Edge-connectivity matrices and their spectra

TL;DR

This paper explores the spectral properties of edge-connectivity matrices in weighted graphs, providing bounds on eigenvalues, analyzing eigenvector structures, and characterizing which matrices can arise as edge-connectivity matrices.

Contribution

It introduces new bounds on eigenvalues, studies eigenvector structures, and characterizes realizable edge-connectivity matrices, advancing understanding of their spectral properties.

Findings

Tight bounds on the smallest eigenvalue and energy of edge-connectivity matrices.

Characterization of nonnegative matrices that can be edge-connectivity matrices.

Insights into when eigenvectors can be derived from matrix entries.

Abstract

The edge-connectivity matrix of a weighted graph is the matrix whose off-diagonal $v$-$w$ entry is the weight of a minimum edge cut separating vertices $v$ and $w$. Its computation is a classical topic of combinatorial optimization since at least the seminal work of Gomory and Hu. In this article, we investigate spectral properties of these matrices. In particular, we provide tight bounds on the smallest eigenvalue and the energy. Moreover, we study the eigenvector structure and show in which cases eigenvectors can be easily obtained from matrix entries. These results in turn rely on a new characterization of those nonnegative matrices that can actually occur as edge-connectivity matrices.