Zeta Functions Of Graphs Research Articles

Quantum entanglement & purity testing: A graph zeta function perspective

We assign an arbitrary density matrix to a weighted graph and associate to it a graph zeta function that is both a generalization of the Ihara zeta function and a special case of the edge zeta function. We show that a recently developed bipartite pure state separability algorithm based on the symmetric group is equivalent to the condition that the coefficients in the exponential expansion of this zeta function are unity. Moreover, there is a one-to-one correspondence between the nonzero eigenvalues of a density matrix and the singularities of its zeta function. Several examples are given to illustrate these findings.

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.

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.

Kac–Ward Formula and Its Extension to Order–Disorder Correlators Through a Graph Zeta Function

A streamlined derivation of the Kac-Ward formula for the planar Ising model's partition function is presented and applied in relating the kernel of the Kac-Ward matrices' inverse with the correlation functions of the Ising model's order-disorder correlation functions. A shortcut for both is facilitated by the Bowen-Lanford graph zeta function relation. The Kac-Ward relation is also extended here to produce a family of non planar interactions on $\mathbb{Z}^2$ for which the partition function and the order-disorder correlators are solvable at special values of the coupling parameters/temperature.

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.

A novel conductivity mechanism of highly disordered carbon systems based on an investigation of graph zeta function

In the previous report (Matsutani and Suzuki, 2000 [21]), by proposing the mechanism under which electric conductivity is caused by the activational hopping conduction with the Wigner surmise of the level statistics, the temperature-dependent of electronic conductivity of a highly disordered carbon system was evaluated including apparent metal–insulator transition. Since the system consists of small pieces of graphite, it was assumed that the reason why the level statistics appears is due to the behavior of the quantum chaos in each granular graphite. In this article, we revise the assumption and show another origin of the Wigner surmise, which is more natural for the carbon system based on a recent investigation of graph zeta function in graph theory. Our method can be applied to the statistical treatment of the electronic properties of the randomized molecular system in general.

Quaternionic Grover Walks and Zeta Functions of Graphs with Loops

For a graph with at most one loop at each vertex, we define a discrete-time quaternionic quantum walk on the graph, which can be viewed as a quaternionic extension of the Grover walk on the graph. We derive the unitary condition for the transition matrix of the quaternionic Grover walk, and discuss the relationship between the right spectra of the transition matrices and zeta functions of graphs.

Deitmar schemes, graphs and zeta functions

In Thas (2014) it was explained how one can naturally associate a Deitmar scheme (which is a scheme defined over the field with one element, F 1 ) to a so-called “loose graph” (which is a generalization of a graph). Several properties of the Deitmar scheme can be proven easily from the combinatorics of the (loose) graph, and known realizations of objects over F 1 such as combinatorial F 1 -projective and F 1 -affine spaces exactly depict the loose graph which corresponds to the associated Deitmar scheme. In this paper, we first modify the construction of loc. cit. , and show that Deitmar schemes which are defined by finite trees (with possible end points) are “defined over F 1 ” in Kurokawa’s sense; we then derive a precise formula for the Kurokawa zeta function for such schemes (and so also for the counting polynomial of all associated F q -schemes). As a corollary, we find a zeta function for all such trees which contains information such as the number of inner points and the spectrum of degrees, and which is thus very different than Ihara’s zeta function (which is trivial in this case). Using a process called “surgery,” we show that one can determine the zeta function of a general loose graph and its associated {Deitmar, Grothendieck}-schemes in a number of steps, eventually reducing the calculation essentially to trees. We study a number of classes of examples of loose graphs, and introduce the Grothendieck ring of F 1 -schemes along the way in order to perform the calculations. Finally, we include a computer program for performing more tedious calculations, and compare the new zeta function to Ihara’s zeta function for graphs in a number of examples.

A functional relation for L-functions of graphs equivalent to the Riemann Hypothesis for Dirichlet L-functions

In this note we define L-functions of finite graphs and study the particular case of finite cycles in the spirit of a previous paper that studied spectral zeta functions of graphs. The main result is a suggestive equivalence between an asymptotic functional equation for these L-functions and the corresponding case of the Generalized Riemann Hypothesis. We also establish a relation between the positivity of such functions and the existence of real zeros in the critical strip of the classical Dirichlet L-functions with the same character.

The Feynman Identity for Planar Graphs

The Feynman identity (FI) of a planar graph relates the Euler polynomial of the graph to an infinite product over the equivalence classes of closed nonperiodic signed cycles in the graph. The main objectives of this paper are to compute the number of equivalence classes of nonperiodic cycles of given length and sign in a planar graph and to interpret the data encoded by the FI in the context of free Lie superalgebras. This solves in the case of planar graphs a problem first raised by S. Sherman and sets the FI as the denominator identity of a free Lie superalgebra generated from a graph. Other results are obtained. For instance, in connection with zeta functions of graphs.

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.

Spectral zeta functions of graphs and the Riemann zeta function in the critical strip

We initiate the study of spectral zeta functions $\zeta_X$ for finite and infinite graphs $X$, instead of the Ihara zeta function, with a perspective towards zeta functions from number theory and connections to hypergeometric functions. The Riemann hypothesis is shown to be equivalent to an approximate functional equation of graph zeta functions. The latter holds at all points where Riemann's zeta function $\zeta(s)$ is non-zero. This connection arises via a detailed study of the asymptotics of the spectral zeta functions of finite torus graphs in the critcal strip and estimates on the real part of the logarithmic derivative of $\zeta(s)$. We relate $\zeta_{\mathbb{Z}}$ to Euler's beta integral and show how to complete it giving the functional equation $\xi_{\mathbb{Z}}(1-s)=\xi_{\mathbb{Z}}(s)$. This function appears in the theory of Eisenstein series although presumably with this spectral intepretation unrecognized. In higher dimensions $d$ we provide a meromorphic continuation of $\zeta_{\mathbb{Z}^d}(s)$ to the whole plane and identify the poles. From our aymptotics several known special values of $\zeta(s)$ are derived as well as its non-vanishing on the line $Re(s)=1$. We determine the spectral zeta functions of regular trees and show it to be equal to a specialization of Appell's hypergeometric function $F_1$ via an Euler-type integral formula due to Picard.

A generalization of the graph theory prime-number theorem of a finite graph

In the first half, we present a generalization of the graph theory prime-number theorem due to Audrey Terras et al. In the last half, we study relations between several densities of a subset consisting of primes in a graph, and by using the first result, we compute a distribution of a subset. This is an answer for the "research problem" in the page 197 of Terras' textbook "Zeta Functions of Graphs."

Seiberg duality, quiver gauge theories, and Ihara’s zeta function

We study Ihara’s zeta function for graphs in the context of quivers arising from gauge theories, especially under Seiberg duality transformations. The distribution of poles is studied as we proceed along the duality tree, in light of the weak and strong graph versions of the Riemann Hypothesis. As a by-product, we find a refined version of Ihara’s zeta function to be the generating function for the generic superpotential of the gauge theory.

A Note on Jacobians, Tutte Polynomials, and Two-Variable Zeta Functions of Graphs

We address questions posed by Lorenzini about relations between Jacobians, Tutte polynomials, and the Brill–Noether theory of finite graphs, as encoded in his two-variable zeta functions. In particular, we give examples showing that none of these invariants is determined by the other two.

A remark on zeta functions of finite graphs via quantum walks

From the viewpoint of quantum walks, the Ihara zeta function of a finite graph can be said to be closely related to its evolution matrix. In this note we introduce another kind of zeta function of a graph, which is closely related to, as to say, the square of the evolution matrix of a quantum walk. Then we give to such a function two types of determinant expressions and derive from it some geometric properties of a finite graph. As an application, we illustrate the distribution of poles of this function comparing with those of the usual Ihara zeta function.

Zeta functions of complexes arising from [formula omitted

In this paper we obtain a closed form expression of the zeta function Z(XΓ,u) of a finite quotient XΓ of the Bruhat–Tits building of PGL3 over a nonarchimedean local field F by a discrete cocompact torsion-free subgroup Γ of PGL3. Analogous to a graph zeta function, Z(XΓ,u) is a rational function with two different expressions and it satisfies the Riemann hypothesis if and only if XΓ is a Ramanujan complex.

Book Review: Zeta functions of graphs: a stroll through the garden

Book Review: Zeta functions of graphs: a stroll through the garden

Zeta functions of graphs: a stroll through the garden

List of illustrations Preface Part I. A Quick Look at Various Zeta Functions: 1. Riemann's zeta function and other zetas from number theory 2. Ihara's zeta function 3. Selberg's zeta function 4. Ruelle's zeta function 5. Chaos Part II. Ihara's Zeta Function and the Graph Theory Prime Number Theorem: 6. Ihara zeta function of a weighted graph 7. Regular graphs, location of poles of zeta, functional equations 8. Irregular graphs: what is the RH? 9. Discussion of regular Ramanujan graphs 10. The graph theory prime number theorem Part III. Edge and Path Zeta Functions: 11. The edge zeta function 12. Path zeta functions Part IV. Finite Unramified Galois Coverings of Connected Graphs: 13. Finite unramified coverings and Galois groups 14. Fundamental theorem of Galois theory 15. Behavior of primes in coverings 16. Frobenius automorphisms 17. How to construct intermediate coverings using the Frobenius automorphism 18. Artin L-functions 19. Edge Artin L-functions 20. Path Artin L-functions 21. Non-isomorphic regular graphs without loops or multiedges having the same Ihara zeta function 22. The Chebotarev Density Theorem 23. Siegel poles Part V. Last Look at the Garden: 24. An application to error-correcting codes 25. Explicit formulas 26. Again chaos 27. Final research problems References Index.

Zeta functions of quantum graphs

In this paper, we construct zeta functions of quantum graphs using a contour integral technique based on the argument principle. We start by considering the special case of the star graph with Neumann matching conditions at the center of the star. We then extend the technique to allow any matching conditions at the center for which the Laplace operator is self-adjoint and finally obtain an expression for the zeta function of any graph with general vertex matching conditions. In the process, it is convenient to work with new forms for the secular equation of a quantum graph that extend the well-known secular equation of the Neumann star graph. In the second half of this paper, we apply the zeta function to obtain new results for the spectral determinant, vacuum energy and heat kernel coefficients of quantum graphs. These have all been topics of current research in their own right and in each case this unified approach significantly expands results in the literature.