Spectral Clustering for Interdisciplinary Research: From Graph Theory to RNA-seq Data Analysis

Spectral clustering is a powerful methodology rooted in graph theory, linear algebra, and probability theory, and is highly effective for unsupervised learning in complex, non-linear data. This article serves as a comprehensive tutorial and guide for interdisciplinary researchers, building a clear connection between the rigorous mathematical framework of spectral clustering, beginning with the continuous Laplacian operator, progressing to its discrete, graph-based counterpart, and finally culminating in a real-world application. We detail the theory through practical examples and apply the framework to bulk RNA-seq data analysis in breast cancer cell lines, demonstrating the method's unique ability to uncover both broad trends and nuanced molecular subtypes. By providing intuitive knowledge on both the theory and the application, this work aims to facilitate collaboration across mathematics, computational science, and life sciences to support robust and sound scientific research.