Asymptotic enumeration of constrained bipartite, directed and oriented graphs by degree sequence

TL;DR

This paper develops asymptotic formulas for counting and analyzing the properties of various constrained graphs, such as bipartite, directed, and oriented graphs, based on degree sequences, with applications to random matrices and graph orientations.

Contribution

It introduces new asymptotic enumeration methods for constrained graphs beyond near-regular cases, including applications to matrix permanents and graph orientations.

Findings

Derived probabilities for edge absence in sparse bipartite graphs

Enumerated loop-free digraphs and oriented graphs with specified degrees

Calculated expected permanents of sparse or dense random matrices

Abstract

In the sufficiently sparse case, we find the probability that a uniformly random bipartite graph with given degree sequence contains no edge from a specified set of edges. This enables us to enumerate loop-free digraphs and oriented graphs with given in-degree and out-degree sequences, and obtain subgraph probabilities. Our theorems are not restricted to the near-regular case. As an application, we determine the expected permanent of sparse or very dense random matrices with given row and column sums; in the regular case, our formula holds over all densities. We also draw conclusions about the degrees of a random orientation of a random undirected graph with given degrees, including its number of Eulerian orientations.