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MiS Preprint Repository

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MiS Preprint
5/2023

Uniform density in matroids, matrices and graphs

Karel Devriendt and Raffaella Mulas

Abstract

We give new characterizations for the class of uniformly dense matroids, and we describe applications 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, and if and only if there exists a measure on the bases such that every element of the ground set has equal probability to be in a random basis with respect to this measure. As one application, we derive new spectral, structural and classification results for uniformly dense graphic matroids. 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 representable matroids can be represented by projection matrices with constant diagonal and that they are parametrized by a subvariety of the real Grassmannian.

Received:
Jun 28, 2023
Published:
Jun 28, 2023

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Preprint
2023 Repository Open Access
Karel Devriendt and Raffaella Mulas

Uniform density in matroids, matrices and graphs