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) ∼ r ζ(3), where ζ(3) = ∑∞ j=1 j −3 ∼ 1.202. Secondly, if G has large girth then there exists an explicitly defined constant cr such that mst(G)∼crn. We find in particular that c3 =9/2−6log2∼0.341.