SOME BASIC PROPERTIES OF THE IDENTITY-COMMUTING GRAPH OF MULTIGROUPS

Abstract

Multigroup is a generalization of a group based on the multiset structure in which repetitions of elements are allowed. The multisets are introduced to remedy the limitations of the classical set which does not allow repetitions of elements. In this paper, we study the multigroup structure $G$ through its associated identity-commuting graph $\Gamma_{i c}(G)$ which is a simple graph whose vertices $V\left(\Gamma_{i c}(G)\right)$ are the elements of the multigroup $G$ and two distinct vertices $x$ and $y$ are adjacent if and only if $x y=y x=e$ and $e$ is adjacent to all vertices of the graph, where $e$ is the identity element. Some basic properties of this graph are studied which include: vertex degree, size, connectedness, completeness among others.

Received: December 18, 2023
Revised: May 30, 2024
Accepted: June 21, 2024