A connected, locally finite graph $\Gamma $ is a Cayley–Abels graph for a totally disconnected, locally compact group G if G acts vertex-transitively on $\Gamma $ with compact, open vertex stabilizers. Define the minimal degree of G as the minimal degree of a Cayley–Abels graph of G. We relate the minimal degree in various ways to the modular function, the scale function and the structure of compact open subgroups. As an application, we prove that if $T_{d}$ denotes the d-regular tree, then the minimal degree of $\mathrm{Aut}(T_{d})$ is d for all $d\geq 2$ .

一个连通的局部有限图 $\Gamma$ 是完全断开的局部紧群G 的Cayley-Abels 图，如果G对 $\Gamma$ 有顶点传递作用，具有紧致的开顶点稳定器。将G的最小次数定义为G的 Cayley–Abels 图的最小次数。我们以各种方式将最小度数与模函数、标度函数和紧开子群的结构联系起来。作为一个应用，我们证明如果 $T_{d}$ 表示d -正则树，那么对于所有 $d\geq 2$， $\mathrm{Aut}(T_{d})$ 的最小度为d .