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arXiv · 2306.15267

Uniform density in matroids, matrices and graphs

Karel Devriendt·Raffaella Mulas
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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.

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Karel Devriendt, Raffaella Mulas. 2026-02-02. Uniform density in matroids, matrices and graphs. https://arxiv.org/abs/2306.15267

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