Graphs defined on algebras

TL;DR

This paper explores extending various graph definitions from groups to general algebras, analyzing which properties carry over and introducing new directed and simplicial complex structures.

Contribution

It systematically investigates how existing algebraic graphs can be generalized to arbitrary algebras, including directed graphs and simplicial complexes, highlighting challenges and extensions.

Findings

Some group-based graph properties extend to algebras unchanged

Certain graphs require specific group properties to generalize

Explicit descriptions provided for independence algebras

Abstract

There has been a great deal of attention recently to graphs whose vertex set is a group, defined using the group structure. (The commuting graph, where two elements are joined if they commute, is the oldest and most famous example.) The purpose of this paper is to investigate extending the definitions of such graphs to general algebras (in the sense of universal algebra). It seems unlikely that such a definition can be made for the commuting graph, or for various others such as the nilpotency and Engel graphs. However, for graphs whose definition depends on the notion of subgroup or subalgebra generated by a subset, the existing definitions work without change. These graphs include several well-studied examples: the power graph, enhanced power graph, generating graph, independence graph, and rank graph. In these cases, some results about groups extend to arbitrary algebras unchanged, but others require specific properties of groups, and pose a challenge to researchers. In the next two sections, I will describe some extensions to directed graphs (the directed power graph and the endomorphism digraph) and to simplicial complexes (the independence and strong independence complexes). The final section gives explicit descriptions of all of these objects for independence algebras.