we recall the construction of the random *multi-graph* known as the *Configuration Model* (illustrated in Fig. \[fig:CM\]). We refer to van der Hofstad [@vdHpr Chapter 7] for further details.

Fix $n\in{\mathbb{N}}$, and let $V=\{v_1,\ldots,v_n\}$. With each vertex $v_i$ we associate a *random degree* $D_i$, in such a way that $D_1,\ldots,D_n \in {\mathbb{N}}$ are i.i.d.with marginal probability distribution $f$ conditional on the event $\{\sum_{i=1}^n D_i
= \mbox{even}\}$. Consider a uniform matching of the elements in the set of *stubs* (also called half-edges), written $$\{x_{i,j}\}_{1 \leq i \leq n,1 \leq j \leq D_i}.$$ By erasing the second label of the stubs, we can associate with it a *multi-graph* ${\mathrm{CM}}_n$ satisfying the requirement that the degree of $v_i$ is $D_i$ for $1 \leq i \leq n$. The total number of edges is $\tfrac12\sum_{i=1}^n D_i$.

Throughout the sequel we use the symbol ${\mathbb{P}}_n$ to denote the law of the random multi-graph ${\mathrm{CM}}_n$ on $n$ vertices generated by the Configuration Model. To avoid degeneracies we assume that $${d_\mathrm{min}}= \min\{k\in{\mathbb{N}}\colon\,f(k)>0\} \geq 3,
\qquad {d_\mathrm{ave}}= \sum_{k\in{\mathbb{N}}} kf(k) < \infty,$$ i.e., all degrees are at least three and the average degree is finite. In this case the graph is connected *with high probability* (w.h.p.), i.e., with a probability 