# Bipartition of graphs based on the normalized cut and spectral methods Source: [Bipartition of graphs based on the normalized cut and spectral methods]() Collection index: [llms.txt]() Data updated: 2025-10-25T10:35:44.309Z Authors: K.K.K.R. Perera, Yoshihiro Mizoguchi Published: 2012-10-26T21:17:48.000Z ## 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. ## Recorded benchmark results A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage. ## Original sources and implementation [Original paper]() [PDF]()