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Enumeration of Tree-like Multigraphs with a Given Number of Vertices, Self-loops and Multiple Edges

Naveed Ahmed Azam, Seemab Hayat

TL;DR

The paper addresses the problem of exactly counting non-isomorphic tree-like multigraphs that may include both self-loops and multiple edges, parameterized by $n$, $s$, and $m$. It proposes a unified dynamic programming framework built on a canonical rooted representation and recursive decomposition of descendant subgraphs, with explicit recurrences and a combinatorial factor $w(f,g,h;y)$ to account for multiple subgraphs. The main contributions are a general DP formulation that unifies prior models handling self-loops and multiple edges separately, along with detailed initialization and boundary conditions that guarantee exact counting without explicit structure generation. This framework provides a foundation for analyzing growth and structural properties of complex polymer/topology graphs and offers pathways to extensions for labeled/weighted or degree-constrained multigraphs with broad applicability in chemistry and network science.

Abstract

Counting non-isomorphic tree-like multigraphs that include self-loops and multiple edges is an important problem in combinatorial enumeration, with applications in chemical graph theory, polymer science, and network modeling. Traditional counting techniques, such as Polya's theorem and branching algorithms, often face limitations due to symmetry handling and computational complexity. This study presents a unified dynamic programming framework for enumerating tree-like graphs characterized by a fixed number of vertices, self-loops, and multiple edges. The proposed method utilizes canonical rooted representations and recursive decomposition of subgraphs to eliminate redundant configurations, ensuring exact counting without the need for explicit structure generation. The framework also provides analytical bounds and recurrence relations that describe the growth behaviour of such multigraphs. This work extends previous models that treated self-loops and multiple edges separately, offering a general theoretical foundation for the enumeration of complex tree-like multigraphs in both mathematical and chemical domains.

Enumeration of Tree-like Multigraphs with a Given Number of Vertices, Self-loops and Multiple Edges

TL;DR

The paper addresses the problem of exactly counting non-isomorphic tree-like multigraphs that may include both self-loops and multiple edges, parameterized by , , and . It proposes a unified dynamic programming framework built on a canonical rooted representation and recursive decomposition of descendant subgraphs, with explicit recurrences and a combinatorial factor to account for multiple subgraphs. The main contributions are a general DP formulation that unifies prior models handling self-loops and multiple edges separately, along with detailed initialization and boundary conditions that guarantee exact counting without explicit structure generation. This framework provides a foundation for analyzing growth and structural properties of complex polymer/topology graphs and offers pathways to extensions for labeled/weighted or degree-constrained multigraphs with broad applicability in chemistry and network science.

Abstract

Counting non-isomorphic tree-like multigraphs that include self-loops and multiple edges is an important problem in combinatorial enumeration, with applications in chemical graph theory, polymer science, and network modeling. Traditional counting techniques, such as Polya's theorem and branching algorithms, often face limitations due to symmetry handling and computational complexity. This study presents a unified dynamic programming framework for enumerating tree-like graphs characterized by a fixed number of vertices, self-loops, and multiple edges. The proposed method utilizes canonical rooted representations and recursive decomposition of subgraphs to eliminate redundant configurations, ensuring exact counting without the need for explicit structure generation. The framework also provides analytical bounds and recurrence relations that describe the growth behaviour of such multigraphs. This work extends previous models that treated self-loops and multiple edges separately, offering a general theoretical foundation for the enumeration of complex tree-like multigraphs in both mathematical and chemical domains.
Paper Structure (8 sections, 6 theorems, 12 equations, 2 figures)

This paper contains 8 sections, 6 theorems, 12 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proposed Method
  4. Canonical representation
  5. Subproblem
  6. Recursive relation
  7. Initial conditions
  8. Conclusion

Key Result

Lemma 1

For any seven integers $n \geq 3$, $f \geq 1$, $s \geq g \geq 0$, $m \geq h+k \geq 0$, and a tree-like multigraph $P\in \mathcal{P}(n, s, m, {f}_{=}, {g}_{=}, {h}_{=}, {k}_{=} ),$ let $y = |\{ v\in N(r_{P}) : P_v \in \mathcal{P}(f,g, h, \,f{-}1_\le,\,g_\le,\,h_\le, h_\le)\}|$. Then it holds that

Figures (2)

  • Figure 1: Example of recursive relation.
  • Figure 2: An illustration of the residual tree depicted with a dashed line, and the descendant subgraph is depicted with solid lines.

Theorems & Definitions (11)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 1 more