Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Catherine Greenhill, Mahdieh Hasheminezhad, Isaiah Iliffe, Brendan D. McKay

TL;DR

The paper develops a unified, asymptotic framework for counting constrained graphs by degree sequence across bipartite, directed, and oriented classes in the sparse regime. Central to the approach is a switching analysis that yields precise exponential corrections for the constrained counts: $B(\boldsymbol{s},\boldsymbol{t},X) = B(\boldsymbol{s},\boldsymbol{t}) \exp(-F/S - 3F^2/(2S^3) + O(\delta_{\max}F/S^2 + F^3/S^5))$ under explicit sparsity conditions. These results translate into subgraph probabilities, enable computation of the expected permanent of random $0$-$1$ matrices with prescribed margins, and provide asymptotics for counts of loop-free, simple oriented, and oriented-from-undirected graphs, including Eulerian orientations. The methods and formulas offer a robust toolset for degree-sequence based enumeration with broad implications for combinatorial probability and random matrix theory.

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.

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

TL;DR

The paper develops a unified, asymptotic framework for counting constrained graphs by degree sequence across bipartite, directed, and oriented classes in the sparse regime. Central to the approach is a switching analysis that yields precise exponential corrections for the constrained counts: under explicit sparsity conditions. These results translate into subgraph probabilities, enable computation of the expected permanent of random - matrices with prescribed margins, and provide asymptotics for counts of loop-free, simple oriented, and oriented-from-undirected graphs, including Eulerian orientations. The methods and formulas offer a robust toolset for degree-sequence based enumeration with broad implications for combinatorial probability and random matrix theory.

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.
Paper Structure (7 sections, 20 theorems, 76 equations, 3 figures)

This paper contains 7 sections, 20 theorems, 76 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Bipartite graphs
  3. Bipartite graph applications
  4. Digraphs without loops
  5. Permanents of random 0-1 matrices
  6. Oriented graphs
  7. Orientations of undirected graphs

Key Result

Theorem 2.1

If $S\to\infty$ and $s_{\mathrm{max}}t_{\mathrm{max}}=o(S^{2/3})$, then the number of bipartite graphs with degrees $\boldsymbol{s},\boldsymbol{t}$ is where

Figures (3)

  • Figure 1: Switching to remove an edge of $X$
  • Figure 2: A digraph and its associated bipartite graph.
  • Figure 3: Switching to remove a 2-cycle.

Theorems & Definitions (35)

  • Theorem 2.1: GMW
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5: GMW,BG2016
  • Lemma 2.6
  • proof
  • proof : Proof of Theorem $\ref{['thm:bip-avoid']}$
  • ...and 25 more