Graphical Enumeration: A Species-Theoretic Approach

TL;DR

This paper introduces a species-theoretic operation called inner plethysm, extends it to n-sorted species and virtual species, and applies these concepts to enumerate regular graphs and digraphs in graph theory.

Contribution

It develops a new species-theoretic operation and extends it to broader contexts, providing novel tools for graph enumeration problems.

Findings

Defined inner plethysm for species and generalized it to n-sorted species

Extended operations to virtual species using polynomial maps

Applied these methods to enumerate regular graphs and digraphs

Abstract

An operation on species corresponding to the inner plethysm of their associated cycle index series is constructed. This operation, the inner plethysm of species, is generalized to n-sorted species. Polynomial maps on species are studied and used to extend inner plethysm and other operations to virtual species. Finally, inner plethysm and other operations on species are applied to various problems in graph theory. In particular, regular graphs, and digraphs in which every vertex has outdegree k, are enumerated.