Geometrical and spectral study of $\beta$-skeleton graphs

We perform an extensive numerical analysis of $\beta$-skeleton graphs, a particular type of proximity graphs. In a $\beta$-skeleton graph (BSG) two vertices are connected if a proximity rule, that depends of the parameter $\beta \in (0,\mathrm{\infty })$, is satisfied. Moreover, for $\beta >1$ 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 $\beta$, between lune-based and circle-based BSGs. Then, within a random matrix theory (RMT) approach, we explore spectral and eigenvector properties of randomly weighted 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 $\beta =1$.