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    Geometrical and spectral study of β-skeleton graphs

    L. Alonso*

    J. A. Méndez-Bermúdez†

    Ernesto Estrada‡

    • Max-Planck-Institut für Physik Komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany

    • Departamento de Matemática Aplicada e Estatística, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo - Campus de São Carlos, Caixa Postal 668, 13560-970 São Carlos, São Paulo, Brazil and Instituto de Física, Benemérita Universidad Autónoma de Puebla, Apartado Postal J-48, Puebla 72570, México

    • Institute of Mathematics and Applications (IUMA), University of Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, Spain and ARAID Foundation, Government of Aragon, 50008 Zaragoza, Spain

    • *lalonso@pks.mpg.de
    • †jmendezb@ifuap.buap.mx
    • ‡estrada66@unizar.es
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    Phys. Rev. E 100, 062309 – Published 19 December, 2019

    DOI: https://doi.org/10.1103/PhysRevE.100.062309

    Abstract

    We perform an extensive numerical analysis of β-skeleton graphs, a particular type of proximity graphs. In a β-skeleton graph (BSG) two vertices are connected if a proximity rule, that depends of the parameter β∈(0,∞), is satisfied. Moreover, for β>1 there exist two different proximity rules, leading to lune-based and circle-based BSGs. First, by computing the average degree of large ensembles of BSGs we detect differences, which increase with the increase of β, between lune-based and circle-based BSGs. Then, within a random matrix theory (RMT) approach, we explore spectral and eigenvector properties of random BSGs by the use of the nearest-neighbor energy-level spacing distribution and the entropic eigenvector localization length, respectively. The RMT analysis allows us to conclude that a localization transition occurs at β=1.

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    Phys. Rev. E 100, 062309– Published 19 December, 2019
    • Received 18 July 2019
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