scholarly journals An Infinite Family of Graphs with the Same Ihara Zeta Function

10.37236/354 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Christopher Storm
Keyword(s):  
Zeta Function ◽  
Generating Functions ◽  
Infinite Family ◽  
Ihara Zeta Function ◽  
One To One

In 2009, Cooper presented an infinite family of pairs of graphs which were conjectured to have the same Ihara zeta function. We give a proof of this result by using generating functions to establish a one-to-one correspondence between cycles of the same length without backtracking or tails in the graphs Cooper proposed. Our method is flexible enough that we are able to generalize Cooper's graphs, and we demonstrate additional families of pairs of graphs which share the same zeta function.

scholarly journals Note on the Hurwitz–Lerch Zeta Function of Two Variables

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1431
Author(s):  
Junesang Choi ◽  
Recep Şahin ◽  
Oğuz Yağcı ◽  
Dojin Kim
Keyword(s):  
Zeta Function ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Zeta Functions ◽  
Integral Representations ◽  
Lerch Zeta Function ◽  
Derivative Formulas ◽  
The One ◽  
Function Of Two Variables

A number of generalized Hurwitz–Lerch zeta functions have been presented and investigated. In this study, by choosing a known extended Hurwitz–Lerch zeta function of two variables, which has been very recently presented, in a systematic way, we propose to establish certain formulas and representations for this extended Hurwitz–Lerch zeta function such as integral representations, generating functions, derivative formulas and recurrence relations. We also point out that the results presented here can be reduced to yield corresponding results for several less generalized Hurwitz–Lerch zeta functions than the extended Hurwitz–Lerch zeta function considered here. For further investigation, among possibly various more generalized Hurwitz–Lerch zeta functions than the one considered here, two more generalized settings are provided.

A new weighted Ihara zeta function for a graph

2019 ◽  
Vol 571 ◽  
pp. 154-179
Author(s):  
Norio Konno ◽  
Hideo Mitsuhashi ◽  
Hideaki Morita ◽  
Iwao Sato
Keyword(s):  
Zeta Function ◽  
Ihara Zeta Function

scholarly journals On Multiple Interpolation Functions of the Nörlund-Typeq-Euler Polynomials

2009 ◽  
Vol 2009 ◽  
pp. 1-14 ◽  
Author(s):  
Mehmet Acikgoz ◽  
Yilmaz Simsek
Keyword(s):  
Zeta Function ◽  
Generating Functions ◽  
Euler Polynomials ◽  
Multiple Interpolation ◽  
Definition Of ◽  
Interpolation Functions

In (2006) and (2009), Kim defined new generating functions of the Genocchi, Nörlund-typeq-Euler polynomials and their interpolation functions. In this paper, we give another definition of the multiple Hurwitz typeq-zeta function. This function interpolates Nörlund-typeq-Euler polynomials at negative integers. We also give some identities related to these polynomials and functions. Furthermore, we give some remarks about approximations of Bernoulli and Euler polynomials.

scholarly journals Multivariate Interpolation Functions of Higher-Orderq-Euler Numbers and Their Applications

2008 ◽  
Vol 2008 ◽  
pp. 1-16 ◽  
Author(s):  
Hacer Ozden ◽  
Ismail Naci Cangul ◽  
Yilmaz Simsek
Keyword(s):  
Zeta Function ◽  
Generating Functions ◽  
Higher Order ◽  
Multivariate Interpolation ◽  
Euler Numbers ◽  
Euler Numbers And Polynomials ◽  
Sums Of Products ◽  
Interpolation Functions

The aim of this paper, firstly, is to construct generating functions ofq-Euler numbers and polynomials of higher order by applying the fermionicp-adicq-Volkenborn integral, secondly, to define multivariateq-Euler zeta function (Barnes-type Hurwitzq-Euler zeta function) andl-function which interpolate these numbers and polynomials at negative integers, respectively. We give relation between Barnes-type Hurwitzq-Euler zeta function and multivariateq-Eulerl-function. Moreover, complete sums of products of these numbers and polynomials are found. We give some applications related to these numbers and functions as well.

scholarly journals Ihara Zeta function, coefficients of Maclaurin series and Ramanujan graphs

2020 ◽  
Vol 31 (10) ◽  
pp. 2050082
Author(s):  
Hau-Wen Huang
Keyword(s):  
Functional Equation ◽  
Zeta Function ◽  
Finite Order ◽  
Undirected Graph ◽  
Number Sequence ◽  
Maclaurin Series ◽  
Ihara Zeta Function ◽  
Ramanujan Graphs ◽  
Weil Bound

Let [Formula: see text] denote a connected [Formula: see text]-regular undirected graph of finite order [Formula: see text]. The graph [Formula: see text] is called Ramanujan whenever [Formula: see text] for all nontrivial eigenvalues [Formula: see text] of [Formula: see text]. We consider the variant [Formula: see text] of the Ihara Zeta function [Formula: see text] of [Formula: see text] defined by [Formula: see text] The function [Formula: see text] satisfies the functional equation [Formula: see text]. Let [Formula: see text] denote the number sequence given by [Formula: see text] In this paper, we establish the equivalence of the following statements: (i) [Formula: see text] is Ramanujan; (ii) [Formula: see text] for all [Formula: see text]; (iii) [Formula: see text] for infinitely many even [Formula: see text]. Furthermore, we derive the Hasse–Weil bound for the Ramanujan graphs.

scholarly journals A trace on fractal graphs and the Ihara zeta function

2009 ◽  
Vol 361 (06) ◽  
pp. 3041-3041 ◽  
Author(s):  
Daniele Guido ◽  
Tommaso Isola ◽  
Michel L. Lapidus
Keyword(s):  
Zeta Function ◽  
Ihara Zeta Function

scholarly journals Priority Queues and Multisets

10.37236/1218 ◽  
1995 ◽  
Vol 2 (1) ◽  
Author(s):  
M. D. Atkinson ◽  
S. A. Linton ◽  
L. A. Walker
Keyword(s):  
Data Structure ◽  
Generating Functions ◽  
Priority Queue ◽  
Priority Queues ◽  
A Priority ◽  
One To One ◽  
Output Component ◽  
Fixed Input ◽  
Symmetry Result

A priority queue, a container data structure equipped with the operations insert and delete-minimum, can re-order its input in various ways, depending both on the input and on the sequence of operations used. If a given input $\sigma$ can produce a particular output $\tau$ then $(\sigma,\tau)$ is said to be an allowable pair. It is shown that allowable pairs on a fixed multiset are in one-to-one correspondence with certain k-way trees and, consequently, the allowable pairs can be enumerated. Algorithms are presented for determining the number of allowable pairs with a fixed input component, or with a fixed output component. Finally, generating functions are used to study the maximum number of output components with a fixed input component, and a symmetry result is derived.

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