Hi,

Define $C(G)$ the number of spanning trees in a graph $G$.

Now, given a graph $G$ with $n$ vertices and $p$ edges, construct a new graph $G'$ which is equal to the graph $G$ but where you add exactly one new edge.

The question is : can we know something about the fraction $\frac{C(G)}{C(G')}$ ?

Perhaps we can use that the number of spanning trees of a graph is equal to the determinant of any cofactor of the laplacian matrix of the graph. (Because the two laplacian matrix are here very close) ? I tried in that way but without success yet.