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We study the distance on the Bruhat-Tits building of the group$\mathrm{SL}_d(\mathbb{Q}_p)$ (and its other combinatorial properties). Codingits vertices by certain matrix representatives, we introduce a way how to buildformulas with combinatorial meanings. In Theorem 1, we give an explicit formulafor 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 smallesttotal distance of a vertex from a given finite set of vertices. In the appendixwe consider the case of $\mathrm{SL}_2(\mathbb{Q}_p)$ and give a formula forthe number of edges shared by two given apartments.

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