scholarly journals A weighted graph zeta function involved in the Szegedy walk

2022 ◽  
Vol 22 (1&2) ◽  
pp. 38-52
Author(s):  
Ayaka Ishikawa ◽  
Norio Konno
Keyword(s):  
Zeta Function ◽  
Characteristic Polynomial ◽  
Transition Matrix ◽  
Weighted Graph ◽  
Finite Graph

We define a new weighted zeta function for a finite graph and obtain its determinant expression. This result gives the characteristic polynomial of the transition matrix of the Szegedy walk on a graph.

scholarly journals On the class of square Petrie matrices induced by cyclic permutations

2004 ◽  
Vol 2004 (31) ◽  
pp. 1617-1622
Author(s):  
Bau-Sen Du
Keyword(s):  
Finite Field ◽  
Zeta Function ◽  
Topological Entropy ◽  
Characteristic Polynomial ◽  
Characteristic Polynomials ◽  
Dynamical Properties

Letn≥2be an integer and letP={1,2,…,n,n+1}. LetZpdenote the finite field{0,1,2,…,p−1}, wherep≥2is a prime. Then every mapσonPdetermines a realn×nPetrie matrixAσwhich is known to contain information on the dynamical properties such as topological entropy and the Artin-Mazur zeta function of the linearization ofσ. In this paper, we show that ifσis acyclicpermutation onP, then all such matricesAσare similar to one another overZ2(but not overZpfor any primep≥3) and their characteristic polynomials overZ2are all equal to∑k=0nxk. As a consequence, we obtain that ifσis acyclicpermutation onP, then the coefficients of the characteristic polynomial ofAσare all odd integers and hence nonzero.

scholarly journals Characteristic polynomials of some weighted graph bundles and its application to links

1994 ◽  
Vol 17 (3) ◽  
pp. 503-510 ◽  
Author(s):  
Moo Young Sohn ◽  
Jaeun Lee
Keyword(s):  
Characteristic Polynomial ◽  
Weighted Graph ◽  
Double Covering ◽  
Characteristic Polynomials ◽  
Weighted Graphs

In this paper, we introduce weighted graph bundles and study their characteristic polynomial. In particular, we show that the characteristic polynomial of a weightedK2(K¯2)-bundles over a weighted graphG?can be expressed as a product of characteristic polynomials two weighted graphs whose underlying graphs areGAs an application, we compute the signature of a link whose corresponding weighted graph is a double covering of that of a given link.

scholarly journals POLYNOMIAL INVARIANTS OF PSEUDO-ANOSOV MAPS

2012 ◽  
Vol 04 (01) ◽  
pp. 13-47 ◽  
Author(s):  
JOAN BIRMAN ◽  
PETER BRINKMANN ◽  
KEIKO KAWAMURO
Keyword(s):  
Characteristic Polynomial ◽  
Transition Matrix ◽  
The Other ◽  
Train Track ◽  
Polynomial Invariants ◽  
Genus 2

We investigate the structure of the characteristic polynomial det (xI - T) of a transition matrix T that is associated to a train track representative of a pseudo-Anosov map [F] acting on a surface. As a result we obtain three new polynomial invariants of [F], one of them being the product of the other two, and all three being divisors of det (xI - T). The degrees of the new polynomials are invariants of [F] and we give simple formulas for computing them by a counting argument from an invariant train-track. We give examples of genus 2 pseudo-Anosov maps having the same dilatation, and use our invariants to distinguish them.

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.

scholarly journals A Characteristic Polynomial for the Transition Probability Matrix of Correlated Random Walks on a Graph

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Takashi Komatsu ◽  
Norio Konno ◽  
Iwao Sato
Keyword(s):  
Random Walk ◽  
Zeta Function ◽  
Characteristic Polynomial ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Regular Graphs ◽  
Correlated Random Walk ◽  
Correlated Random Walks ◽  
Transition Probability Matrices ◽  
Probability Matrices

We define a correlated random walk (CRW) induced from the time evolution matrix (the Grover matrix) of the Grover walk on a graph $G$, and present a formula for the characteristic polynomial of the transition probability matrix of this CRW by using a determinant expression for the generalized weighted zeta function of $G$. As an application, we give the spectrum of the transition probability matrices for the CRWs induced from the Grover matrices of regular graphs and semiregular bipartite graphs. Furthermore, we consider another type of the CRW on a graph. 

Graph Theory in Chemistry II. Graph-Theoretical Description of Heteroconjugated Molecules

1975 ◽  
Vol 30 (12) ◽  
pp. 1696-1699 ◽  
Author(s):  
A. Graovac ◽  
O. E. Polansky ◽  
N. Trinajstić ◽  
N. Tyutyulkov
Keyword(s):  
Graph Theory ◽  
Characteristic Polynomial ◽  
Weighted Graph ◽  
Theoretical Description ◽  
Weighted Graphs ◽  
Conjugated Molecules ◽  
Molecular Graphs

Abstract The Sachs' formula which relates the structure of a (vertex-and edge) -weighted graph and its characteristic polynomial is given. Weighted graphs are used to represent conjugated molecules containing the variety of heteroatoms and heterobonds. Vertices and edges in such molecular graphs have different weights following the deviation of the corresponding (Hiickel) Coulomb and resonance integrals from the standard (benzene) values.

scholarly journals Zeta Functions with Respect to General Coined Quantum Walk of Periodic Graphs

10.37236/9104 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Takashi Komatsu ◽  
Norio Konno ◽  
Iwao Sato
Keyword(s):  
Zeta Function ◽  
Time Evolution ◽  
Zeta Functions ◽  
Quantum Walk ◽  
Finite Graph ◽  
Periodic Graphs ◽  
Periodic Graph

We define a zeta function of a graph by using the time evolution matrix of a general coined quantum walk on it, and give a determinant expression for the zeta function of a finite graph. Furthermore, we present a determinant expression for the zeta function of an (inifinite) periodic graph. 

Sign in / Sign up

Export Citation Format

Share Document