Subgraph counts for dense random graphs with specified degrees

TL;DR

This paper derives asymptotic formulas for counting subgraphs, including copies and induced copies, as well as spanning trees, in dense random graphs with specified degrees, expanding understanding of their structural properties.

Contribution

It introduces new estimates for exponential functions of random permutations and subsets, enabling precise asymptotic counts of subgraphs in dense degree-constrained random graphs.

Findings

Asymptotic formulas for subgraph counts in dense random graphs

Expected number of spanning trees in the model

Degree range includes $d_j = \lambda n + O(n^{1/2+\varepsilon})$

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