Let $G _ { 0 } = \left( S _ { 0 } , A _ { 0 } \right)$ be a particular fixed graph with $s _ { 0 } = s _ { G _ { 0 } }$, $a _ { 0 } = a _ { G _ { 0 } }$, $s_0 \geq 2$, $a_0 \geq 1$. We now assume that $\lim _ { n \rightarrow + \infty } \left( n ^ { \omega _ { 0 } } p _ { n } \right) = + \infty$.\\
Show that the expectation $\mathbf { E } \left( \left( X _ { n } ^ { 0 } \right) ^ { 2 } \right)$ satisfies :
$$\mathbf { E } \left( \left( X _ { n } ^ { 0 } \right) ^ { 2 } \right) = \sum _ { \left( H , H ^ { \prime } \right) \in C _ { 0 } ^ { 2 } } \mathbf { P } \left( H \cup H ^ { \prime } \subset G \right) = \sum _ { \left( H , H ^ { \prime } \right) \in C _ { 0 } ^ { 2 } } p _ { n } ^ { 2 a _ { 0 } - a _ { H \cap H ^ { \prime } } } .$$