any cofactor of a Laplacian of a weighted graph will give the sum of all weighted spanning trees, lets denote it by $A$. The same can be calculated for spanning trees which avoid certain edge $e$, denote it by $A_e$. There, $P[e \in T]=1\frac{A_e}{A}$, the probability that an edge $e$ being part of a random spanning tree.
I would like to calculate the average probability of an edge $e$ being part of a random spanning tree of a weighted graph.
Would be glad for any help.
Thank you!
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$\begingroup$ mathoverflow.net/questions/100209/… is quite a similar question. $\endgroup$ – Brendan McKay Sep 14 '16 at 14:07

$\begingroup$ I'm looking for an average probability. $\endgroup$ – Daniel Sep 14 '16 at 14:10

$\begingroup$ Average over what? Do you mean average over $e$? $\endgroup$ – Brendan McKay Sep 14 '16 at 14:12

$\begingroup$ Average over all $P_e $ $\endgroup$ – Daniel Sep 14 '16 at 14:16

$\begingroup$ If $P_e$ is what you called $P[e\in T]$, that is just a number. What probability space do you want to average it over? $\endgroup$ – Brendan McKay Sep 14 '16 at 14:23