  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
$\beta\in(0,\infty)$, 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 $\beta=1$.
