Bartholdi zeta and L-functions of weighted digraphs, their coverings and products

Choe, Youngbin; Kwak, Jin Ho; Park, Yong Sung; Sato, Iwao

doi:10.1016/j.aim.2007.01.013

Cited by 13 publications

(10 citation statements)

References 28 publications

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“…Finally, zeta functions and L-functions have been well studied in the case of weighted graphs and as covering graphs. See [2,8,14,15,16,17]. A natural question to ask is how properties of the natural zeta and L-functions arising from these weighted covering graphs compare to those in the above cases.…”

Section: Further Questionsmentioning

confidence: 99%

“…Our construction relies on realizing the graphs as a special type of covering graph called weighted covering graphs which generalize standard covering graphs for simple graphs. Weighted covering graphs are similar in structure to multi-edged covering graphs studied in [2,4].…”

Section: Introductionmentioning

confidence: 99%

“…See [2,4,8,14] among others. While authors have considered zeta functions of both weighted and standard covering graphs, the definitions we present here are different in the use of weights.…”

Section: Introductionmentioning

confidence: 99%

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Ramanujan graphs arising as weighted Galois covering graphs

Minei¹,

Skogman

2018

EJGTA

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We give a new construction of Ramanujan graphs using a generalized type of covering graph called a weighted covering graph. For a given prime p the basic construction produces bipartite Ramanujan graphs with 4p vertices and degrees 2N where roughly p+1− √ 2p < N ≤ p. We then give generalizations to produce Ramanujan graphs of other sizes and degrees as well as general results about base graphs which have weighted covers that satisfy their Ramanujan bounds. To do the construction, we define weighted covering graphs and distinguish a subclass of Galois weighted covers that allows for block diagonalization of the adjacency matrix. The specific construction allows for easy computation of the resulting blocks. The Gershgorin Circle Theorem is then used to compute the Ramanujan bounds on the spectra.

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Section: Further Questionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Ramanujan graphs arising as weighted Galois covering graphs

Minei¹,

Skogman

2018

EJGTA

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“…Bartholdi also showed that some results for the Ihara zeta function extend to this new zeta function. We quote [5,18,19,20] for further results and generalizations of the Bartholdi zeta function. The extension to the case of infinite periodic simple graphs is contained in [12], where a functional equation for regular graphs and a determinant formula are proved.…”

mentioning

confidence: 99%

Zeta Functions for Infinite Graphs and Functional Equations

Guido¹,

Isola²

2013

Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics II

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Abstract. The definitions and main properties of the Ihara and Bartholdi zeta functions for infinite graphs are reviewed. The general question of the validity of a functional equation is discussed, and various possible solutions are proposed. IntroductionIn this paper, we review the main results concerning the Ihara zeta function and the Bartholdi zeta function for infinite graphs. Moreover, we propose various possible solutions to the problem of the validity of a functional equation for those zeta functions.The zeta function associated to a finite graph by Ihara, Sunada, Hashimoto and others, combines features of Riemann's zeta function, Artin L-functions, and Selberg's zeta function, and may be viewed as an analogue of the Dedekind zeta function of a number field [3,14,15,16,17,25,26]. It is defined by an Euler product over proper primitive cycles of the graph.A main result for the Ihara zeta function Z X (z) associated with a graph X, is the so called determinant formula, which shows that the inverse of this function can be written, up to a polynomial, as det(I − Az + Qz 2 ), where A is the adjacency matrix and Q is the diagonal matrix corresponding to the degree minus 1. As a consequence, for a finite graph, Z X (z) is indeed the inverse of a polynomial, hence can be extended meromorphically to the whole plane.A second main result is the fact that, for (q + 1)-regular graphs, namely graphs with degree constantly equal to (q + 1), Z X , or better its so called completion ξ X , satisfies a functional equation, namely is invariant under the transformation z → 1 qz . The first of the mentioned results has been proved for infinite (periodic or fractal) graphs in [11,10], by introducing the analytic determinant for operator 2000 Mathematics Subject Classification. 05C25; 05C38; 46Lxx; 11M41.

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“…Further results and generalizations of the Bartholdi zeta function are contained, for example, in [3,12,13,14].…”

Section: Introductionmentioning

confidence: 99%