Complex Martingales and Regular Graphs

Joint work with Mikhail Isaev (Monash University & Moscow Institute of Physics and Technology)

The asymptotic enumeration of regular graphs, and more generally of graphs with specified degree sequences, has occupied combinatorialists for a long time.

We will start with a survey of previous results, with a quick summary of the methods used. Then we will focus on the application of complex-analytic methods to degree sequences that are sufficiently dense. Next we will show how a general theorem on complex martingales can be applied to the estimation of the necessary integrals. Then we will apply a theorem of Isaev on cumulant expansions to find an asymptotic expansion for the number of regular graphs of degree at least n^t for some t>0.