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We perform an extensive numerical analysis of $\beta$-skeleton graphs, aparticular type of proximity graphs. In a $\beta$-skeleton graph (BSG) twovertices are connected if a proximity rule, that depends of the parameter$\beta\in(0,\infty)$, is satisfied. Moreover, for $\beta>1$ there exist twodifferent proximity rules, leading to lune-based and circle-based BSGs. First,by computing the average degree of large ensembles of BSGs we detectdifferences, which increase with the increase of $\beta$, between lune-basedand circle-based BSGs. Then, within a random matrix theory (RMT) approach, weexplore spectral and eigenvector properties of randomly weighted BSGs by theuse of the nearest-neighbor energy-level spacing distribution and the entropiceigenvector localization length, respectively. The RMT analysis allows us toconclude that a localization transition occurs at $\beta=1$.

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