We introduce the Density Formula for (topological) drawings of graphs in the plane or on the sphere, which relates the number of edges, vertices, crossings, and sizes of cells in the drawing. We demonstrate its capability by providing several applications: we prove tight upper bounds on the edge density of various beyond-planar graph classes, including so-called $k$-planar graphs with $k=1,2$, fan-crossing / fan-planar graphs, $k$-bend RAC-graphs with $k=0,1,2$, and quasiplanar graphs. In some cases ($1$-bend and $2$-bend RAC-graphs and fan-crossing / fan-planar graphs), we thereby obtain the first tight upper bounds on the edge density of the respective graph classes. In other cases, we give new streamlined and significantly shorter proofs for bounds that were already known in the literature. Thanks to the Density Formula, all of our proofs are mostly elementary counting and mostly circumvent the typical intricate case analysis found in earlier proofs. Further, in some cases (simple and non-homotopic quasiplanar graphs), our alternative proofs using the Density Formula lead to the first tight lower bound examples.

翻译：我们针对平面或球面上的（拓扑）图绘制引入了密度公式，该公式关联了绘制中的边数、顶点数、交叉数以及单元大小。通过展示若干应用，我们证明了该公式的能力：我们推导了多种超越平面图类的边密度紧上界，包括所谓的 $k=1,2$ 时的 $k$-平面图、扇形交叉图/扇形平面图、$k=0,1,2$ 时的 $k$-弯曲RAC图以及准平面图。在某些情形（1-弯曲和2-弯曲RAC图、扇形交叉图/扇形平面图）中，我们由此首次获得了相应图类边密度的紧上界；在其他情形中，我们为文献中已知的界提供了简化且显著更短的证明。得益于密度公式，我们的所有证明主要基于初等计数，基本避免了早期证明中典型的复杂情形分析。此外，在某些情形（简单与非同伦准平面图）中，我们使用密度公式的替代证明首次给出了紧下界实例。