Regularity in Weighted Graphs: A Symmetric Function Approach

TL;DR

This paper explores the properties of k-regular multigraphs with controlled edge multiplicities, using symmetric functions to analyze their generating functions and conditions for D-finiteness.

Contribution

It introduces a symmetric function approach to express generating functions of regular multigraphs and establishes conditions for their D-finiteness.

Findings

Derived generating functions using symmetric functions

Provided conditions for D-finiteness of graph classes

Systematic expression via symmetric species results

Abstract

In this note we consider $k$-regular multigraphs, where the possible edge multiplicities are controlled. These structures are considered in a question recently posed by Brendan McKay. We express the generating functions using the scalar product of symmetric functions, and consequently give conditions on when the classes are D-finite. We appeal to symmetric species results of Mendez to write the expressions in a systematic way.