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

## TL;DR

This paper investigates the geometrical and spectral properties of $eta$-skeleton graphs, revealing a localization transition at $eta=1$ through numerical analysis and random matrix theory techniques.

## Contribution

It provides the first comprehensive spectral analysis of $eta$-skeleton graphs, identifying a localization transition at $eta=1$ and comparing lune-based and circle-based variants.

## Key findings

- Differences in average degree increase with $eta$
- Spectral analysis shows a localization transition at $eta=1$
- Lune-based and circle-based BSGs exhibit distinct properties

## Abstract

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 $\beta=1$.