scholarly journals Edge Reconstruction of the Ihara Zeta Function

10.37236/5909 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Gunther Cornelissen ◽  
Janne Kool
Keyword(s):  
Zeta Function ◽  
Spectral Properties ◽  
Average Degree ◽  
Ihara Zeta Function ◽  
Edge Reconstruction ◽  
Simple Part ◽  
Pass Through ◽  
Natural Symmetry

We show that if a graph $G$ has average degree $\overline d \geq 4$, 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 $\overline d>4$, 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 both directions at least once.

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

Null trajectories for the symmetrized Hurwitz zeta function

2006 ◽  
Vol 463 (2077) ◽  
pp. 303-319 ◽  
Author(s):  
Ross C McPhedran ◽  
Lindsay C Botten ◽  
Nicolae-Alexandru P Nicorovici
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Critical Line ◽  
Hurwitz Zeta Function ◽  
Trigonometric Functions ◽  
Asymptotic Results ◽  
End Points ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Pass Through

We consider the Hurwitz zeta function ζ ( s , a ) and develop asymptotic results for a = p / q , with q large, and, in particular, for p / q tending to 1/2. We also study the properties of lines along which the symmetrized parts of ζ ( s , a ), ζ + ( s , a ) and ζ − ( s , a ) are zero. We find that these lines may be grouped into four families, with the start and end points for each family being simply characterized. At values of a =1/2, 2/3 and 3/4, the curves pass through points which may also be characterized, in terms of zeros of the Riemann zeta function, or the Dirichlet functions L −3 ( s ) and L −4 ( s ), or of simple trigonometric functions. Consideration of these trajectories enables us to relate the densities of zeros of L −3 ( s ) and L −4 ( s ) to that of ζ ( s ) on the critical line.

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 Quaternionic quantum walks of Szegedy type and zeta functions of graphs

2017 ◽  
Vol 17 (15&16) ◽  
pp. 1349-1371
Author(s):  
Norio Konno ◽  
Kaname Matsue ◽  
Hideo Mitsuhashi ◽  
Iwao Sato
Keyword(s):  
Zeta Function ◽  
Spectral Properties ◽  
Transition Matrix ◽  
Zeta Functions ◽  
Quantum Walks ◽  
Spectral Mapping Theorem ◽  
Spectral Mapping ◽  
Weighted Matrix ◽  
The Right ◽  
The Way

We define a quaternionic extension of the Szegedy walk on a graph and study its right spectral properties. The condition for the transition matrix of the quaternionic Szegedy walk on a graph to be quaternionic unitary is given. In order to derive the spectral mapping theorem for the quaternionic Szegedy walk, we derive a quaternionic extension of the determinant expression of the second weighted zeta function of a graph. Our main results determine explicitly all the right eigenvalues of the quaternionic Szegedy walk by using complex right eigenvalues of the corresponding doubly weighted matrix. We also show the way to obtain eigenvectors corresponding to right eigenvalues derived from those of doubly weighted matrix.

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