Counting unlabelled multigraphs with three nodes

TL;DR

This paper develops a mathematical method to count and generate all connected, unlabeled multigraphs with three nodes and fixed degrees, aiding applications in biology and physics.

Contribution

It introduces polynomial formulas for enumerating and efficiently generating all such multigraphs based on adjacency matrix counting.

Findings

Derived polynomial expressions for multigraph enumeration.

Provided an efficient method for generating multigraphs.

Facilitated applications in biological and physical network analysis.

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.