We study the distance on the Bruhat–Tits building of the group ${\mathrm{\text{SL}}}_d({\mathbb{Q}}_p)$ (and its other combinatorial properties). Coding its vertices by certain matrix representatives, we introduce a way how to build formulas with combinatorial meanings. In Theorem 1, we give an explicit formula for the graph distance $\delta (\alpha ,\beta )$ of two vertices $\alpha$ and $\beta$ (without having to specify their common apartment). Our main result, Theorem 2, then extends the distance formula to a formula for the smallest total distance of a vertex from a given finite set of vertices. In the appendix we consider the case of ${\mathrm{\text{SL}}}_2({\mathbb{Q}}_p)$ and give a formula for the number of edges shared by two given apartments.

我们研究了该集团 Bruhat-Tits 大楼的距离${\mathrm{\text{SL}}}_d（{\mathbb{Q}}_p）$（及其其他组合属性）。通过某些矩阵代表对其顶点进行编码，我们介绍了一种如何构建具有组合意义的公式的方法。在定理1中，我们给出了图距离的明确公式$\delta （\alpha ,\beta ）$两个顶点的$\alpha$和$\beta$（无需指定他们的共同公寓）。我们的主要结果，定理 2，然后将距离公式扩展到顶点与给定有限顶点集的最小总距离的公式。在附录中，我们考虑以下情况${\mathrm{\text{SL}}}_2（{\mathbb{Q}}_p）$并给出两个给定公寓共享的边数的公式。