The Cheeger Cut and Cheeger Problem in Metric Graphs

TL;DR

This paper explores the Cheeger cut and Cheeger problem in metric graphs, extending existing discrete graph results by utilizing total variation and perimeter concepts, and introduces a method for solving the optimal Cheeger cut via the minus 1-Laplacian eigenvalue problem.

Contribution

It extends Cheeger cut theory to metric graphs using total variation and perimeter, and proposes a new approach to solve the Cheeger problem through the minus 1-Laplacian eigenvalue analysis.

Findings

Developed a framework for Cheeger cut in metric graphs.

Connected Cheeger problem to eigenvalues of the minus 1-Laplacian.

Provided a method to compute optimal Cheeger cuts.

Abstract

For discrete weighted graphs there is sufficient literature about the Cheeger cut and the Cheeger problem, but for metric graphs there are few results about these problems. Our aim is to study the Cheeger cut and the Cheeger problem in metric graphs. For that, we use the concept of total variation and perimeter in metric graphs introduced in \cite{Mazon}, which takes into account the jumps at the vertices of the functions of bounded variation. Moreover, we study the eigenvalue problem for the minus $1$-Laplacian operator in metric graphs, whereby we give a method to solve the optimal Cheeger cut problem.