Spectral Graph Theory: Eigenvalues and Graph Properties for Network Analysis

Abstract

Spectral graph theory explores the relationship between the spectrum of a graph’s adjacency matrix or Laplacian matrix and various structural properties of the graph. Eigenvalues and eigenvectors of these matrices have been shown to provide valuable insights into the connectivity, robustness, and dynamics of networks. In this paper, we review key concepts in spectral graph theory and explore their applications in network analysis, including community detection, clustering, centrality, and network synchronization. the use of spectral methods for analysing large-scale real-world networks such as social, biological, and transportation systems.