Subgraph counts for dense random graphs with specified degrees. (5th May 2021) - Greenhill, Catherine; Isaev, Mikhail; McKay, Brendan D.
Abstract: We prove two estimates for the expectation of the exponential of a complex function of a random permutation or subset. Using this theory, we find asymptotic expressions for the expected number of copies and induced copies of a given graph in a uniformly random graph with degree sequence( d 1, …, d n ) as n → ∞. We also determine the expected number of spanning trees in this model. The range of degrees covered includes d j = λ n + O ( n 1/2+ ε ) for some λ bounded away from 0 and 1.