Number of spanning trees which contain a given edge 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 $c(d,d')$?
 A: First of all, I think it's important to note that the number of spanning trees containing a given edge may depend on the global properties of a graph rather than just local properties like vertex degrees.  For instance, if an edge is a cut-edge (also known as a bridge) then every spanning tree will contain it, but the vertex degrees won't necessarily tell you if that's the case.
To every graph $G$ we can associate a bivarite polynomial over $\mathbb{Z}$ called the Tutte polynomial $T_G(x,y)$.  Any standard text on algebraic graph theory (e.g., Norman Biggs' Algebraic Graph Theory; Bela Bollobas' Modern Graph Theory, etc) will contain a treatment of it.  The Tutte polynomial encodes a really surprising amount of combinatorial information about a graph, particularly regarding its connectivity and cycle structure.  Among other things:
(1) $T_G(1,1)$ is equal to the number of spanning trees of $G$ (which is always positive, if $G$ is connected).
(2) If $e$ is any edge of $G$, then $T_{G}(x,y) = T_{G\backslash e}(x,y) + T_{G*e}(x,y)$.  $G\backslash e$ is, as in your notation, $G$ with $e$ excised and $G*e$ is the graph obtained by merging the two vertices of $e$ together (and keeping any loops that form).  With some base conditions, this is often taken as the definition of $T_G$.
Notice that $1 - T_{G*e}(1,1)/T_G(1,1) = T_{G-e}(1,1)/T_G(1,1)$.  If I understand your problem correctly, you're interested in a choice of $e$ that minimizes the right hand side.  So it might be helpful to consider maximizing the quotient on the left hand side, but both seem pretty opaque to me.  Also I should warn you that Tutte polynomial computations, as you might expect, are NP-hard in general.
Hopefully this is of some use.
A: The probability that an edge $e=(u,v)$ is part of a uniform spanning tree is equal to the resistance between $u$ and $v$ when the graph is considered as an electric network (see the book by Lyons with Peres, section 4.2). The bounds you get (in term of the degrees $d_u,d_v$) are
$$ \frac{1}{\min(d_u,d_v)} \le R_{eff}(u \leftrightarrow v) \le 1$$
when you allow multiple edges, or 
$$ \frac{(d_u-1)+(d_v-1)}{(d_u-1)+(d_v-1)+(d_u-1)(d_v-1)} < R_{eff}(u \leftrightarrow v) \le 1$$
when the graph is simple, and these bounds are sharp.
A: The assertion below regarding O(1/n) is in contradiction with another posted answer, so I leave the construction available while I check the assertion.
Let M_n be the (graph of the Hasse diagram of the) modular lattice on (n+2) elements.  This will have 2n edges.  Add two more edges on either side. Call the leaves u and v, and let us add an edge (the problem edge, called e) between u and v.  I have a u-v gadget with 2n+3 edges on n+4 vertices, and n-many cycles of length 5.  However, u and v have degree 2.
Now to the u side of this gadget, add an edge and then dangle whatever favorite nonempty graph off this edge, and choose a disjoint graph to dangle off of v.  In this graph, u and v have degree 3.  However, any spanning tree that contains e can be modified to one of at least some number of other spanning trees;  the analysis is more complicated than I originally imagined, but I think one can use this to show the ratio for this edge is at most O(1/n).  So if d and d' are at least 3, I see no useful lower bound for the fraction in terms of the degrees themselves regarding edge e.
Gerhard "Ask Me About System Design" Paseman, 2011.11.18
