Taichi Kousaka
We establish a generalized Ihara zeta function formula for simple graphs with bounded degree. This is a generalization of the formula obtained by G. Chinta, J. Jorgenson and A. Karlsson from a vertex-transitive graph.
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The theory of Ihara zeta functions is extended to infinite graphs which are weighted and of finite total weight. In this case one gets meromorphic instead of rational functions and the classical determinant formulas of Bass and Ihara hold true with Fredholm determinants.
The infinite grid is the Cayley graph of $\mathbb{Z} \times \mathbb{Z}$ with the usual generators. In this paper, the Ihara zeta function for the infinite grid is computed using elliptic integrals and theta functions. The zeta function of the grid extends to an analytic, multivalued function which satisfies a functional equation. The set of singularities in its domain is finite. The grid zeta function is the first computed example which is non-elementary, and which takes infinitely many values at each point of its domain. It is also the limiting value of the normalized sequence of Ihara zeta f…
The Ihara expression of a weighted zeta function for a general finite digraph is given. It unifies all the Ihara expressions obtained for known zeta functions for finite digraphs. Any digraph in this paper permits multi-edges and multi-loops.
We define a new weighted zeta function for a finite digraph and obtain its determinant expression called the Ihara expression. The graph zeta function is a generalization of the weighted graph zeta function introduced in previous research. That is, our result makes it possible to derive the Ihara expressions of the previous graph zeta functions for any finite digraphs.
On the Ihara expression for the generalized weighted zeta function
本文研究了有限有向图的广义加权zeta函数，证明在特定逆弧定义下该函数具有Ihara表达式（一种图zeta函数的行列式表达式），且允许重弧和重环。
We show that if a graph $G$ has average degree $\bar d \geq 4$, then the Ihara zeta function of $G$ is edge-reconstructible. We prove some general spectral properties of the edge adjacency operator $T$: it is symmetric for an indefinite form and has a "large" semi-simple part (but it can fail to be semi-simple in general). We prove that this implies that if $\bar d>4$, one can reconstruct the number of non-backtracking (closed or not) walks through a given edge, the Perron-Frobenius eigenvector of $T$ (modulo a natural symmetry), as well as the closed walks that pass through a given edge in bo…
We derive combinatorial proofs of the main two evaluations of the Ihara-Selberg Zeta function associated with a graph. We give three proofs of the first evaluation all based on the algebra of Lyndon words. In the third proof it is shown that the first evaluation is an immediate consequence of Amitsur's identity on the characteristic polynomial of a sum of matrices. The second evaluation of the Ihara-Selberg Zeta function is first derived by means of a sign-changing involution technique. Our second approach makes use of a short matrix-algebra argument.
The definition and main properties of the Ihara zeta function for graphs are reviewed, focusing mainly on the case of periodic simple graphs. Moreover, we give a new proof of the associated determinant formula, based on the treatment developed by Stark and Terras for finite graphs.
In this paper, using matrix techniques, we compute the Ihara-zeta function and the number of spanning trees of the join of two semi-regular bipartite graphs. Furthermore, we show that the spectrum and the zeta function of the join of two semi-regular bipartite graphs can determine each other.
Chinta, Jorgenson and Karlsson introduced a generalized version of the determinant formula for the Ihara zeta function associated to finite or infinite regular graphs. On the other hand, Konno and Sato obtained a formula of the characteristic polynomial of the Grover matrix by using the determinant expression for the second weighted zeta function of a finite graph. In this paper, we focus on a relationship between the Grover walk and the generalized Ihara zeta function. That is to say, we treat the generalized Ihara zeta function of the one-dimensional integer lattice as a limit of the Ihara z…
Starting with Ihara's work in 1968, there has been a growing interest in the study of zeta functions of finite graphs, by Sunada, Hashimoto, Bass, Stark and Terras, Mizuno and Sato, to name just a few authors. Then, Clair and Mokhtari-Sharghi have studied zeta functions for infinite graphs acted upon by a discrete group of automorphisms. The main formula in all these treatments establishes a connection between the zeta function, originally defined as an infinite product, and the Laplacian of the graph. In this article, we consider a different class of infinite graphs. They are fractal graphs, …
In this article, we introduce an entropy based on the formal power series expansion of the Ihara Zeta function. We find a number of inequalities based on the values of the Ihara zeta function. These new entropies are applicable in symbolic dynamics and the dynamics of billiards.
We generalize the Ihara-Selberg zeta function to hypergraphs in a natural way. Hashimoto's factorization results for biregular bipartite graphs apply, leading to exact factorizations. For $(d,r)$-regular hypergraphs, we show that a modified Riemann hypothesis is true if and only if the hypergraph is Ramanujan in the sense of Winnie Li and Patrick Solé. Finally, we give an example to show how the generalized zeta function can be applied to graphs to distinguish non-isomorphic graphs with the same Ihara-Selberg zeta function.
Determinant formulas for Ihara zeta functions via simple cycles
基于迹幺半群代数框架，通过Cayley行列式建立有向图Ihara zeta函数的新行列式公式，并证明其倒数可表示为线图无回溯简单环数量的行列式。
Ihara zeta functions for some simple graph families
本文简化了 Ihara zeta 函数系数的计算，证明其对秩二图是完全不变量，并给出完全图、完全二部图、Möbius 梯子图、鸡尾酒会图及五阶以下图的 zeta 函数，进而用特殊值计算生成树数。
In this paper, we give a more direct proof of the results by Clair and Mokhtari-Sharghi on the zeta functions of periodic graphs. In particular, using appropriate operator-algebraic techniques, we establish a determinant formula in this context and examine its consequences for the Ihara zeta function. Moreover, we answer in the affirmative one of the questions raised by Grigorchuk and Zuk. Accordingly, we show that the zeta function of a periodic graph with an amenable group action is the limit of the zeta functions of a suitable sequence of finite subgraphs.
We establish a new formula for the heat kernel on regular trees in terms of classical I-Bessel functions. Although the formula is explicit, and a proof is given through direct computation, we also provide a conceptual viewpoint using the horocyclic transform on regular trees. From periodization, we then obtain a heat kernel expression on any regular graph. From spectral theory, one has another expression for the heat kernel as an integral transform of the spectral measure. By equating these two formulas and taking a certain integral transform, we obtain several generalized versions of the dete…
Using recent results on the concentration of the largest eigenvalue and maximal vertex degree of large random graphs, we show that the infinite sequence of Erd\H os-Rényi random graphs $G(n,ρ_n/n)$ such that $ρ_n/\log n$ infinitely increases as $n\to\infty$ verifies a version of the graph theory Riemann Hypothesis.
We consider the ensemble of $N\times N$ real random symmetric matrices $H_N^{(R)}$ obtained from the determinant form of the Ihara zeta function associated to random graphs $Γ_N^{(R)}$ of the long-range percolation radius model with the edge probability determined by a function $ϕ(t)$. We show that the normalized eigenvalue counting function of $H_N^{( R)}$ weakly converges in average as $N,R\to\infty$, $R=o(N)$ to a unique measure that depends on the limiting average vertex degree of $Γ_N^{(R)}$ given by $ϕ_1 = \int ϕ(t) dt$. This measure converges in the limit of infinite $ϕ_1$ to a shift of…
We put forward the concept of measure graphs. These are (possibly uncountable) graphs equipped with an action of a groupoid and a measure invariant under this action. Examples include finite graphs, periodic graphs, graphings and percolation graphs. Making use of Connes' non-commutative integration theory we construct a Zeta function and present a determinant formula for it. We further introduce a notion of weak convergence of measure graphs and show that our construction is compatible with it. The approximation of the Ihara Zeta function via the normalized version on finite graphs in the sens…
We consider the ensemble of real symmetric random matrices $H^{(n,ρ)}$ obtained from the determinant form of the Ihara zeta function of random graphs that have $n$ vertices with the edge probability $ρ/n$. We prove that the normalized eigenvalue counting function of $H^{(n,ρ)}$ weakly converges in average as $n,ρ\to\infty$ and $ρ=o(n^α)$ for any $α>0$ to a shift of the Wigner semi-circle distribution. Our results support a conjecture that the large Erdos-Rényi random graphs satisfy in average the weak graph theory Riemann Hypothesis.
Let $X$ denote a connected $(q+1)$-regular undirected graph of finite order $n$. The graph $X$ is called Ramanujan whenever $$ |λ|\leq 2q^{\frac{1}{2}} $$ for all nontrivial eigenvalues $λ$ of $X$. We consider the variant $Ξ(u)$ of the Ihara zeta function $Z(u)$ of $X$ defined by \begin{gather*} Ξ(u)^{-1} = \left\{ \begin{array}{ll} (1-u)(1-qu)(1-q^{\frac{1}{2}} u)^{2n-2}(1-u^2)^{\frac{n(q-1)}{2}} Z(u) \qquad &\hbox{if $X$ is nonbipartite}, (1-q^2u^2) (1-q^{\frac{1}{2}} u)^{2n-4} (1-u^2)^{\frac{n(q-1)}{2}+1} Z(u) \qquad &\hbox{if $X$ is bipartite}. \end{array} \right. \end{gather*} The functio…
Residues of weighted dynamical Ihara zeta functions on finite regular graphs
本文在有限正则图上构造加权动力Ihara zeta函数，证明其亚纯延拓并计算留数，将其与不变Ruelle分布关联，推导出Patterson-Sullivan和Wigner公式，为局部对称空间情形提供有限图类比。
Stark and Terras introduced the edge zeta function of a finite graph in 1996. The edge zeta function is the reciprocal of a polynomial in twice as many variables as edges in the graph and can be computed in polynomial time. We look at graph properties which we can determine using the edge zeta function. In particular, the edge zeta function is enough to deduce the clique number, the number of Hamiltonian cycles, and whether a graph is perfect or chordal. Actually computing these properties takes exponential time. Finally, we present a new example illustrating that the Ihara zeta function canno…
We consider the Ihara zeta function $ζ(u,X//G)$ and Artin-Ihara $L$-function of the quotient graph of groups $X//G$, where $G$ is a group acting on a finite graph $X$ with trivial edge stabilizers. We determine the relationship between the primes of $X$ and $X//G$ and show that $X\to X//G$ can be naturally viewed as an unramified Galois covering of graphs of groups. We show that the $L$-function of $X//G$ evaluated at the regular representation is equal to $ζ(u,X)$ and that $ζ(u,X//G)$ divides $ζ(u,X)$. We derive two-term and three-term determinant formulas for the zeta and $L$-functions, and …
Conjecturally, almost all graphs are determined by their spectra. This problem has also been studied for variants such as the spectra of the Laplacian and signless Laplacian. Here we consider the problem of determining graphs with Ihara and Bartholdi zeta functions, which are also computable in polynomial time. These zeta functions are geometrically motivated, but can be viewed as certain generalizations of characteristic polynomials. After discussing some graph properties determined by zeta functions, we show that large classes of cospectral graphs can be distinguished with zeta functions and…
Artin-Ihara L-functions for hypergraphs
本研究将图的Artin-Ihara L函数推广到超图领域，借助超图关联的二分图的Artin-Ihara L函数开展研究，证明了超图Artin-Ihara L函数的多项性质，包括超图Ihara zeta函数的Artin-Ihara L函数乘积分解。
Chamber zeta function and closed galleries in the standard non-uniform complex from $\operatorname{PGL}_3$
本文引入了chamber zeta函数，通过欧拉乘积扩展了Ihara-Bass框架，并证明了其与chamber转移算子特征多项式的关系，从而获得闭gallery类的计数结果。
The definitions and main properties of the Ihara and Bartholdi zeta functions for infinite graphs are reviewed. The general question of the validity of a functional equation is discussed, and various possible solutions are proposed.
We establish the quaternionic weighted zeta function of a graph and its Study determinant expressions. For a graph with quaternionic weights on arcs, we define a zeta function by using an infinite product which is regarded as the Euler product. This is a quaternionic extension of the square of the Ihara zeta function. We show that the new zeta function can be expressed as the exponential of a generating function and that it has two Study determinant expressions, which are crucial for the theory of zeta functions of graphs.
We introduce a generalized Bartholdi zeta function for simple graphs with bounded degree. This zeta function is a generalization of both the Bartholdi zeta function which was introduced by L.~Bartholdi and the Ihara zeta function which was introduced by G.~Chinta, J.~Jorgenson and A.~Karlsson. Furthermore, we establish a Bartholdi type formula of this Bartholdi zeta function for simple graphs with bounded degree. Moreover, for regular graphs, we give a new expression of the heat kernel which is regarded as a one-parameter deformation of the expression obtained by G.~Chinta, J.~Jorgenson and A.…
On the Zeta functions of supersingular isogeny graphs and modular curves
本文证明了模曲线的 Hasse--Weil zeta 函数与配备级结构的超奇异椭圆曲线同源图的 Ihara zeta 函数之间的关系公式，推广了 Sugiyama 的结果。
Spanning trees and their relations in Galois covers
本文推广了Galois覆盖中生成树数量的关系公式，利用Ihara zeta函数和Artin--Ihara L-函数证明了Kuroda公式和Brauer--Kuroda关系的图论类比，并指出循环Galois群时不存在生成树公式。
We establish a generalization of the second weighted zeta function of a graph to the case of quaternions. For an arc-weighted graph whose weights are quaternions, we define the second weighted zeta function by using the Study determinant that is a quaternionic determinant for quaternionic matrices defined by Study. This definition is regarded as a quaternionic analogue of the determinant expression of Hashimoto type for the Ihara zeta function of a graph. We derive the Study determinant expression of Bass type and the Euler product for the quaternionic second weighted zeta function.
Let $π=\{V_1,\dots,V_r\}$ be an equitable partition of the vertex set of a directed graph (digraph) $X$. It is well known that the characteristic polynomial $ϕ(X/π,x)$ of a quotient graph $X/π$ divides that of $X$, but the remainder part is not well investigated. In this paper, we define a deletion graph $X\backslashπ$ over an equitable partition $π$, which is a signed directed graph defined for a fixed set of deleting vertices $\{\bar{v}_i\in V_i, i=1,\cdots,r\}$, and give a similarity transformation exchanging the adjacency matrix $A(X)$ which is compatible with the equitable partition for a…
In this article, we define an information-theoretic entropy based on the Ihara zeta function of a graph which is called the Ihara entropy. A dynamical system consists of a billiard ball and a set of reflectors correspond to a combinatorial graph. The reflectors are represented by the vertices of the graph. Movement of the billiard ball between two reflectors is represented by the edges. The prime cycles of this graph generate the bi-infinite sequences of the corresponding symbolic dynamical system. The number of different prime cycles of a given length can be expressed in terms of the adjacenc…
We initiate the study of spectral zeta functions $ζ_{X}$ for finite and infinite graphs $X$, instead of the Ihara zeta function, with a perspective towards zeta functions from number theory and connections to hypergeometric functions. The Riemann hypothesis is shown to be equivalent to an approximate functional equation of graph zeta functions. The latter holds at all points where Riemann's zeta function $ζ(s)$ is non-zero. This connection arises via a detailed study of the asymptotics of the spectral zeta functions of finite torus graphs in the critcal strip and estimates on the real part of …
We introduce a generalized Grover matrix of a graph and present an explicit formula for its characteristic polynomial. As a corollary, we give the spectra for the generalized Grover matrix of a regular graph. Next, we define a zeta function and a generalized zeta function of a graph $G$ with respect to its generalized Grover matrix as an analog of the Ihara zeta function and present explicit formulas for their zeta functions for a vertex-transitive graph. As applications, we express the limit on the generalized zeta functions of a family of finite vertex-transitive regular graphs by an integra…
We consider the alternating zeta function and the alternating $L$-function of a graph $G$, and express them by using the Ihara zeta function of $G$. Next, we define a generalized alternating zeta function of a graph, and express the generalized alternating zeta function of a vertex-transitive regular graph by spectra of the transition probability matrix of the symmetric simple random walk on it and its Laplacian. Furthermore, we present an integral expression for the limit of the generalized alternating zeta functions of a series of vertex-transitive regular graphs. As an example, we treat the…
Our previous work presented explicit formulas for the generalized zeta function and the generalized Ihara zeta function corresponding to the Grover walk and the positive-support version of the Grover walk on the regular graph via the Konno-Sato theorem, respectively. This paper extends these walks to a class of walks including random walks, correlated random walks, quantum walks, and open quantum random walks on the torus by the Fourier analysis.
We discuss two combinatorical ways of generalizing the definition of expander graphs and Ramanujan graphs, to quotients of buildings of higher dimension. The two possible definitions are equivalent for affine buildings, giving the notion of an Lp-expander complex. We calculate explicit spectral gaps on many combinatorical operators, on any Lp-expander complex. We associate with any complex a natural "zeta function", generalizing the Ihara-Hashimoto zeta function of a finite graph. We generalize a well known theorem of Hashimoto, showing that a complex is Ramanujan if and only if the zeta funct…
In this paper we study spectral zeta functions associated to finite and infinite graphs. First we establish a meromorphic continuation of these functions under some general conditions. Then we study special values in the case of standard lattice graphs associated to free abelian groups. In particular we connect it to Catalan numbers in several ways, and obtain some non-trivial special values and functional symmetries. Furthermore we relate the values at the negative integers with the more studied Ihara zeta functions, and prove a few minor results that seem not to have been recorded before. Fi…
Recently the Ihara zeta function for the finite graph was extended to infinite one by Clair and Chinta et al. In this paper, we obtain the same expressions by a different approach from their analytical method. Our new approach is to take a suitable limit of a sequence of finite graphs via the Konno-Sato theorem. This theorem is related to explicit formulas of characteristic polynomials for the evolution matrix of the Grover walk. The walk is one of the most well-investigated quantum walks which are quantum counterpart of classical random walks. We call the relation between the Grover walk and …
Generating functions of non-backtracking walks on weighted digraphs: radius of convergence and Ihara's theorem
本文利用多项式与有理矩阵理论，研究了一般有向和/或加权图上非回溯游走生成函数的收敛半径，给出了有向无权图的刻画及加权图的精确公式，并证明了回溯降权游走下的Ihara定理。
This paper is about the determinantal identities associated with the Ihara (Ih) zeta function of a non directed graph and the Bowen-Lanford (BL) zeta function of a directed graph. They will be called the Ih and the BL identities in this paper. We show that the Witt identity (WI) is a special case of the BL identity and inspired by the links the WI has with Lie algebras and combinatorics we investigate similar aspects of the Ih and BL identities. We show that they satisfy generalizations of the Strehl identity and Carlitz, Metropolis-Rota relations and each one of them can be interpreted as the…
Fermions and Zeta Function on the Graph
本文提出一种图上的费米子模型，其配分函数为图zeta函数的倒数，系数计数费米子圈数，并推广至网格图，证明无费米子加倍，可构造重叠费米子，且与统计模型及规范理论统一关联。
Sum formulas for Schur multiple zeta values
本文研究 Schur 多重 zeta 值的求和公式，推广多重 zeta(-star) 值求和公式，并针对不同角数的带状形给出有界型表达式及相关恒等关系。
Given a finite abelian $p$-group $F$, we prove an efficient recursive formula for $σ_a(F)=\sum_{\substack{H\leq F}}|H|^a$ where $H$ ranges over the subgroups of $F$. We infer from this formula that the $p$-component of the corresponding zeta-function on groups of $p$-rank bounded by some constant $r$ is rational with a simple denominator. We also provide two explicit examples in rank $r=3$ and $r=4$ as well as a closed formula for $σ_a(F)$.
Iwasawa theory for vertex-weighted graphs
本文推广完善前人工作，开创顶点加权图的岩泽理论。将矩阵树定理推广到任意域值顶点加权图，完善证明分解公式，据此证明了岩泽型公式等，能估计\(\mathbb{Z}_p^d\)-塔中加权复杂度的根向增长并给出数值例子。
PageRank on Lubotzky--Phillips--Sarnak graphs
本研究比较LPS图与无限正则树上随机游走的返回概率，利用Ihara zeta函数修正环的影响，在平均步长与最短环长相当的情形下给出上界，并在广义黎曼猜想下得到与顶点数倒数同阶的修正。
Edge Zeta Functions and Eigenvalues for Buildings of Finite Groups of Lie Type
研究有限李型群建筑的边ζ函数，证明非零边特征值在特定指数下成为q的幂，扩展了之前关于类型A和对立图的结果。
This paper generalizes Bass' work on zeta functions for uniform tree lattices. Using the theory of von Neumann algebras, machinery is developed to define the zeta function of a discrete group of automorphisms of a bounded degree tree. The main theorems relate the zeta function to determinants of operators defined on edges or vertices of the tree. A zeta function associated to a non-uniform tree lattice with appropriate Hilbert representation is defined. Zeta functions are defined for infinite graphs with a cocompact or finite covolume group action.
We prove an analogue of Kirchhoff's matrix tree theorem for computing the volume of the tropical Prym variety for double covers of metric graphs. We interpret the formula in terms of a semi-canonical decomposition of the tropical Prym variety, via a careful study of the tropical Abel-Prym map. In particular, we show that the map is harmonic, determine its degree at every cell of the decomposition, and prove that its global degree is $2^{g-1}$. Along the way, we use the Ihara zeta function to provide a new proof of the analogous result for finite graphs. As a counterpart, the appendix by Sebast…
Functional Equations and Pole Structure of the Bartholdi Zeta Function
本文研究了正则有向图上 Bartholdi zeta 函数的函数方程、临界带边界及极点结构，揭示了其与 Ihara zeta 函数的等价关系，并给出了极点阶数在特定参数下的增强条件。
Galois coverings of Schreier graphs of groups generated by bounded automata
本文研究了由有界自动机生成的群的施雷jer图的格罗-陈覆盖特性，并探讨了几个特定群的施雷jer图的齐次函数和L函数。
We consider minimal, aperiodic symbolic subshifts and show how to characterize the combinatorial property of bounded powers by means of a metric property. For this purpose we construct a family of graphs which all approximate the subshift space, and define a metric on each graph which extends to a metric on the subshift space. The characterization of bounded powers is then given by the Lipschitz equivalence of a suitably defined infimum metric with the corresponding supremum metric. We also introduce zeta-functions and relate their abscissa of convergence to various exponents of complexity of …
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