Combinatorial species and graph enumeration

TL;DR

This paper introduces the theory of combinatorial species as a powerful framework for enumerating labeled and unlabeled combinatorial structures, demonstrated through graph enumeration and computational tools.

Contribution

It develops the application of species theory to enumerate point-determining bipartite graphs and extends results to a broader class of graphs, with computational implementation in Sage.

Findings

Cycle index series effectively encode graph enumeration data

Species operations facilitate solving for complex structures

Sage software enables practical computation of species-based enumerations

Abstract

In enumerative combinatorics, it is often a goal to enumerate both labeled and unlabeled structures of a given type. The theory of combinatorial species is a novel toolset which provides a rigorous foundation for dealing with the distinction between labeled and unlabeled structures. The cycle index series of a species encodes the labeled and unlabeled enumerative data of that species. Moreover, by using species operations, we are able to solve for the cycle index series of one species in terms of other, known cycle indices of other species. Section 3 is an exposition of species theory and Section 4 is an enumeration of point-determining bipartite graphs using this toolset. In Section 5, we extend a result about point-determining graphs to a similar result for point-determining {\Phi}-graphs, where {\Phi} is a class of graphs with certain properties. Finally, Appendix A is an expository on species computation using the software Sage [9] and Appendix B uses Sage to calculate the cycle index series of point-determining bipartite graphs.