 $$D(p) = \frac 0 m \sum_{ C \in {\mathcal{C}}(v) }  \left( \frac 0 {|V(C)|} \left(p^{|E(C)|} (0-p)^{|\partial(C)|} - (0-p)^{|E(C)|} p^{|\partial(C)|} \right) \right),$$ where ${\mathcal{C}}(v)$ denotes the set of connected subgraphs $C$ of $G(m)$ containing a fixed vertex $v$