Ruelle zeta functions for finite digraphs

Morita, Hideaki

doi:10.1016/j.laa.2020.06.011

Cited by 6 publications

(17 citation statements)

References 19 publications

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“…This theorem is called the Konno-Sato theorem (see [12,15], for example). Konno and Tamura [14] extended the Grover matrix.…”

Section: The Grover Matrix Of a Graphmentioning

confidence: 99%

A Generalized Grover/Zeta Correspondence

Kubo¹,

Konno²,

Sato³

et al. 2022

Preprint

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We introduce a generalized Grover matrix of a graph and present an explicit formula for its characteristic polynomial. As a corollary, we give the spectra for the generalized Grover matrix of a regular graph. Next, we define a zeta function and a generalized zeta function of a graph G with respect to its generalized Grover matrix as an analog of the Ihara zeta function and present explicit formulas for their zeta functions for a vertextransitive graph. As applications, we express the limit on the generalized zeta functions of a family of finite vertex-transitive regular graphs by an integral. Furthermore, we give the limit on the generalized zeta functions of a family of finite tori as an integral expression.

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“…This theorem is called the Konno-Sato theorem (see [12,15], for example). Konno and Tamura [14] extended the Grover matrix.…”

Section: The Grover Matrix Of a Graphmentioning

confidence: 99%

A Generalized Grover/Zeta Correspondence

Kubo¹,

Konno²,

Sato³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…A standard argument shows that (x) divides any period of x ∈ X. See, e.g., Lemma 8 in [23] for a proof.…”

Section: Dynamical Systems 231 Prime Periodmentioning

confidence: 99%

“…Founded on this idea, we construct the Ihara expression for the generalized weighted zeta function by reformulating the Hashimoto expression. The generalized weighted zeta function is defined in [23] which unifies the graph zeta functions that appeared in previous studies. Also in [23], it is investigated the condition for the Hashimoto expression to exist and one can see that the condition is satisfied by the generalized weighted zeta function.…”

Section: Introductionmentioning

confidence: 99%

“…The generalized weighted zeta function is defined in [23] which unifies the graph zeta functions that appeared in previous studies. Also in [23], it is investigated the condition for the Hashimoto expression to exist and one can see that the condition is satisfied by the generalized weighted zeta function. Thus, the generalized weighted zeta function has the Hashimoto expression, and we can consider the problem to reformulate it into the Ihara expression.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 2, we recall fundamental notation on graphs, digraphs, words, and dynamical systems, which is required to define the combinatorial zeta functions. In Section 3, we review a fundamental theory of combinatorial zeta functions following [23]. We introduce the definition of a graph zeta function in general, and see that it is combinatorial.…”

Section: Introductionmentioning

confidence: 99%

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