Multiway Spectral Graph Partitioning: Cut Functions, Cheeger Inequalities, and a Simple Algorithm

SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 112-133, March 2024.
Abstract. The problem of multiway partitioning of an undirected graph is considered. A spectral method is used, where the [math] largest eigenvalues of the normalized adjacency matrix (equivalently, the [math] smallest eigenvalues of the normalized graph Laplacian) are computed. It is shown that the information necessary for partitioning is contained in the subspace spanned by the [math] eigenvectors. The partitioning is encoded in a matrix [math] in indicator form, which is computed by approximating the eigenvector matrix by a product of [math] and an orthogonal matrix. A measure of the distance of a graph to being [math]-partitionable is defined, as well as two cut (cost) functions, for which Cheeger inequalities are proved; thus the relation between the eigenvalue and partitioning problems is established. Numerical examples are given that demonstrate that the partitioning algorithm is efficient and robust.