldots , {e}_m^{~k}, \end{aligned}$$

where $v_i^{j}=(v_i,j)$ and ${e}_i^{~j}=({e}_i, j)$. The adjacency matrix $A(B_{\overline{{H}}})$ of $B_{\overline{{H}}}$ is considered under this order.

From Lemma 5, it is known that there is a k-fold covering $B_{\overline{{H}}}$ over $B_{{H}}$ brings into correspondence with $B_{{H}}^{\phi }$. For any $g\in \Gamma _1$, the $(n+m) \times (n+m)$ matrix $\mathbf {A}_{g}=(a^{g}_{ij})$ is defined by

$$\begin{aligned} a_{ij}^{g}= {\left\{ \begin{array}{ll} 1~~~~~~~{\hbox {if}}~e=(v_i, {e}_j)\in E_{B_{{H}}} ~or~ e=({e}_i, v_j) \in E_{B_{{H}}}, ~and~\phi (e)=g,\\ 0~~~~~~~{\hbox {otherwise}}. \end{array}\right. } \end{aligned}$$

(2)

It is easily checked that

$$\begin{aligned} A(B_{{H}})=\sum _{g\in \Gamma _1}A_{g}. \end{aligned}$$

(3)

Therefore, we have

$$\begin{aligned} A(B_{\overline{{H}}})=\sum _{g\in \Gamma _1}(A_{g}\otimes \mathbf {P}_g). \end{aligned}$$

(4)

By Lemma 19, there exists a nonsingular matrix T such that $T^{-1}\mathbf {P}_gT=m_1\circ \rho _1(g)\oplus m_2\circ \rho _2(g)\oplus \cdots \oplus m_s\circ \rho _s(g)$ for all $g\in \Gamma _1$, where $\rho _1(g)=I_1$. Set $S=I_{m+n}\otimes T$, then we have

$$\begin{aligned}&S^{-1}A(B_{\overline{{H}}})S \nonumber \\= & {} (I_{m+n}\otimes T)^{-1}[\sum _{g\in \Gamma _1}(A_{g}\otimes \mathbf {P}_g)](I_{m+n}\otimes T) ~~~~\text {(by Eq. (4))}\nonumber \\= & {} \sum _{g\in \Gamma _1}(A_{g}\otimes T^{-1}\mathbf {P}_gT)\nonumber \\= & {} \sum _{g\in \Gamma _1} [A_{g}\otimes (\oplus _{i=1}^{s}m_i\circ \rho _i(g))]\nonumber \\= & {} \bigoplus _{i=1}^s \{\Sigma _{g\in \Gamma _1}[A_{g}\otimes (m_i\circ \rho _i(g))] \} \nonumber \\= & {} [\Sigma _{g\in \Gamma _1}(A_{g}\otimes I_{m_1})]\bigoplus \oplus _{i=2}^s\{\Sigma _{g\in \Gamma _1}[\mathbf {A}_{g} \otimes (m_i\circ \rho _i(g))]\}\nonumber \\= & {} (A(B_{{H}})\otimes I_{m_1})\bigoplus \oplus _{i=2}^s\{m_i\circ [\Sigma _{g\in \Gamma _1}(\mathbf {A}_{g} \otimes \rho _i(g))]\}~~ \text {(by Eq. (3))}. \end{aligned}$$

(5)

Note that $\sum _{i=1}^sm_if_i=k$ and $f_1=1$. It follows that

$$\begin{aligned}&\varsigma _{\overline{{H}}}(u)^{-1} \\= & {} \det (I_{m+n}\otimes I_k-\sqrt{u}A(B_{\overline{{H}}})) ~~~~~~ \text {(by Theorem 9)} \\= & {} \det (S^{-1} (I_{m+n}\otimes I_k-\sqrt{u}A(B_{\overline{{H}}}))S) \\= & {} \det \big (I_{m+n}\bigotimes I_{\sum _{i=1}^sm_if_i}-(\sqrt{u}A(B_{{H}})\otimes I_{m_1})\bigoplus \oplus _{i=2}^s(\sqrt{u}m_i\circ (\Sigma _{g\in \Gamma _1}(\mathbf{A} _{g} \otimes \rho _i(g)))\big )\\&\text {(by Eq. (5))}\\= & {} \det \big ((I_{m+n}\otimes I_{m_1}-\sqrt{u}A(B_{{H}})\otimes I_{m_1})\bigoplus \oplus _{i=2}^s(I_{m+n}\otimes I_{m_if_i}-\sqrt{u}m_i\circ (\Sigma _{g\in \Gamma _1}(\mathbf {A}_{g} \otimes \rho _i(g))))\big )\\= & {} \det (I_{m+n}-\sqrt{u}A(B_{{H}}))^{m_1}\prod _{i=2}^s\det \big (I_{f_i(m+n)} -\sqrt{u}\Sigma _{g\in \Gamma _1}(\mathbf {A}_{g} \otimes \rho _i(g))\big )^{m_i}\\= & {} \varsigma _{{H}}(u)^{-m_1} \prod _{i=2}^s\det \big (I_{f_i(m+n)} -\sqrt{u}\Sigma _{g\in \Gamma _1}(\mathbf {A}_{g} \otimes \rho _i(g))\big )^{m_i} ~~~~~~ \text {(by Theorem 9)} \end{aligned}$$

$\square$

Theorem 21

Let $\overline{{H}}$ be a k-fold dihypergraph covering over ${H}$ associated with a strong permutation voltage assignment ${\widetilde{\phi }}$ on $G{H}_c$. Then, the zeta function of $\overline{{H}}$ can be decomposed as follows:

$$\begin{aligned} \varsigma _{\overline{{H}}}(u)^{-1}=\varsigma _{{H}}(u)^{-m_1}\prod _{i=2}^s\det (I_{lf_i} -u\Sigma _{g\in \Gamma _2}( \rho _i(g) \otimes \mathbf {A}_{g}))^{m_i}, \end{aligned}$$

where $\mathbf {A}_{g}$ is defined by Eq. (6), and l is the number of directed edges of $G{H}_c$.

Proof

Suppose that $E_{G{H}_c}=\{ \alpha _1, \alpha _2, \ldots ,\alpha _l\}$ and $G{H}_c^{{\widetilde{\phi }}}$ is a k-fold covering over $G{H}_c$ associated with ${\widetilde{\phi }}$ . Arrange arcs of $G{H}_c^{{\widetilde{\phi }}}$ into k blocks:

$$\begin{aligned} \alpha _1^{1}, \ldots , \alpha _l^{1}; \alpha _1^{2}, \ldots , \alpha _l^{2}; \ldots ; \alpha _1^{k}, \ldots , \alpha _l^{k}, \end{aligned}$$

where $\alpha _{i}^{j}=(\alpha _i, j)$.

Note that the vertices of the oriented line graph ${H}^{{\widetilde{\phi }}}_L$ of ${H}^{{\widetilde{\phi }}}$ is the arcs of $G{H}_c^{{\widetilde{\phi }}}$. We consider the adjacency matrix $A({H}_{L}^{{\widetilde{\phi }}})$ under this order. For any $g\in \Gamma _2$, the $l\times l$ matrix $\mathbf {A}_g=(a_{ij}^{g})$ is defined by

$$\begin{aligned} a_{ij}^{g}= {\left\{ \begin{array}{ll} 1~~~~~~~{\hbox {if}}~(\alpha _i, \alpha _j)\in E_{{H}_L}~ and~{\widetilde{\phi }}(\alpha _i)=g,\\ 0~~~~~~~{\hbox {otherwise}}. \end{array}\right. } \end{aligned}$$

(6)

Thus, we have

$$\begin{aligned} A({H}_L)=\sum _{g\in \Gamma _2}\mathbf {A}_{g}, \end{aligned}$$

(7)

and it follows that

$$\begin{aligned} A({H}_{L}^{{\widetilde{\phi }}})=\sum _{g\in \Gamma _2}(\mathbf {P}_g\otimes \mathbf {A}_{g}). \end{aligned}$$

(8)

By Lemma 19, there exists a nonsingular matrix T such that $T^{-1}\mathbf {P}_gT=m_1\circ \rho _1(g)\oplus m_2\circ \rho _2(g)\oplus \cdots \oplus m_s\circ \rho _s(g)$ for all $g\in \Gamma _2$, where $\rho _1(g)=I_1$. Note that $\sum _{i=1}^sm_if_i=k$ and $f_1=1$. We have

$$\begin{aligned}&(T\otimes I_{l})^{-1}(I_{lk}-uA({H}_{L}^{{\widetilde{\phi }}}))(T\otimes I_{l})\nonumber \\= & {} I_{l}\otimes I_{k}-u(T\otimes I_{l})^{-1}[\sum _{g\in \Gamma _2}(\mathbf {P}_g\otimes A_{g})](T \otimes I_{l})~~~~~~ \text {(by Eq. (8))}\nonumber \\= & {} I_{l}\otimes I_{\sum _{i=1}^sm_if_i}-u\sum _{g\in \Gamma _2}(T^{-1}\mathbf {P}_gT\otimes A_{g} ) \nonumber \\= & {} I_{l}\otimes I_{\sum _{i=1}^sm_if_i}-u\sum _{g\in \Gamma _2} [(\oplus _{i=1}^{s}m_i\circ \rho _i(g))\otimes A_{g}] \nonumber \\= & {} I_{l}\otimes I_{\sum _{i=1}^sm_if_i}-u\bigoplus _{i=1}^s \{\Sigma _{g\in \Gamma _2}[(m_i\circ \rho _i(g))\otimes A_{g}] \} \nonumber \\= & {} I_{l}\otimes I_{\sum _{i=1}^sm_if_i}-u[\Sigma _{g\in \Gamma _2}(I_{m_1}\otimes A_{g})]\bigoplus \oplus _{i=2}^s\{\Sigma _{g\in \Gamma _2}[(m_i\circ \rho _i(g)) \otimes \mathbf {A}_{g}]\} \nonumber \\= & {} [I_{l}\otimes I_{m_1}-u(m_1\circ A({H}_L))]\bigoplus \oplus _{i=2}^s\{I_{l}\otimes I_{m_if_i}-u m_i\circ [\Sigma _{g\in \Gamma _2}( \rho _i(g))\otimes \mathbf {A}_{g}]\} \nonumber \\&\text {(by Eq. (7))}\nonumber \\= & {} m_1\circ (I_{l}- uA({H}_L))\bigoplus \oplus _{i=2}^s\{m_i\circ [I_{lf_i}-u\Sigma _{g\in \Gamma _2}(\rho _i(g) \otimes \mathbf {A}_{g})]\}. \end{aligned}$$

(9)

The proof is completed by Theorem 11 and Eq. (9). $\square$

From Theorems 20 and 21, the following corollary can be obtained immediately:

Corollary 22

Let ${H}$ be a dihypergraph and $\overline{{H}}$ a dihypergraph covering over ${H}$; then the zeta function of ${H}$ divides the zeta function of $\overline{{H}}$.