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.

翻译：本文为一致稠密拟阵类提供了新的刻画，并研究了这些刻画在图拟阵与实可表拟阵中的应用。我们证明：一个拟阵是一致稠密的，当且仅当其基多面体包含一个具有恒定坐标的点。作为主要应用，我们推导出了一致稠密图的新颖谱性质、结构性质与分类结果。特别地，我们证明了连通的、正则的一致稠密图是$1$-坚韧的，因而包含一个（近）完美匹配。作为第二个应用，我们证明了严格一致稠密的实可表拟阵可由具有恒定对角线的投影矩阵表示，并且它们由格拉斯曼流形的一个子簇所参数化。