Geometrical and spectral study of β-skeleton graphs.

Prior research has explored various types of proximity graphs to model spatial relationships between points. However, the specific behavior of β-skeleton graphs (BSGs) under different β values remains unclear. Established knowledge includes the general structure of proximity graphs, but no prior work had resolved how β affects connectivity patterns in BSGs. This gap motivated the need for a detailed numerical and spectral analysis of BSGs. Understanding how β influences graph properties could improve applications in network science and spatial modeling. The distinction between lune-based and circle-based BSGs has not been fully quantified. Prior studies have used random matrix theory to analyze graph spectra, but not specifically for BSGs. This paper's contribution lies in its focus on β-skeleton graphs and their spectral properties. The study aims to clarify how β affects the structure and randomness of BSGs.

The aim of this study is to investigate how the parameter β influences the structure and spectral properties of β-skeleton graphs. The specific problem is the lack of understanding about how β affects connectivity and randomness in BSGs. The motivation comes from the need to better characterize these graphs for applications in spatial modeling and network science. The study addresses whether lune-based and circle-based BSGs differ in connectivity as β increases. It also explores whether a localization transition occurs in the graph's eigenvectors. The study seeks to determine how β affects the average degree of BSGs. The researchers propose to use random matrix theory to analyze spectral properties. The goal is to detect structural and spectral changes as β varies.

The researchers used numerical analysis to study β-skeleton graphs. They computed the average degree of large ensembles of BSGs for different β values. They compared lune-based and circle-based BSGs to detect differences in connectivity. The study applied random matrix theory to analyze the spectral properties of BSGs. The nearest-neighbor energy-level spacing distribution was used to assess spectral randomness. The entropic eigenvector localization length was calculated to study eigenvector properties. The researchers examined how β affects the structure of BSGs. The study focused on β values in the range (0, ∞). The approach involved comparing results for β < 1 and β > 1.

The strongest finding is that differences between lune-based and circle-based BSGs increase with higher β values. The average degree of BSGs changes significantly as β increases. The study detected a localization transition in eigenvectors at β = 1. The nearest-neighbor energy-level spacing distribution showed deviations from random matrix theory predictions. The entropic eigenvector localization length decreased as β increased. The results suggest that β-skeleton graphs transition from delocalized to localized states at β = 1. The average degree of BSGs was found to be sensitive to the choice of proximity rule. The study confirmed that β-skeleton graphs exhibit distinct spectral properties depending on β.

The authors concluded that β-skeleton graphs exhibit structural differences between lune-based and circle-based rules as β increases. They observed a localization transition in eigenvectors at β = 1. The study found that the average degree of BSGs is highly dependent on β. The researchers propose that β-skeleton graphs undergo a transition in spectral properties at β = 1. The results suggest that the choice of proximity rule affects the connectivity of BSGs. The study supports the use of random matrix theory to analyze BSG spectra. The authors suggest that β-skeleton graphs may be useful for modeling spatial relationships. The findings imply that β-skeleton graphs are sensitive to changes in β.