Bipartition of graphs based on the normalized cut and spectral methods

TL;DR

This paper surveys Laplacian matrices, derives eigenvalues for specific graph classes, and identifies graphs where spectral clustering and normalized cuts yield different partitions.

Contribution

It unifies known results with simple proofs and finds graphs where spectral clustering and normalized cuts differ in partitioning.

Findings

Derived eigenvalue formulas for paths and cycles.

Identified graphs with differing spectral and normalized cut partitions.

Provided characteristic polynomials for specific Laplacian matrices.

Abstract

In the first part of this paper, we survey results that are associated with three types of Laplacian matrices:difference, normalized, and signless. We derive eigenvalue and eigenvector formulaes for paths and cycles using circulant matrices and present an alternative proof for finding eigenvalues of the adjacency matrix of paths and cycles using Chebyshev polynomials. Even though each results is separately well known, we unite them, and provide uniform proofs in a simple manner. The main objective of this study is to solve the problem of finding graphs, on which spectral clustering methods and normalized cuts produce different partitions. First, we derive a formula for a minimum normalized cut for graph classes such as paths, cycles, complete graphs, double-trees, cycle cross paths, and some complex graphs like lollipop graph $LP_{n,m}$, roach type graph $R_{n,k}$, and weighted path $P_{n,k}$. Next, we provide characteristic polynomials of the normalized Laplacian matrices ${\mathcal L}(P_{n,k})$ and ${\mathcal L}(R_{n,k})$. Then, we present counter example graphs based on $R_{n,k}$, on which spectral methods and normalized cuts produce different clusters.