Weighted Zeta Function Research Articles

Zeta functions of signed graphs

We introduce the zeta function of a signed graph and give a determinant expression for it in terms of the signed oriented line graph, and moreover, we obtain some properties of the zeta function of a signed graph. As applications we affirm the conjecture proposed by Sato [Weighted zeta functions of graph coverings, Electronic J. Combin. 13 (2006), #91] and give a method to construct a family of cospectral signed digraphs. Additionally, we show that two connected regular signed graphs have the same zeta function if and only if they are cospectral.

Resonances and Weighted Zeta Functions for Obstacle Scattering via Smooth Models

AbstractWe consider a geodesic billiard system consisting of a complete Riemannian manifold and an obstacle submanifold with boundary at which the trajectories of the geodesic flow experience specular reflections. We show that if the geodesic billiard system is hyperbolic on its trapped set and the latter is compact and non-grazing, the techniques for open hyperbolic systems developed by Dyatlov and Guillarmou (Ann Henri Poincaré 17(11):3089–3146, 2016) can be applied to a smooth model for the discontinuous flow defined by the non-grazing billiard trajectories. This allows us to obtain a meromorphic resolvent for the generator of the billiard flow. As an application we prove a meromorphic continuation of weighted zeta functions together with explicit residue formulae. In particular, our results apply to scattering by convex obstacles in the Euclidean plane.

The Ihara expression of a generalization of the weighted zeta function on a finite digraph

ABSTRACT We define a new weighted zeta function for a finite digraph and obtain its determinant expression called the Ihara expression. The graph zeta function is a generalization of the weighted graph zeta function introduced in previous research. That is, our result makes it possible to derive the Ihara expressions of the previous graph zeta functions for any finite digraphs.

Meromorphic Continuation of Weighted Zeta Functions on Open Hyperbolic Systems

In this article we prove meromorphic continuation of weighted zeta functions Z_f in the framework of open hyperbolic systems by using the meromorphically continued restricted resolvent of Dyatlov and Guillarmou (Ann Henri Poincaré 17(11):3089–3146, 2016). We obtain a residue formula proving equality between residues of Z_f and invariant Ruelle distributions. We combine this equality with results of Guillarmou et al. (Ann Henri Lebesgue 4:81–119, 2021) in order to relate the residues to Patterson–Sullivan distributions. Finally we provide proof-of-principle results concerning the numerical calculation of invariant Ruelle distributions for 3-disc scattering systems.

Semiclassical formulae for Wigner distributions

In this paper we give an overview over some aspects of the modern mathematical theory of Ruelle resonances for chaotic, i.e. uniformly hyperbolic, dynamical systems and their implications in physics. First we recall recent developments in the mathematical theory of resonances, in particular how invariant Ruelle distributions arise as residues of weighted zeta functions. Then we derive a correspondence between weighted and semiclassical zeta functions in the setting of negatively curved surfaces. Combining this with results of Hilgert, Guillarmou and Weich yields a high frequency interpretation of invariant Ruelle distributions as quantum mechanical matrix coefficients in constant negative curvature. We finish by presenting numerical calculations of phase space distributions in the more physical setting of three-disk scattering systems.

A weighted graph zeta function involved in the Szegedy walk

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.

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

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.

The Ihara expression for the generalized weighted zeta function of a finite simple graph

Graph zeta functions have two kinds of determinant expressions, the Hashimoto expression and the Ihara expression. The main theorem of the present article gives the Ihara expression for graph zeta functions with a certain generalized weight. It unifies all the Ihara expressions known for the existing zeta functions of finite simple graphs.

Quaternionic quantum walks of Szegedy type and zeta functions of graphs

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.

The Discrete-time Quaternionic Quantum Walk and the Second Weighted Zeta Function on a Graph

We define the quaternionic quantum walk on a finite graph and investigate its properties. This walk can be considered as a natural quaternionic extension of the Grover walk on a graph. We explain the way to obtain all the right eigenvalues of a quaternionic matrix and a notable property derived from the unitarity condition for the quaternionic quantum walk. Our main results determine all the right eigenvalues of the quaternionic quantum walk by using complex eigenvalues of the quaternionic weighted matrix which is easily derivable from the walk. Since our derivation is owing to a quaternionic generalization of the determinant expression of the second weighted zeta function, we explain the second weighted zeta function and the relationship between the walk and the second weighted zeta function.

The quaternionic second weighted zeta function of a graph and the Study determinant

We establish a generalization of the second weighted zeta function of a graph to the case of quaternions. For an arc-weighted graph whose weights are quaternions, we define the second weighted zeta function by using the Study determinant that is a quaternionic determinant for quaternionic matrices defined by Study. This definition is regarded as a quaternionic analogue of the determinant expression of Hashimoto type for the Ihara zeta function of a graph. We derive the Study determinant expression of Bass type and the Euler product for the quaternionic second weighted zeta function.

The quaternionic weighted zeta function of a graph

We establish the quaternionic weighted zeta function of a graph and its Study determinant expressions. For a graph with quaternionic weights on arcs, we define a zeta function by using an infinite product, which is regarded as the Euler product. This is a quaternionic extension of the square of the Ihara zeta function. We show that the new zeta function can be expressed as the exponential of a generating function and that it has two Study determinant expressions, which are crucial for the theory of zeta functions of graphs.

Euler flag enumeration of Whitney stratified spaces

The flag vector contains all the face incidence data of a polytope, and in the poset setting, the chain enumerative data. It is a classical result due to Bayer and Klapper that for face lattices of polytopes, and more generally, Eulerian graded posets, the flag vector can be written as a cd-index, a non-commutative polynomial which removes all the linear redundancies among the flag vector entries. This result holds for regular CW complexes.We relax the regularity condition to show the cd-index exists for Whitney stratified manifolds by extending the notion of a graded poset to that of a quasi-graded poset. This is a poset endowed with an order-preserving rank function and a weighted zeta function. This allows us to generalize the classical notion of Eulerianness, and obtain a cd-index in the quasi-graded poset arena. We also extend the semi-suspension operation to that of embedding a complex in the boundary of a higher dimensional ball and study the simplicial shelling components.

Some weighted Bartholdi zeta function of a digraph

Related to the weighted bond scattering matrix of a graph, we define some weighted Bartholdi zeta function of a digraph D, and give a determinant expression for it. We present a decomposition formula for the weighted zeta function of a group covering of D. Furthermore, we define an L-function of D, and give a determinant expression for it. As a corollary, we express the weighted Bartholdi zeta function of a group covering of D by means of its L-functions.

On the relation between quantum walks and zeta functions

We present an explicit formula for the characteristic polynomial of the transition matrix of the discrete-time quantum walk on a graph via the second weighted zeta function. As applications, we obtain new proofs for the results on spectra of the transition matrix and its positive support.

Weighted zeta functions for quotients of regular coverings of graphs

Let G be a connected graph. We reformulate Stark and Terras' Galois Theory for a quotient H of a regular covering K of a graph G by using voltage assignments. As applications, we show that the weighted Bartholdi L-function of H associated to the representation of the covering transformation group of H is equal to that of G associated to its induced representation in the covering transformation group of K. Furthermore, we express the weighted Bartholdi zeta function of H as a product of weighted Bartholdi L-functions of G associated to irreducible representations of the covering transformation group of K. We generalize Stark and Terras' Galois Theory to digraphs, and apply to weighted Bartholdi L-functions of digraphs.

The Scattering Matrix of a Graph

Recently, Smilansky expressed the determinant of the bond scattering matrix of a graph by means of the determinant of its Laplacian. We present another proof for this Smilansky's formula by using some weighted zeta function of a graph. Furthermore, we reprove a weighted version of Smilansky's formula by Bass' method used in the determinant expression for the Ihara zeta function of a graph.

The stochastic weighted complexity of a group covering of a digraph

We consider the (stochastic) weighted complexity of a digraph D and a stochastic function from A(D) to nonnegative real numbers. Furthermore, we consider the weighted zeta function of a digraph, and present a determinant expression of it. We present a decomposition formula for the weighted zeta function of a group covering of a digraph. As an application, we give an explicit formula for the stochastic weighted complexity of a group covering of a digraph by using its stochastic weighted complexity.

Weighted Zeta Functions of Graph Coverings

We present a decomposition formula for the weighted zeta function of an irregular covering of a graph by its weighted $L$-functions. Moreover, we give a factorization of the weighted zeta function of an (irregular or regular) covering of a graph by equivalence classes of prime, reduced cycles of the base graph. As an application, we discuss the structure of balanced coverings of signed graphs.

Weighted zeta functions of graphs

We give a decomposition formula for the weighted zeta function of a regular covering of a graph.