Distance formulas in Bruhat-Tits building of ${\mathrm{S}\mathrm{L}}_d({\mathbb{Q}}_p)$

We study the distance on the Bruhat-Tits building of the group ${\mathrm{S}\mathrm{L}}_d({\mathbb{Q}}_p)$ (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 ${\mathrm{S}\mathrm{L}}_2({\mathbb{Q}}_p)$ and give a formula for the number of edges shared by two given apartments.