A BIPARTITE GRAPH ASSOCIATED WITH IRREDUCIBLE
ELEMENTS AND GROUP OF UNITS IN Zn
Augustine Musukwa1, Khumbo Kumwenda2
1Department of Mathematics
University of Trento
Via Della Malpensada 140, Trento TN, ITALY
1,2Department of Mathematics
Mzuzu University, MALAWI

Abstract

A nonzero nonunit $a$ of a ring $R$ is called an irreducible element if, for some $b,c\in R$, $a = bc$ implies that either $b$ or $c$ (not both) is a unit. We construct a bipartite graph in which the union of the set of irreducible elements and group of units is a vertex-set and an edge-set is the set of pairs between irreducible elements and their unit factors in the ring of integers modulo $n$. Many properties of this constructed bipartite graph are studied. We show that this bipartite graph contains components which are isomorphic. We also note that each component of this bipartite graph can be presented in some form which we call star form presentation. Some examples of graphs in star form presentation are provided for illustration purposes. Furthermore, we prove that the girth of this bipartite graph is 8. Most of the results in this paper are arrived at via group action.

Citation details of the article



Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 31
Issue: 1
Year: 2018

DOI: 10.12732/ijam.v31i1.10

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