DefinePK

Abstract

This comparative analysis investigates eigenvalues of graph theory based on scale-free, random, and regular graphs in an effort to comprehend their structural characteristics. Eigenvalues give useful information on network activity and hence affect applications such as resilience, connectivity, and randomness. Regular graphs have standard eigenvalue characteristics because of the homogenization of node degrees, and random graphs have unordered spectral distributions consistent with their randomness. Scale-free graphs characterized by power-law degree distributions exhibit how high-degree nodes determine eigenvalue landscapes in a bid to understand network stability.
By studying actual networks—social, biological, and technical networks—this paper closes the gap between theory and practice. The results affect network design, optimization, and anomaly detection and provide insights to researchers, engineers, and practitioners. This research also advances our knowledge of spectral properties in general graph structures to lay the groundwork for future research and innovative application in complex system analysis.