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

TL;DR

This paper derives differential equations for generating functions of 5-, 6-, and 7-regular labeled graphs using a reduction-based approach with Gr"obner bases, extending to variants like multigraphs and loops.

Contribution

It introduces a robust method employing Gr"obner bases in Weyl algebras to compute differential equations for regular graphs' generating functions, expanding previous results.

Findings

Derived differential equations for 5-, 6-, and 7-regular graphs

Method applies to graphs with multiple edges and loops

Extends to graphs with fixed degree sets

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.