Let $G$ be 2-generated group. The generating graph $\Gamma(G)$ of $G$ is the graph whose vertices are the elements of $G$ and where two vertices $g$ and $h$ are adjacent if $G = \langle g, h \rangle.$ This definition can be extended to a 2-generated profinite group $G,$ considering in this case topological generation. We prove that the set $V(G)$ of non-isolated vertices of $\Gamma(G)$ is closed in $G$ and that, if $G$ is prosoluble, then the graph $\Delta(G)$ obtained from $\Gamma(G)$ by removing its isolated vertices is connected with diameter at most 3. However we construct an example of a 2-generated profinite group $G$ with the property that $\Delta(G)$ has $2^{\aleph_0}$ connected components. This implies that the so called "swap conjecture" does not hold for finitely generated profinite groups. We also prove that if an element of $V(G)$ has finite degree in the graph $\Gamma(G),$ then $G$ is finite.

中文翻译：

---

一个profinite群的生成图

令 $G$ 为 2-生成组。$G$ 的生成图$\Gamma(G)$ 是顶点为$G$ 的元素且$g$ 和$h$ 两个顶点相邻的图，如果$G = \langle g, h \rangle .$ 在这种情况下，考虑到拓扑生成，这个定义可以扩展到一个 2-生成的 profinite 群 $G,$。我们证明 $\Gamma(G)$ 的非孤立顶点的集合 $V(G)$ 在 $G$ 中是封闭的，并且如果 $G$ 是可解的，那么图 $\Delta(G)$通过去除 $\Gamma(G)$ 的孤立顶点而获得的与直径最多为 3 的连接。然而，我们构造了一个 2-生成的超限群 $G$ 的例子，其性质 $\Delta(G)$ 有 $2 ^{\aleph_0}$ 连通分量。这意味着所谓的“交换猜想”不适用于有限生成的profinite 群。