= Foster's theorem
{c}
{title2=$\sum_e c_eR_{\mathrm{eff}}(e)=|V|-1$}
= Foster theorem
{c}
{synonym}
For a finite connected unweighted loopless <graph> with $n$ <graph vertices>,
$$
\sum_{e\in E}R_{\mathrm{eff}}(e)=n-1.
$$
The <edge-inclusion formula for a uniform spanning tree> gives the left-hand side as $\mathbb E|T|$, and every <spanning tree> has $n-1$ <edges>. On the same finite <connected graph> with positive <edge> conductances $c_e>0$, the corresponding identity is $\sum_e c_eR_{\mathrm{eff}}(e)=n-1$.
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