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Universal graph series, chromatic functions, and their index theory

Tsuyoshi Miezaki, Akihiro Munemasa, Yusaku Nishimura, Tadashi Sakuma, Shuhei Tsujie

TL;DR

This paper introduces universal graph series and a family of invariants based on H-chromatic functions $X_H(G)$ to achieve complete graph invariants, addressing limitations of the chromatic polynomial and Stanley's chromatic symmetric function. It develops a general framework using universal graphs (notably Kneser and Paley families) to derive expansion formulas, including a power-sum expansion for $X_{K_{\mathbb{N},k}}(G)$ and a robust theory of weak homomorphisms. The authors establish both trivial and nontrivial upper bounds for Paley-induced indices, prove pancyclicity properties of Paley graphs, and derive concrete results for cycles, paths, and complete bipartite graphs, illustrating the practical power of the invariant and its potential to distinguish graphs beyond the chromatic polynomial. The work provides a foundation for complete graph invariants via universal graph series and opens several questions on the distinguishing power of these invariants for trees and broader graph families, with implications for graph isomorphism and combinatorial invariant theory.

Abstract

In the present paper, we introduce the concept of universal graph series. We then present four invariants of graphs and discuss some of their properties. In particular, one of these invariants is a generalization of the chromatic symmetric function and a complete invariant for graphs.

Universal graph series, chromatic functions, and their index theory

TL;DR

This paper introduces universal graph series and a family of invariants based on H-chromatic functions to achieve complete graph invariants, addressing limitations of the chromatic polynomial and Stanley's chromatic symmetric function. It develops a general framework using universal graphs (notably Kneser and Paley families) to derive expansion formulas, including a power-sum expansion for and a robust theory of weak homomorphisms. The authors establish both trivial and nontrivial upper bounds for Paley-induced indices, prove pancyclicity properties of Paley graphs, and derive concrete results for cycles, paths, and complete bipartite graphs, illustrating the practical power of the invariant and its potential to distinguish graphs beyond the chromatic polynomial. The work provides a foundation for complete graph invariants via universal graph series and opens several questions on the distinguishing power of these invariants for trees and broader graph families, with implications for graph isomorphism and combinatorial invariant theory.

Abstract

In the present paper, we introduce the concept of universal graph series. We then present four invariants of graphs and discuss some of their properties. In particular, one of these invariants is a generalization of the chromatic symmetric function and a complete invariant for graphs.
Paper Structure (11 sections, 20 theorems, 104 equations, 1 figure)

This paper contains 11 sections, 20 theorems, 104 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm:main1']}
  3. Functions regarding homomorphisms and weak homomorphisms
  4. Expansion formulas for $X_{H}(G)$ and $W_{H}(G)$
  5. Power sum expansion of $X_{K_{\mathbb{N},k}}(G,x)$
  6. Evaluation of the Paley-induced index
  7. A trivial upper bound for the Paley-induced index
  8. Lemmas for a non-trivial upper bound for some Paley-induced indices
  9. Non-trivial upper bound of the Paley-induced index for some graphs
  10. The value of the Paley index for cycles and paths
  11. Remarks and questions

Key Result

Theorem 1.10

Figures (1)

  • Figure 1: Graphs $G_{1}$ and $G_{2}$

Theorems & Definitions (59)

  • Definition 1.1
  • Definition 1.2: Sta95
  • Remark 1.3
  • Conjecture 1.4: Sta95
  • Definition 1.5: Rado
  • Definition 1.6
  • Definition 1.7
  • Definition 1.8
  • Remark 1.9
  • Theorem 1.10
  • ...and 49 more