Uniform density in matroids, matrices and graphs

TL;DR

This work provides a complete polyhedral and measure-theoretic framework for uniformly dense matroids, proving that $P(M)$ contains the uniform point $$(\rho^{-1},\dots,\rho^{-1})$$ if and only if $\rho(A)\le\rho(E)$ for all $A$, equivalent to the existence of an $E$-uniform basis measure; it also distinguishes strict uniform density via relative interior membership. In graphic matroids, uniform density implies strong connectivity properties, with a sharp toughness bound $\rho(G)/\Delta(G)$ and, for regular connected graphs, a guaranteed 1-toughness and near-perfect matchings, plus spectral characterizations via Laplacians. For real representable matroids, strictly uniformly dense cases admit determinantal $E$-uniform measures and projection representations with constant diagonal $\rho(M)^{-1}$, culminating in a moduli-space description: strictly UD real matroids are parameterized by a Grassmannian subvariety $\mathcal{V}(n,k)$, equivalently by projection matrices with fixed diagonal. Overall, the paper links matroid density to polyhedral, spectral, and algebro-geometric structures, yielding practical criteria for testing uniform density and a unifying view across matroids, graphs, and matrices.

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.

Figures (8)

• Figure 1: A graph $G$ and its spanning forests. The cycle matroid of $G$ has ground set $E=[5]$ and bases given by the three-element subsets indicated under the spanning forests.
• Figure 2: From left to right: (Example \ref{['ex: tadpole graph']}) The tadpole graph has density $\rho=4/3$ and is not uniformly dense. (Example \ref{['ex: tree graph']}) The tree graph has density $\rho=1$ and is uniformly dense. (Example \ref{['ex: edge-transitive graphs']}) The $4$-cycle graph has density $\rho=4/3$ and is uniformly dense, the complete graph on $4$ vertices has density $\rho=2$ and is uniformly dense and the hypercube graph of dimension $3$ has density $\rho=12/7$ and is uniformly dense. The latter four graphs are edge-transitive.
• Figure 3: A planar graph $G$ consisting of paths of length $L_1=3,L_2=3,L_3=5$. The three vertices of the dual graph $G^\star$ correspond to the two bounded and one unbounded face of $G$.
• Figure 4: The three graphs $G,H$ and $K$ from Example \ref{['example: bicyclic graphs example']}. $G$ and $H$ are uniformly dense and $K$ is not uniformly dense.
• Figure 5: A graph $G$ and spanning tree $T\subset E(G)$. The graph $G'$ and $F'\subset E(G')$ are obtained by removing the vertices in $U$ from $G$. Note that in this example $F'$ is not a spanning forest of $G'$, and $c(F')> c(G')$.

Theorems & Definitions (46)

• Theorem 1.1
• Theorem 1.1
• Theorem 1.1
• Lemma 2.1
• Theorem 2.1
• Example 2.2: Tadpole matroid
• Example 2.3: Uniform matroid
• Example 2.4: Self-dual matroids
• Theorem 2.5
• Theorem 2.6: narayanan_1981_molecular