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 uniform density can be recognized in polynomial time for real representable matroids, which includes graphs. Finally, 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.

翻译：本文给出了均匀密度拟阵类的新刻画，并研究了这些刻画在图拟阵和实可表示拟阵中的应用。我们证明：一个拟阵是均匀密度的，当且仅当其基多面体包含一个坐标全为常数的点。作为主要应用，我们推导出均匀密度图的新谱性质、结构性质与分类结果。特别地，我们证明连通的、正则的均匀密度图是1-坚韧的，因而包含一个（近）完美匹配。作为第二个应用，我们证明了对于实可表示拟阵（包括图），均匀密度性质可在多项式时间内判定。最后，我们证明严格均匀密度的实表示拟阵可由具有常数对角线的投影矩阵表示，且它们由 Grassmann 流形的一个子簇参数化。