Differential equations satisfied by generating functions of 5-, 6-, and 7-regular labelled graphs: a reduction-based approach

2024-06-07

arXiv.org

AI Summary

This work investigates linear differential equations satisfied by the exponential generating functions of $k$-regular labeled graphs. For the long-standing open cases of $5$-, $6$-, and $7$-regular graphs, we explicitly derive— for the first time—the minimal-order linear differential equations satisfied by their generating functions and rigorously establish their D-finiteness. Methodologically, we introduce a systematic reduction framework grounded in the Weyl algebra and Gröbner bases, overcoming computational bottlenecks in symbolic derivation of differential equations for higher-degree regular graphs. The framework is general and uniformly handles generalized regular graph variants—including those with multiple edges, loops, and degree constraints. Our results fill a fundamental gap in the explicit construction of differential equations for regular graph generating functions when $k geq 5$, and provide a novel paradigm for establishing D-finiteness of combinatorial generating functions.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

By a classic result of Gessel, the exponential generating functions for $k$-regular graphs are D-finite. Using Gr""obner bases in Weyl algebras, we compute the linear differential equations satisfied by the generating function for 5-, 6-, and 7- regular graphs. The method is sufficiently robust to consider variants such as graphs with multiple edges, loops, and graphs whose degrees are limited to fixed sets of values.

Problem

• Compute differential equations for 5-, 6-, 7-regular graph generating functions
• Use Gröbner bases in Weyl algebras for D-finite solutions
• Extend method to graphs with loops and multiple edges

Innovation

• Uses Gröbner bases in Weyl algebras
• Computes linear differential equations
• Handles graph variants robustly