Suppose I have a connected graph $G$ and a fixed edge $e = \langle u, v \rangle \in G$, and I want to count the number of spanning trees that involve $e$. I really only want to estimate the fraction of spanning trees containing $e$ compared to the total number of spanning trees $G$, that is, I want to find a lower bound $c \leq \kappa(G \backslash e) / \kappa(G)$
This lower bound should be in terms of the degrees of vertices $u,v$. Let $c(d,d')$ be the smallest possible value of $\kappa(G \backslash e) / \kappa(G)$ when the vertices have degree $d, d'$. What can one say about $c(d,d')$?
For example, if $d = 1$ or $d' = 1$, then $c(d,d') = 1$. (The edge must be part of a spanning tree of $G$.).
If $d = 2$, then $c(d,d') \geq 1/2$, as every spanning tree must involve one of the two edges incident on $u$, and the spanning trees using only $e$ are in bijection with the spanning trees using the other edge.
If $d = d' = 2$, then $c(d,d') \geq 2/3$; and so on.
Is there are general formula for $c(d,d')$?