Realizability problem for commuting graphs

TL;DR

This paper characterizes which finite graphs can be realized as commuting graphs of groups or semigroups, providing classifications and constructions for various graph types.

Contribution

It offers a complete classification of graphs that are commuting graphs of groups or semigroups, including those with isolated vertices or edges.

Findings

All graphs with at least two vertices and no universal vertex are commuting graphs of some semigroup.

Classified graphs with isolated vertices or edges that are commuting graphs of groups.

Identified cycles that are commuting graphs of centrefree semigroups.

Abstract

We investigate properties which ensure that a given finite graph is the commuting graph of a group or semigroup. We show that all graphs on at least two vertices such that no vertex is adjacent to all other vertices is the commuting graph of some semigroup. Moreover, we obtain a complete classification of the graphs with an isolated vertex or edge that are the commuting graph of a group and the cycles that are the commuting graph of a centrefree semigroup.