Geometrical and spectral study of β-skeleton graphs.

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.