Spectral Graph Theory and Its Applications in Network Clustering

Abstract

Spectral graph theory provides a rigorous mathematical foundation for analyzing the structural properties of complex networks through the study of eigenvalues and eigenvectors of graph-associated matrices. One of its most impactful applications lies in network clustering, where the objective is to partition a network into meaningful groups of nodes with dense internal connections and sparse external links. Spectral clustering leverages the eigen structure of the graph Laplacian to transform clustering into a low-dimensional representation problem, enabling efficient detection of communities even in high-dimensional, nonlinear, or noisy data. This approach has proven effective across diverse domains, including social networks, biological systems, image segmentation, and information retrieval. Recent advances integrate spectral methods with machine learning models and graph neural networks, improving scalability and robustness. This paper explores the theoretical foundations, methodologies, and applications of spectral graph theory in clustering, highlighting its strengths, limitations, and future research directions.