The Ihara zeta function of the complement of a semiregular bipartite graph

2021 ◽  
Vol 344 (12) ◽  
pp. 112598
Author(s):  
Deqiong Li ◽  
Yaoping Hou ◽  
Dijian Wang
Keyword(s):  
Zeta Function ◽  
Bipartite Graph ◽  
Ihara Zeta Function

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 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 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 Limiting eigenvalue distribution of random matrices of Ihara zeta function of long-range percolation graphs

2018 ◽  
Vol 07 (03) ◽  
pp. 1850007
Author(s):  
O. Khorunzhiy
Keyword(s):  
Graph Theory ◽  
Zeta Function ◽  
Long Range ◽  
Riemann Hypothesis ◽  
Counting Function ◽  
Symmetric Matrices ◽  
Ihara Zeta Function ◽  
Unique Measure ◽  
Edge Probability ◽  
Average Vertex

We consider the ensemble of [Formula: see text] real random symmetric matrices [Formula: see text] obtained from the determinant form of the Ihara zeta function associated to random graphs [Formula: see text] of the long-range percolation radius model with the edge probability determined by a function [Formula: see text]. We show that the normalized eigenvalue counting function of [Formula: see text] weakly converges in average as [Formula: see text], [Formula: see text] to a unique measure that depends on the limiting average vertex degree of [Formula: see text] given by [Formula: see text]. This measure converges in the limit of infinite [Formula: see text] to a shift of the Wigner semi-circle distribution. We discuss relations of these results with the properties of the Ihara zeta function and weak versions of the graph theory Riemann Hypothesis.

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