The average number of spanning trees in sparse graphs with given degrees
  We give an asymptotic expression for the expected number of spanning trees in a random graph with a given degree sequence $\boldsymbol{d}=(d_1,\ldots, d_n)$, provided that the number of edges is at least $n + \textstyle{\frac{1}{2}} d_{\max}^4$, where $d_{\max}$ is the maximum degree. A key part of our argument involves establishing a concentration result for a certain family of functions over random trees with given degrees, using Pr\"ufer codes. 
Catherine Greenhill, Mikhail Isaev, Matthew Kwan and Brendan D. McKay
math.CO