On the Ihara expression for the generalized weighted zeta function Ayaka Ishikawa

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On the Ihara expression for the generalized

weighted zeta function

Ayaka Ishikawa∗

Ritsumeikan University

Noji-higashi, Kusatsu 525-85771, Japan

Hideaki Morita†

Muroran Institute of Technology

Mizumoto, Muroran 050-8585, Japan

Abstract

We consider the generalized weighted zeta function for a ﬁnite digraph, and show

that it has the Ihara expression, a determinant expression of graph zeta functions,

with a certain speciﬁed deﬁnition for inverse arcs. A ﬁnite digraph in this paper allows

multi-arcs or multi-loops.

1 Introduction

A graph zeta function is a formal power series associated with a ﬁnite graph. It enumerates

the closed paths of a given length, exposes the primes, or depicts the cycles in a ﬁnite graph.

The prototype of the graph zeta functions was introduced by Y. Ihara [9] in 1966 from

a number-theoretical point of view. Ihara’s zeta was subsequently pointed out by J. -P.

Serre [21] that it can be formulated in terms of ﬁnite graphs, and is now called the Ihara

zeta function for a ﬁnite graph [8, 16, 23]. In the paper of H. Bass [2], as was implied by

Ihara [9], the Ihara zeta is provided the determinant expression described by the adjacency

matrix and the degree matrix of the corresponding graph. This determinant expression is

now called the Ihara expression [10, 13] (see also [18]), and the theorem is called the Bass-

Ihara theorem. The Ihara expression is one of the main interests in the study of graph zeta

functions, and many researches have pursued this subject [1, 5, 7, 10, 13, 17, 19, 20, 22] . It

is also necessary to mention that the Ihara expression was recently provided a new point of

view from quantum walk theory [11, 14], and thus the signiﬁcance of the Ihara expression is

now increasing in related areas.

∗a-iskw@fc.ritsumei.ac.jp

†morita@mmm.muroran-it.ac.jp

arXiv:2210.13262v3 [math.CO] 2 Apr 2023

The subject of the present paper is the Ihara expression for the generalized weighted zeta

function. The generalized weighted zeta function was introduced in [18] as a single scheme

which uniﬁes the graph zetas appeared in previous studies, for instance, the Ihara zeta [9],

the Bartholdi zeta [1], the Mizuno-Sato zeta [17] and the Sato zeta [20]. Graph zetas may

have in general four expressions called the exponential expression, the Euler expression, the

Hashimoto expression and the Ihara expression. It is veriﬁed in [18] that the ﬁrst three

expressions are equivalent for those graph zeta functions. The last two expressions are both

determinant expressions, where the size of the matrices used in the latter one, the adjacency

matrix and the degree matrix, is smaller in general than the other one, the edge matrix.

For known examples of graph zetas, the Ihara expression is obtained by transforming

the Hashimoto expression. In this paper, we will show that the generalized weighted zeta

function also have the Ihara expression as in the same manner with those graph zetas.

In particular, we verify the main theorem for the case where the underlying graph is an

arbitrary ﬁnite digraph. Graph zetas have usually been deﬁned via the symmetric digraph

of a given ﬁnite graph, so it is natural to deﬁne for ﬁnite digraphs rather than ﬁnite graphs.

In addition, a ﬁnite digraph in this paper allows multi-arcs and multi-loops, and one will see

in the procedure that it is an unavoidable issue how one deﬁnes the inverses for each arc of

a digraph. For this, we can consider two extreme ways. One is the case where all the arcs

with inverse direction to an arc aare deﬁned to be the inverses of a, and the other one is the

case where a single arc with inverse direction, if exists, deﬁned to be the inverse of a. In [13],

we treat the former case. In the present article, we treat the latter case. These two cases

are natural generalizations for the case where the underlying graph is a ﬁnite simple graph.

Therefore, the main result in this paper provides a way to generalize the developments in

previous researches on the Ihara expression, and gives a uniﬁed method to handle it.

Throughout this paper, we use the following notation. The ring of integers is denoted by

Z. The ﬁeld of rational numbers and complex numbers are denoted by Qand Crespectively.

For a set X, the cardinality of Xis denoted by |X|. The Kronecker delta is denoted by δxy,

which returns 1 if x=y, 0 otherwise. The symbol Istands for the identity matrix.

2 Preliminaries

2.1 Graphs and Digraphs

Adigraph is a pair ∆ = (V, A) of a set Vand a multi-set Aconsisting of ordered pairs (u, v)

of elements u, v in V. If the cardinalities of Vand Aare ﬁnite, then ∆ is called a ﬁnite

digraph. An element of V(resp. A) is called a vertex (resp. an arc) of ∆. An arc a= (u, v)

is depicted by an arrow from uto v. The vertex uis called the tail of a, and vthe head of

a, which are denoted by t(a) and h(a) respectively. Let u, v ∈V. We denote by Auv the set

{a∈A|t(a) = u, h(a) = v}

of arcs with the tail uand the head v. An arc lof the form (u, u) is called a loop. The set

of loops is denoted by L. Hence L=tu∈VAuu. Note that |Auv| ≥ 1 may occur in general.

Thus an arc a∈ Auv sometimes called a multi-arc, and the cardinality |Auv|is called the

multiplicity of a. Similarly, a loop l∈ Auu may be called a multi-loop. The cardinality |Auu|

is called the multiplicity of l. Set Au∗=tv∈VAuv and A∗v=tu∈VAuv.A digraph ∆ = (V, A)

is called simple if Auu =∅for any u∈Vand |Auv |= 1 if Auv 6=∅. Let A(u, v) = Auv ∪ Avu

denote the set of arcs lying between vertices uand v. A digraph ∆ is called connected if

A(u, v)6=∅for any distinct u,v. A digraph in this paper is always assumed to be connect

otherwise stated.

Let ∆ = (V, A) be a ﬁnite digraph, and u,vtwo distinct vertices. We may assume that

|Auv| ≤ |Avu|. If Auv and Avu are both not empty, then one can ﬁx an injection

ιuv :Auv → Avu.

In this case, we say that an arc a∈ Auv has inverse, and the arc ιuv(a)∈ Avu is the inverse

arc, or simple the inverse of a, denoted by a−1; and vice versa, ais the inverse of a−1. An

arc a0∈ Avu not lying in the image of ιuv has no inverse. In the case where Auv =∅, any

arc a0∈ Avu is deﬁned to have no inverse. If u=v, then ιuu is deﬁned to be the identity

map. By this deﬁnition, a loop l∈ Auu satisﬁes l−1=l, that is, each loop is self-inverse.

Alternatively, one can also deﬁne any arc belonging to Avu to be inverse of an arc of Auv.

This alternate deﬁnition also works, and the development with this deﬁnition will be found

in [13].

Agraph is a pair Γ = (V, E) of a set Vand a multi-set Econsisting of 2-subsets {u, v}

of V. If Vand Eare ﬁnite (multi-)sets, then the graph Γ is called ﬁnite. An element

{u, v} ∈ Eis called a edge. In particular, an edge of the form l={u, u}is called an loop.

The set of loops is denoted by L. Obviously, {u, v} ∈ E\Limplies u6=v. We also suppose

that an edge or a loop has multiplicity. Hence these are sometimes called an multi-edge and

multi-loop, respectively. In other words, if we denote by E(u, v) the set of multi-edges lying

between vertices u, v ∈V, then we assume that |E(u, v)| ≥ 1 for u, v ∈Vwith E(u, v)6=∅.

Note that E(u, u) denotes the set of loops of the form {u, u}. The cardinality |E(u, v)|is

called the multiplicity of an edge {u, v}. A graph is called simple if it has no loops and the

multiplicity of any edge is at most one. The matrix

AΓ= (|E(u, v)|)u,v∈V

is called the adjacency matrix of Γ. For a vertex u∈V, the number of edges {u, v}(v∈V)

is called the degree of u, denoted by du. Thus we have du=Pv∈V|E(u, v)|for u∈V. The

diagonal matrix

DΓ= (δuvdu)u,v∈V

is called the degree matrix of Γ.

Let Γ = (V, E) be a ﬁnite graph. We recall the deﬁnition of the symmetric digraph of

Γ. We assign for each edge {u, v} ∈ E\L, two arcs (u, v) and (v, u) in mutually reverse

direction. For a loop {u, u} ∈ L, we assign a single directed loop (u, u). Then we have an

set of arcs

A={(u, v),(v, u)| {u, v} ∈ E\L}t{(u, u)| {u, u} ∈ L}.

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OntheIharaexpressionforthegeneralizedweightedzetafunctionAyakaIshikawa*RitsumeikanUniversityNoji-higashi,Kusatsu525-85771,JapanHideakiMoritaMuroranInstituteofTechnologyMizumoto,Muroran050-8585,JapanAbstractWeconsiderthegeneralizedweightedzetafunctionforanitedigraph,andshowthatithastheIharaexpressi...

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On the Ihara expression for the generalized weighted zeta function Ayaka Ishikawa

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