Algebraic Constructions for the Digraph Routing Problems

TL;DR

This paper explores algebraic methods, using quasigroups and permutation groups, to analyze and construct digraphs, providing new algebraic characterizations and expressions for complex graphs like the Hoffman-Singleton graph.

Contribution

It introduces algebraic frameworks for studying digraph routing problems, including definitions and representations using quasigroups and permutations, and offers compact algebraic descriptions of important graphs.

Findings

Algebraic definitions for key graphs derived from quasigroup properties

Representation of groupoids by disjoint permutations

A single algebraic expression for the Hoffman-Singleton graph

Abstract

Efficiency of routing on a regular digraph often involves finding opitmal properties of the graph. For example, the diameter of a digraph is the maximum distance between any two vertices. We show how we can study these problems algebraically in terms of quasigroups, 1-factors, and permutation groups. Our investigation originated from the study of graphs as the Cayley graphs of groupoids with $d$ generators, a left identity, and right cancellation; that is, a right quasigroup. This enables us to provide compact algebraic definitions for some important graphs that are either given as explicit edge lists or as the Cayley coset graphs of groups larger than the graph. One such example is a single expression for the Hoffman-Singleton graph. From there, we notice that the groupoids can be represented uniquely by a set of disjoint permutations and we explore the consequences of that observation.