平面规则网格中的逐代点普查
A Census of Vertices by Generations in Regular Tessellations of the Plane
摘要：
我们认为平面的规则镶嵌是无限图，其中$q$边和$q$面在每个顶点相遇，并且其中$p$边和$p$顶点包围每个面。对于$1/p + 1/q = 1/2$，它们是欧几里得平面的平铺;对于$1/p + 1/q < 1/2 $，它们是双曲平面的平铺。我们选择一个顶点作为原点，并根据顶点到原点的距离（由最短路径中的边数测量）将顶点分类为几代。对于所有的$p\ge 3$和$q \ge 3$与$1/p + 1/q \le 1/2 $，我们确定的有理生成函数，在每一代的顶点数。
英文摘要：
We consider regular tessellations of the plane as infinite graphs in which $q$ edges and $q$ faces meet at each vertex, and in which $p$ edges and $p$ vertices surround each face. For $1/p + 1/q = 1/2$, these are tilings of the Euclidean plane; for $1/p + 1/q < 1/2 $, they are tilings of the hyperbolic plane. We choose a vertex as the origin, and classify vertices into generations according to their distance (as measured by the number of edges in a shortest path) from the origin. For all $p\ge 3$ and $q \ge 3$ with $1/p + 1/q \le 1/2 $, we determine the rational generating function giving the number of vertices in each generation.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
