The Imbedding Sum of a Graph

TL;DR

This paper introduces the embedding sum of a graph, a polynomial that encodes information about its labeled and unlabeled imbeddings, including their symmetries and coloring possibilities.

Contribution

It defines the embedding sum polynomial and demonstrates its utility in enumerating graph imbeddings and their symmetries.

Findings

The embedding sum polynomial encodes symmetry information.

It can count vertex colorings of graph imbeddings.

Provides a new tool for analyzing graph embeddings.

Abstract

The automorphisms of a graph act naturally on its set of labeled imbeddings to produce its unlabeled imbeddings. The imbedding sum of a graph is a polynomial that contains useful information about a graph's labeled and unlabeled imbeddings. In particular, the polynomial enumerates the number of different ways the unlabeled imbeddings can be vertex colored and enumerates the labeled and unlabeled imbeddings by their symmetries.