Notes on Elementary Spectral Graph Theory. Applications to Graph Clustering Using Normalized Cuts

These are notes on the method of normalized graph cuts and its applications to graph clustering. I provide a fairly thorough treatment of this deeply original method due to Shi and Malik, including complete proofs. I include the necessary background on graphs and graph Laplacians. I then explain in detail how the eigenvectors of the graph Laplacian can be used to draw a graph. This is an attractive application of graph Laplacians. The main thrust of this paper is the method of normalized cuts. I give a detailed account for K = 2 clusters, and also for K > 2 clusters, based on the work of Yu and Shi. Three points that do not appear to have been clearly articulated before are elaborated: 1. The solutions of the main optimization problem should be viewed as tuples in the K-fold cartesian product of projective space RP^{N-1}. 2. When K > 2 , the solutions of the relaxed problem should be viewed as elements of the Grassmannian G(K,N). Disciplines Computer Engineering | Computer Sciences Comments University of Pennsylvania Department of Computer and Information Science Technical Report No. MSCIS-13-09. This technical report is available at ScholarlyCommons: http://repository.upenn.edu/cis_reports/986 Notes on Elementary Spectral Graph Theory Applications to Graph Clustering Using Normalized Cuts Jean Gallier Department of Computer and Information Science University of Pennsylvania Philadelphia, PA 19104, USA e-mail: [email protected]