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：我们引入了平面或球面上图的（拓扑）绘制的密度公式，该公式建立了图中边数、顶点数、交叉数以及胞腔尺寸之间的关系。通过若干应用展示了其能力：我们证明了多种超越平面图类边密度的紧上界，包括所谓的$k=1,2$的$k$-平面图、扇交叉图/扇平面图、$k=0,1,2$的$k$-弯RAC图以及准平面图。在某些情形（1-弯和2-弯RAC图、扇交叉图/扇平面图）下，我们首次获得了相应图类边密度的紧上界。在其他情形下，我们为文献中已知的界给出了简洁且显著缩短的新证明。得益于密度公式，我们的证明主要依赖初等计数方法，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）下，我们利用密度公式的替代证明首次导出了紧下界实例。