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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