Subgraph counts for dense random graphs with specified degrees

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 $(d_1,\dots ,d_n)$ as $n\to \infty$. We also determine the expected number of spanning trees in this model. The range of degrees covered includes $d_j=\lambda n+O(n^{1/2+\epsilon })$ for some $\lambda$ bounded away from $0$ and $1$.

To appear in Combinatorics, Probability and Computing

References in corpus (1)

• Complex martingales and asymptotic enumeration

Cited by in corpus (2)

• Counting triangles in power-law uniform random graphs
• Distribution of tree parameters by martingale approach