Combinatorial Fiedler Theory and Graph Partition

TL;DR

This paper introduces a novel graph partitioning approach using the $\, ext{l}_1$-norm, leading to a new parameter $b(G)$ that relates to the sparsest cut problem, with theoretical bounds and explicit formulas for trees.

Contribution

It extends spectral graph theory by exploring $\, ext{l}_1$-norm based partitioning, establishing bounds, connectivity results, and explicit formulas for specific graph classes.

Findings

The new parameter $b(G)$ relates to the sparsest cut problem.

Explicit formulas for $b(G)$ are derived for trees.

Connectivity results and bounds for $b(G)$ are established.

Abstract

Partition problems in graphs are extremely important in applications, as shown in the Data science and Machine learning literature. One approach is spectral partitioning based on a Fiedler vector, i.e., an eigenvector corresponding to the second smallest eigenvalue $a(G)$ of the Laplacian matrix $L_G$ of the graph $G$. This problem corresponds to the minimization of a quadratic form associated with $L_G$, under certain constraints involving the $\ell_2$-norm. We introduce and investigate a similar problem, but using the $\ell_1$-norm to measure distances. This leads to a new parameter $b(G)$ as the optimal value. We show that a well-known cut problem arises in this approach, namely the sparsest cut problem. We prove connectivity results and different bounds on this new parameter, relate to Fiedler theory and show explicit expressions for $b(G)$ for trees. We also comment on an $\ell_{\infty}$-norm version of the problem.