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Graph polynomials: some questions on the edge

Graham Farr, Kerri Morgan

TL;DR

This work surveys the landscape of graph polynomials, focusing on reduction relations, levels of recursion, and how algebraic structure encodes graph properties, while introducing new polynomials based on partial colourings. It connects classical Tutte–Whitney theory to partition functions from statistical physics, and extends reduction concepts beyond graphs via λ-reductions and binary functions, illustrating with Go polynomials and partial-colouring polynomials. The paper also investigates the origins of graph polynomials, the phenomenon of extending reductions to wider object classes, and the role of certificates in explaining equivalence and factorisation, suggesting a broader, SOL-definable framework for a comparative theory. Collectively, these ideas push toward a unified theory of graph polynomials, with implications for computation, complexity, and structural graph insights.

Abstract

We raise some questions about graph polynomials, highlighting concepts and phenomena that may merit consideration in the development of a general theory. Our questions are mainly of three types: When do graph polynomials have reduction relations (simple linear recursions based on local operations), perhaps in a wider class of combinatorial objects? How many levels of reduction relations does a graph polynomial need in order to express it in terms of trivial base cases? For a graph polynomial, how are properties such as equivalence and factorisation reflected in the structure of a graph? We illustrate our discussion with a variety of graph polynomials and other invariants. This leads us to reflect on the historical origins of graph polynomials. We also introduce some new polynomials based on partial colourings of graphs and establish some of their basic properties.

Graph polynomials: some questions on the edge

TL;DR

This work surveys the landscape of graph polynomials, focusing on reduction relations, levels of recursion, and how algebraic structure encodes graph properties, while introducing new polynomials based on partial colourings. It connects classical Tutte–Whitney theory to partition functions from statistical physics, and extends reduction concepts beyond graphs via λ-reductions and binary functions, illustrating with Go polynomials and partial-colouring polynomials. The paper also investigates the origins of graph polynomials, the phenomenon of extending reductions to wider object classes, and the role of certificates in explaining equivalence and factorisation, suggesting a broader, SOL-definable framework for a comparative theory. Collectively, these ideas push toward a unified theory of graph polynomials, with implications for computation, complexity, and structural graph insights.

Abstract

We raise some questions about graph polynomials, highlighting concepts and phenomena that may merit consideration in the development of a general theory. Our questions are mainly of three types: When do graph polynomials have reduction relations (simple linear recursions based on local operations), perhaps in a wider class of combinatorial objects? How many levels of reduction relations does a graph polynomial need in order to express it in terms of trivial base cases? For a graph polynomial, how are properties such as equivalence and factorisation reflected in the structure of a graph? We illustrate our discussion with a variety of graph polynomials and other invariants. This leads us to reflect on the historical origins of graph polynomials. We also introduce some new polynomials based on partial colourings of graphs and establish some of their basic properties.
Paper Structure (18 sections, 6 theorems, 59 equations, 4 figures)

This paper contains 18 sections, 6 theorems, 59 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Definitions and notation
  3. Some graph polynomials
  4. Tutte-Whitney polynomials
  5. Some partition functions
  6. Interpolating between contraction and deletion
  7. Go polynomials
  8. Polynomials for partial colourings
  9. Origins of graph polynomials
  10. Reduction relations
  11. Counting edge-colourings
  12. The symmetric Ashkin-Teller model
  13. Go polynomials
  14. Polynomials based on partial colourings
  15. Questions
  16. ...and 3 more sections

Key Result

Theorem 1

For any binary function $f:2^E\rightarrow\mathbb{C}$ and any $e\in E$,

Figures (4)

  • Figure 1: The two Gray graphs, from tutte74.
  • Figure 2: An Ashkin-Teller model configuration $(\sigma,\tau)$ for a graph.
  • Figure 3: Good (thick, black) and bad (thin, red) edges for the three Ising-type configurations in the Ashkin-Teller model configuration in Figure \ref{['fig:ashkin-teller-col']}.
  • Figure 4: Certificate of Tutte Equivalence. Graph $G$ is Tutte Equivalent to Graph $H$.

Theorems & Definitions (6)

  • Theorem 1: farr04
  • Proposition 2
  • Theorem 3: farr07b
  • Theorem 4
  • Theorem 5
  • Theorem 6