or for $p(N-1)$ (canonical ensemble). The parameter $\alpha$ is the average connectivity, the average number of neighbors of a given vertex in the random graph.

If $N \left( p(1-p) \right)^{1/2} \to \infty$ and $L \sim
p\frac{N(N-1)}{2}$, the thermodynamic properties of a $G(N,L)$ in the microcanonical ensemble and of a $G(N,p)$ in the canonical ensemble are the same. This is in particular true if $pN = \alpha$ is kept fixed as $N \to
\infty$.

Leaf removal process and the core of a graph {#sec:core}
============================================

Our aim is to define, for any (random or not) finite graph $G$, a remarkable subgraph which we call the *core* of $G$. It is obtained by *leaf removal*, an operation that we define now.

**Leaf**: A leaf of a graph $G$ is a couple of vertices $(v,w)$ such that $\{v,w\}$ is an edge of $G$ and $w$ belongs to no other edge of $G$. Note that this is not the most standard definition and that $(v,w)$ and $(w,v)$ are both leaves if and only if $\{v,w\}$ is a connected component of $G$.

**Bunch of leaves**: A bunch of leaves is a maximal family of leaves with the same first vertex. The leaves of a graph 