Uniform density in matroids, matrices and graphs

TL;DR

This paper introduces new characterizations of uniformly dense matroids and explores their implications for graphs and matrix representations, revealing structural properties and classification results.

Contribution

It provides novel characterizations of uniformly dense matroids and applies these to derive spectral, structural, and classification results for related graphs and matrices.

Findings

Connected regular uniformly dense graphs are 1-tough and contain near-perfect matchings.

Strictly uniformly dense real represented matroids can be represented by projection matrices with constant diagonals.

Uniform density in matroids relates to base polytopes containing constant coordinate points.

Abstract

We give new characterizations for the class of uniformly dense matroids and study applications of these characterizations to graphic and real representable matroids. We show that a matroid is uniformly dense if and only if its base polytope contains a point with constant coordinates. As a main application, we derive new spectral, structural and classification results for uniformly dense graphs. In particular, we show that connected regular uniformly dense graphs are $1$-tough and thus contain a (near-)perfect matching. As a second application, we show that strictly uniformly dense real represented matroids can be represented by projection matrices with a constant diagonal and that they are parametrized by a subvariety of the Grassmannian.