Random minimum length spanning trees in regular graphs

Consider a connected r-regular n-vertex graph G with random independent edge lengths, each uniformly distributed on (0,1). Let mst(G) be the expected length of a minimum spanning tree. We show that mst(G) can be estimated quite accurately under two distinct circumstances. Firstly, if r is large and G has a modest edge expansion property then mst(G) similar to n/r zeta(3), where zeta(3) = Sigma(j=1)(infinity) j(-3) similar to 1.202. Secondly, if G has large girth then there exists an explicitly defined constant c(r) such that mst(G) similar to c(r)n. We find in particular that c(3) = 9/2 - 6log2 similar to 0.341.