Diameter of random spanning trees in a given graph

We show that a random spanning tree formed in a general graph G (such as a power law graph) has diameter much larger than the diameter of G. We show, with high probability the diameter of a random spanning tree of G is shown to be between c √ n and c′ √ n log n, where c and c′ depend on the spectral gap of G and the ratio of the moments of the degree sequence. For the special case of regular graphs, this result improves the previous lower bound by Aldous by a factor of log n.