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Counting unlabelled multigraphs with three nodes

Andrea Bonato

TL;DR

This work addresses the enumeration of unlabeled, undirected, connected and traversable multigraphs with three nodes under minimum node-degree constraints $\mathcal{L}_i\ge3$, motivated by applications to chromatin network topology. It develops a framework that maps graphs to symmetric $3\times3$ adjacency matrices and imposes node-ordering to factor out isomorphisms, yielding polynomial counting expressions for distinct, double-equal, and triple-equal degree sequences; the detailed formulas are provided in Appendix A. The authors present concrete counts and examples for $N$ up to at least $15$, including explicit results for all-distinct, two-equal, and three-equal degree cases, together with tables and illustrative graphs. The method yields analytic tools to predict statistically favored motifs in folded chromatin networks via configurational entropy and path counting, and it lays groundwork for extending enumeration to larger graphs or alternative structural constraints.

Abstract

Unlabeled multigraphs have diverse applications across scientific fields, from transportation and social networks to polymer physics. In particular, multigraphs are essential for studying the relationship between the spatial organization and biological function of chromatin, which is often folded into complex polymer networks whose structure is closely tied to patterns of gene expression. A fundamental yet challenging aspect in applying graph theory to these areas is the enumeration of multigraphs, especially under structural constraints For example, when coupled with the statistical mechanics of polymer networks, the ability to identify traversable and connected multigraphs provides powerful tools for predicting statistically favored motifs that may arise within chromatin networks. In this work, by counting the adjacency matrices, we derive polynomial expressions that enumerate all connected, undirected, and unlabeled multigraphs with three nodes and fixed degree, and provide a method to efficiently generate them.

Counting unlabelled multigraphs with three nodes

TL;DR

This work addresses the enumeration of unlabeled, undirected, connected and traversable multigraphs with three nodes under minimum node-degree constraints , motivated by applications to chromatin network topology. It develops a framework that maps graphs to symmetric adjacency matrices and imposes node-ordering to factor out isomorphisms, yielding polynomial counting expressions for distinct, double-equal, and triple-equal degree sequences; the detailed formulas are provided in Appendix A. The authors present concrete counts and examples for up to at least , including explicit results for all-distinct, two-equal, and three-equal degree cases, together with tables and illustrative graphs. The method yields analytic tools to predict statistically favored motifs in folded chromatin networks via configurational entropy and path counting, and it lays groundwork for extending enumeration to larger graphs or alternative structural constraints.

Abstract

Unlabeled multigraphs have diverse applications across scientific fields, from transportation and social networks to polymer physics. In particular, multigraphs are essential for studying the relationship between the spatial organization and biological function of chromatin, which is often folded into complex polymer networks whose structure is closely tied to patterns of gene expression. A fundamental yet challenging aspect in applying graph theory to these areas is the enumeration of multigraphs, especially under structural constraints For example, when coupled with the statistical mechanics of polymer networks, the ability to identify traversable and connected multigraphs provides powerful tools for predicting statistically favored motifs that may arise within chromatin networks. In this work, by counting the adjacency matrices, we derive polynomial expressions that enumerate all connected, undirected, and unlabeled multigraphs with three nodes and fixed degree, and provide a method to efficiently generate them.
Paper Structure (18 sections, 2 theorems, 34 equations, 9 figures, 10 tables)

This paper contains 18 sections, 2 theorems, 34 equations, 9 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Connected unlabeled traversable multigraphs
  3. Counting multigraphs with three nodes
  4. Traversability
  5. Connectivity
  6. Ordering the nodes
  7. Different degrees
  8. Example with different degrees
  9. Two equal degrees
  10. Example with two equal degrees
  11. Three equal degrees
  12. Example with three equal degrees
  13. Conclusions
  14. Summation to polynomial expressions
  15. $\mathcal{L}_1 \neq \mathcal{L}_2 \neq \mathcal{L}_3$
  16. ...and 3 more sections

Key Result

Theorem 1

(Trudeau2013) A disconnected graph is clearly not traversable, and it is well known that a connected graph is traversable if and only if exactly 0 or 2 of its nodes are of odd degree.

Figures (9)

  • Figure 1: Examples of A labeled (top) and unlabeled (bottom) (multi)graphs, B graphs with a singleton ($\mathcal{L} = 2$, top) or an end ($\mathcal{L} = 1$, bottom), C connected (top) and disconnected (bottom) graphs, D traversable (top) and not traversable (bottom) graphs.
  • Figure 2: Example of mapping from adjacency matrices to multigraphs, and of isomorphic multigraphs and permutation-similar adjacency matrices. Each line (or column) of the matrix corresponds to a node of the graph, and the entries of that line to how that node is connected to the other nodes (i.e. number of connecting edges).
  • Figure 3: Connected unlabeled graphs with $\mathcal{L}_1=3$, $\mathcal{L}_2=4$ and $\mathcal{L}_3=7$.
  • Figure 4: Connected unlabeled graphs with $\mathcal{L}_1=3$, $\mathcal{L}_2=5$ and $\mathcal{L}_3=6$.
  • Figure 5: Connected unlabeled graphs with $\mathcal{L}_1=4$ and $\mathcal{L}_2 = \mathcal{L}_3=5$.
  • ...and 4 more figures

Theorems & Definitions (8)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Theorem 1
  • Definition 6
  • Theorem 2