Bipartition of graphs based on the normalized cut and spectral methods

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.

💡 Research Summary

The paper presents a unified treatment of three Laplacian matrices—difference (combinatorial) Laplacian (L = D - A), normalized Laplacian (\mathcal{L} = I - D^{-1/2} A D^{-1/2}), and signless Laplacian (Q = D + A)—and uses this framework to investigate when spectral clustering and the normalized cut (Ncut) criterion yield different partitions. The authors first derive eigenvalues and eigenvectors for path graphs (P_n) and cycle graphs (C_n) by exploiting the circulant structure of their adjacency matrices. They also give an alternative derivation based on Chebyshev polynomials (T_k) and (U_k), showing that both approaches lead to the same closed‑form expressions (\lambda_k = 2\bigl(1-\cos\frac{k\pi}{n+1}\bigr)) for paths and (\lambda_k = 2\bigl(1-\cos\frac{2\pi k}{n}\bigr)) for cycles. By presenting these results together, the paper provides a compact, self‑contained reference for the spectra of these basic graph families.

The second part of the work focuses on the minimum normalized cut value for a variety of graph families. For simple structures—paths, cycles, complete graphs—the authors obtain explicit formulas: for a path of length (n) the optimal Ncut is (\frac{2}{n+1}), for a cycle it is (\frac{4}{n}), and for a complete graph (K_n) the best cut approaches (\frac{n-1}{n}). More intricate families such as double‑trees, cycle‑cross‑paths, and several composite graphs are handled by recursively decomposing the graph into sub‑components, computing the contribution of each component to the cut, and then combining them analytically.

A major contribution is the derivation of the characteristic polynomials of the normalized Laplacian for two parameterized families: the weighted path (\mathcal{L}(P_{n,k})) and the “roach” graph (\mathcal{L}(R_{n,k})). The roach graph consists of a central cycle of length (n) with (k) pendant paths attached to distinct vertices. Its normalized Laplacian matrix exhibits a block structure that leads to a polynomial of the form \