#### DMCA

## Ihara zeta functions for periodic SIMPLE GRAPHS (2008)

Citations: | 11 - 4 self |

### Citations

252 |
Discrete groups, expanding graphs and invariant measures (Birkhäuser
- Lubotzky
- 2010
(Show Context)
Citation Context ...e graph, RX be the radius of the greatest circle of convergence of ZX. Denote by πn the number of prime cycles which have length n. If g.c.d.{|C| : C ∈ P} = 1, then πn ∼ R−n X , n → ∞. n Theorem 2.9. =-=[17, 19, 26]-=- Let X be a finite graph which is (q + 1)-regular, i.e. deg(v) = q + 1 for all v ∈ V X. Then the following are equivalent (i) (RH) � ZX(q −s ) −1 = 0 Rs ∈ (0, 1) =⇒ Rs = 1 2 . (ii) X is a Ramanujan gr... |

96 | A survey on spectra of infinite graphs
- Mohar, Woess
- 1989
(Show Context)
Citation Context ... ∈ Γ} ′ of bounded operators on ℓ2 (V X) commuting with the action of Γ inherits a trace given by TrΓ(T) = � x∈F T(x, x), for T ∈ N(X, Γ). Let us denote by A the adjacency matrix of X. Then (by [21], =-=[22]-=-) �A� ≤ d := sup v∈V X deg(v) < ∞, and it is easy to see that A ∈ N(X, Γ). For any m ∈ N, let us denote by Am(x, y) the number of proper paths in X, of length m, with initial vertex x and terminal ver... |

92 |
The Ihara-Selberg zeta function of a tree lattice
- Bass
(Show Context)
Citation Context ...ted determinant formula, based on the treatment developed by Stark and Terras for finite graphs. 1. Introduction The zeta functions associated to finite graphs by Ihara [17], Hashimoto [12, 13], Bass =-=[2]-=- and others, combine features of Riemann’s zeta function, Artin L-functions, and Selberg’s zeta function, and may be viewed as analogues of the Dedekind zeta functions of a number field. They are defi... |

91 |
Tree lattices
- Bass, Lubotzky
- 2001
(Show Context)
Citation Context ... formula has been proved. They deduce this result as a specialization of the treatment of group actions on trees (the so-called theory of tree lattices, as developed by Bass, Lubotzky and others, see =-=[3]-=-). The purpose of this work is to give a more direct proof of that result, for the case of periodic simple graphs with a free action. We hope that our treatment, being quite elementary, could be usefu... |

91 |
Zeta functions of finite graphs and coverings.
- Stark, Terras
- 1996
(Show Context)
Citation Context ...to a meromorphic function satisfying a functional equation. They can be expressed as the determinant of a perturbation of the graph Laplacian and, for Ramanujan graphs, satisfy the Riemann hypothesis =-=[26]-=-. The first attempt in this context to study infinite graphs was made by Grigorchuk and ˙ Zuk [9], who considered graphs obtained as a suitable limit of a sequence of finite graphs. They proved that t... |

78 |
Determinant theory in finite factors
- Fuglede, Kadison
- 1952
(Show Context)
Citation Context ...on 6 to prove a determinant formula for the zeta function. BsIHARA ZETA FUNCTION FOR PERIODIC SIMPLE GRAPHS 9 P Figure 5. A cycle with |ΓC| = 4 P Figure 6. A cycle with |ΓC| = 2 In a celebrated paper =-=[8]-=-, Fuglede and Kadison defined a positive-valued determinant for finite factors (i.e. von Neumann algebras with trivial center and finite trace). Such a determinant is defined on all invertible element... |

75 |
On discrete subgroups of the two by two projective linear group over p-adic fields
- Ihara
- 1966
(Show Context)
Citation Context ...give a new proof of the associated determinant formula, based on the treatment developed by Stark and Terras for finite graphs. 1. Introduction The zeta functions associated to finite graphs by Ihara =-=[17]-=-, Hashimoto [12, 13], Bass [2] and others, combine features of Riemann’s zeta function, Artin L-functions, and Selberg’s zeta function, and may be viewed as analogues of the Dedekind zeta functions of... |

74 |
Zeta functions of finite graphs and representations of p-adic groups. Automorphic forms and geometry of arithmetic varieties,
- Hashimoto
- 1989
(Show Context)
Citation Context ... of the associated determinant formula, based on the treatment developed by Stark and Terras for finite graphs. 1. Introduction The zeta functions associated to finite graphs by Ihara [17], Hashimoto =-=[12, 13]-=-, Bass [2] and others, combine features of Riemann’s zeta function, Artin L-functions, and Selberg’s zeta function, and may be viewed as analogues of the Dedekind zeta functions of a number field. The... |

70 |
Répartition asymptotique des valeurs propres de l’opérateur de Hecke Tp
- Serre
- 1997
(Show Context)
Citation Context ... (0, 1) =⇒ Rs = 1 2 . (ii) X is a Ramanujan graph, i.e. λ ∈ σ(A), |λ| < q + 1 =⇒ |λ| ≤ 2 √ q.sIHARA ZETA FUNCTION FOR PERIODIC SIMPLE GRAPHS 5 More results on the Ihara zeta function are contained in =-=[25, 27, 28]-=- and in various papers by Mizuno and Sato. In closing this section, we mention a generalization of the Ihara zeta function recently introduced by Bartholdi [1] and studied by Mizuno and Sato (see [20]... |

39 | Zeta functions of finite graphs.
- Kotani, Sunada
- 2000
(Show Context)
Citation Context ... graphs; subsequently, through the efforts of Sunada [29], Hashimoto [12, 13] and Bass [2], that result was proved in full generality. Nowadays, there exist many different proofs of Theorem 2.5, e.g. =-=[26, 7, 18]-=-. To state it, we need to introduce some more notation. Let us denote by A = [A(v, w)], v, w ∈ V X, the adjacency matrix of X, that is, A(v, w) = � 1 {v, w} ∈ EX 0 otherwise.s4 DANIELE GUIDO, TOMMASO ... |

30 |
Counting paths in graphs. Enseign
- Bartholdi
- 1999
(Show Context)
Citation Context ... zeta function are contained in [25, 27, 28] and in various papers by Mizuno and Sato. In closing this section, we mention a generalization of the Ihara zeta function recently introduced by Bartholdi =-=[1]-=- and studied by Mizuno and Sato (see [20] and references therein). 3. Periodic simple graphs Let X = (V X, EX) be a simple graph, which we assume to be (countable and) with bounded degree, i.e. the de... |

25 |
The spectrum of an infinite graph
- Mohar
- 1982
(Show Context)
Citation Context ...γ) : γ ∈ Γ} ′ of bounded operators on ℓ2 (V X) commuting with the action of Γ inherits a trace given by TrΓ(T) = � x∈F T(x, x), for T ∈ N(X, Γ). Let us denote by A the adjacency matrix of X. Then (by =-=[21]-=-, [22]) �A� ≤ d := sup v∈V X deg(v) < ∞, and it is easy to see that A ∈ N(X, Γ). For any m ∈ N, let us denote by Am(x, y) the number of proper paths in X, of length m, with initial vertex x and termin... |

24 | A combinatorial proof of Bass’s evaluation of the Ihara-Selberg zeta function for graphs
- Foata, Zeilberger
- 1999
(Show Context)
Citation Context ... graphs; subsequently, through the efforts of Sunada [29], Hashimoto [12, 13] and Bass [2], that result was proved in full generality. Nowadays, there exist many different proofs of Theorem 2.5, e.g. =-=[26, 7, 18]-=-. To state it, we need to introduce some more notation. Let us denote by A = [A(v, w)], v, w ∈ V X, the adjacency matrix of X, that is, A(v, w) = � 1 {v, w} ∈ EX 0 otherwise.s4 DANIELE GUIDO, TOMMASO ... |

19 |
On zeta and L-functions of finite graphs
- Hashimoto
- 1990
(Show Context)
Citation Context ...nt formula. We obtain ZX(u) −1 = (1 − u 2 ) 2 (1 − u)(1 − 2u)(1 + u + 2u 2 ) 3 . Figure 3. A graph The zeta function has been used to establish some properties of the graphs. For example Theorem 2.7. =-=[13, 14, 2, 23, 18]-=- Let X be a finite graph, r = |EX| − |V X|+1 the rank of the fundamental group π1(X, x0). Then r is the order of the pole of ZX(u) at u = 1. If r > 1 lim u→1− ZX(u)(1 − u) r 1 = − 2r , (r − 1)κX where... |

18 |
The Ihara zeta function of infinite graphs, the KNS spectral measure and integrable maps, in: “Random Walks and Geometry
- Grigorchuk, A
- 2004
(Show Context)
Citation Context ...nt of a perturbation of the graph Laplacian and, for Ramanujan graphs, satisfy the Riemann hypothesis [26]. The first attempt in this context to study infinite graphs was made by Grigorchuk and ˙ Zuk =-=[9]-=-, who considered graphs obtained as a suitable limit of a sequence of finite graphs. They proved that their definition does not depend on the approximating sequence in case of Cayley graphs of finitel... |

15 | What are zeta functions of graphs and what are they good for
- Horton, Stark, et al.
- 2006
(Show Context)
Citation Context ...f the fundamental group π1(X, x0). Then r is the order of the pole of ZX(u) at u = 1. If r > 1 lim u→1− ZX(u)(1 − u) r 1 = − 2r , (r − 1)κX where κX is the number of spanning trees in X. Theorem 2.8. =-=[15, 16]-=- Let X be a finite graph, RX be the radius of the greatest circle of convergence of ZX. Denote by πn the number of prime cycles which have length n. If g.c.d.{|C| : C ∈ P} = 1, then πn ∼ R−n X , n → ∞... |

14 |
Selberg–Ihara’s zeta function for p-adic discrete groups, in “Automorphic Forms and Geometry of Arithmetic Varieties
- Hashimoto, Hori
- 1989
(Show Context)
Citation Context ... of the associated determinant formula, based on the treatment developed by Stark and Terras for finite graphs. 1. Introduction The zeta functions associated to finite graphs by Ihara [17], Hashimoto =-=[12, 13]-=-, Bass [2] and others, combine features of Riemann’s zeta function, Artin L-functions, and Selberg’s zeta function, and may be viewed as analogues of the Dedekind zeta functions of a number field. The... |

13 | A trace on fractal graphs and the Ihara zeta function
- Guido, Isola, et al.
(Show Context)
Citation Context ... in Section 5. In the final section, we establish several functional equations. In closing this introduction, we note that the operator-algebraic techniques used here are introduced by the authors in =-=[10]-=- in order to study the Ihara zeta functions attached to a new class of infinite graphs, called self-similar fractal graphs. 2. Zeta function for finite graphs The Ihara zeta function is defined by mea... |

12 |
A note on the zeta function of a graph
- Northshield
- 1998
(Show Context)
Citation Context ...nt formula. We obtain ZX(u) −1 = (1 − u 2 ) 2 (1 − u)(1 − 2u)(1 + u + 2u 2 ) 3 . Figure 3. A graph The zeta function has been used to establish some properties of the graphs. For example Theorem 2.7. =-=[13, 14, 2, 23, 18]-=- Let X be a finite graph, r = |EX| − |V X|+1 the rank of the fundamental group π1(X, x0). Then r is the order of the pole of ZX(u) at u = 1. If r > 1 lim u→1− ZX(u)(1 − u) r 1 = − 2r , (r − 1)κX where... |

11 | Zeta functions of discrete groups acting on trees
- Clair, Mokhtari-Sharghi
(Show Context)
Citation Context ...er graphs of a pair (G, H) of a finitely generated group G and a separable subgroup H. The definition of the zeta function was extended to (countable) periodic graphs by Clair and Mokhtari-Sharghi in =-=[4]-=-, where the determinant formula has been proved. They deduce this result as a specialization of the treatment of group actions on trees (the so-called theory of tree lattices, as developed by Bass, Lu... |

10 | Ihara’s zeta function for periodic graphs and its approximation in the amenable case
- Guido, Isola, et al.
(Show Context)
Citation Context ...he case of periodic simple graphs with a free action. We hope that our treatment, being quite elementary, could be useful for someone seeking an introduction to the subject. In a sequel to this paper =-=[11]-=-, we shall prove that for periodic amenable graphs, the Ihara zeta function can be approximated by the zeta functions of a Date: August 9, 2006. 2000 Mathematics Subject Classification. 05C25; 05C38; ... |

10 |
Artin type L-functions and the density theorem for prime cycles on finite graphs
- Hashimoto
- 1992
(Show Context)
Citation Context ...f the fundamental group π1(X, x0). Then r is the order of the pole of ZX(u) at u = 1. If r > 1 lim u→1− ZX(u)(1 − u) r 1 = − 2r , (r − 1)κX where κX is the number of spanning trees in X. Theorem 2.8. =-=[15, 16]-=- Let X be a finite graph, RX be the radius of the greatest circle of convergence of ZX. Denote by πn the number of prime cycles which have length n. If g.c.d.{|C| : C ∈ P} = 1, then πn ∼ R−n X , n → ∞... |

6 | Convergence of zeta functions of graphs
- Clair, Mokhtari-Sharghi
(Show Context)
Citation Context ...or v = (v1, v2) ∈ V X = Z2 . X P Figure 4. A periodic graph X with its quotient B = X/Γ The interested reader can find the computation of the Ihara zeta function for several periodic simple graphs in =-=[9, 4, 5, 6]-=-. 5. An analytic determinant for von Neumann algebras with a finite trace In this section, we define a determinant for a suitable class of not necessarily normal operators in a von Neumann algebra wit... |

3 |
Bartholdi zeta functions of some graphs
- Mizuno, Sato
(Show Context)
Citation Context ... 28] and in various papers by Mizuno and Sato. In closing this section, we mention a generalization of the Ihara zeta function recently introduced by Bartholdi [1] and studied by Mizuno and Sato (see =-=[20]-=- and references therein). 3. Periodic simple graphs Let X = (V X, EX) be a simple graph, which we assume to be (countable and) with bounded degree, i.e. the degree of the vertices is uniformly bounded... |

3 |
L-functions in geometry and applications, Springer Lecture Notes in Math. 1201
- Sunada
- 1986
(Show Context)
Citation Context ...(Zeta function). ZX(u) := � (1 − u |C| ) −1 , u ∈ C. C∈P Ihara also proved the main result of this theory, though in the particular case of regular graphs; subsequently, through the efforts of Sunada =-=[29]-=-, Hashimoto [12, 13] and Bass [2], that result was proved in full generality. Nowadays, there exist many different proofs of Theorem 2.5, e.g. [26, 7, 18]. To state it, we need to introduce some more ... |

2 |
Zeta functions of graphs with Z actions, preprint
- Clair
- 2006
(Show Context)
Citation Context ...or v = (v1, v2) ∈ V X = Z2 . X P Figure 4. A periodic graph X with its quotient B = X/Γ The interested reader can find the computation of the Ihara zeta function for several periodic simple graphs in =-=[9, 4, 5, 6]-=-. 5. An analytic determinant for von Neumann algebras with a finite trace In this section, we define a determinant for a suitable class of not necessarily normal operators in a von Neumann algebra wit... |