A BIPARTITE GRAPH ASSOCIATED WITH IRREDUCIBLE
ELEMENTS AND GROUP OF UNITS IN Zn

# Abstract

A nonzero nonunit of a ring is called an irreducible element if, for some , implies that either or (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 . 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

# References

1. [1]A. Benjamin, G. Chartrand, P. Zhang, The Fascinating World of Graph Theory, Princeton University Press, Oxford (2015).
2. [2] G. Chartrand, P. Zhang, Chromatic Graph Theory, Chapman and Hall/CRC, London (2009).
3. [3] J. Coykendall, D.E. Dobbs, and B. Mullins, Exploring irreducible elements: An abstract algebra project, Alabama Journal of Mathematics (Education), Fall 2002.
4. [4] J.A. Gallian, Contemporary Abstract Algebra, 7th Edition, Richard Stratton, United Kingdom (2010).
5. [5] G.A. Jones, J.A. Jones, Elementary Number Theory, Springer Science, United Kingdom (2005).
6. [6] T.W. Judson, Abstract Algebra, Theory and Application, Cambridge University Press, London (2009).
7. [7] A. Musukwa, K. Kumwenda, A Special Subring Associated with Irreducible Elements in Zn, MAYFEB Journal of Mathematics, 2 (2017), 1-7.
8. [8] M. Pranjali, M. Acharya, Graphs Associated with Finite Zero Ring, Gen. Math. Notes, 24, No 2 (2014), 53-69.
9. [9] B. Steinberg, Representation Theory of Finite Group: An Introductory Approach, Springer Science, New York (2012)
10. [10] J.J. Tattersall, Elementary Number Theory in Nine Chapters, 2nd Edition, Cambridge University Press, New York (2005).
11. [11] V.I. Voloshin, Introduction to Graph Theory, Nova Science, New York (2009).