A Normal Graph Algebra

TL;DR

This paper introduces a normal graph algebra inspired by genetics, explores its properties, and applies it to the Petersen graph, revealing connections to automorphisms and unique algebraic features.

Contribution

It defines a novel algebraic framework for graphs, analyzes its properties, and demonstrates its application to the Petersen graph, uncovering new algebraic and automorphism insights.

Findings

Normal algebra often highlights key graph features

Unique normal algebra for the Petersen graph

Connections to automorphisms of Sym(6)

Abstract

We define a normal graph algebra modeled on algebras used in genetics. Although the algebra does not always determine its graph, it often highlights special features. After developing basic properties of the algebra, we examine those of certain minimal graphs. We then apply the results to the Petersen graph, finding connections between some of its many aspects. For example, the outer automorphisms of Sym(6) emerge naturally. The normal algebra of the Petersen graph is unique among normal graph algebras.