On constructing rational spanning tree edge densities

Let τ(G) and τG(e) denote the number of spanning trees of a graph G and the number of spanning trees of G containing edge e of G, respectively. Ferrara, Gould, and Suffel asked if, for every rational 0 < p/q < 1 there existed a graph G with edge e ∈ E(G) such that τG(e)/τ(G) = p/q. In this note we provide constructions that show this is indeed the case. Moreover, we show this is true even if we restrict G to claw-free graphs, bipartite graphs, or planar graphs. Let dep(G) = maxe∈G τG(e)/τ(G). Ferrara et al. also asked if, for every rational 0 < p/q < 1 there existed a graph G with dep(G) = p/q. For the claw-free construction, we are also able to answer this question in the affirmative.