Some Basic Properties of The Identity-Commuting Graph of Multigroups

Abstract

Multigroups are generalization of groups which are based on the multiset structure in which repetition of elements are allowed. The multisets were introduced to remedy the handicap idea of the classical set which does not allow repetitions of elements. In this paper, we study the multigroup G structure through its associated identity-commuting graph Γ_ic(G) which is a simple graph whose vertices V (Γ_ic(G)) are the elements of the multigroup G and two distinct vertices x and y are adjacent if and only if xy = yx = e and e is adjacent to all vertices of the graph, where e is the identity element. Some basic properties of this graph were studied which include: vertex degree, size, connectedness, completeness among others. It was shown that the graph is connected but not complete and condition for completeness was obtained.