The Zero Divisor Graph of the Ring Zqp.

  • Nazar H. Shuker University of Mosul
  • Payman A. Rashed University of Salahaddin
Keywords: Commutative rings, zero divisor graphs, star zero divisor graph, crossing number, girth of the graph

Abstract

In this paper we construct a star zero divisor graph from the zero divisor graph of the ring Zqp. The star zero divisor graph is obtained by removing some vertices from the zero divisor graph Γ(Zqp), in different ways , but the best way to get star zero divisor graph SΓ(Zqp) is by removing an odd number of zero divisors. Finally the crossing number, the girth and the diameter of this graph is also determined.

Author Biographies

Nazar H. Shuker, University of Mosul

College of computer Science and Mathematics, University of Mosul, Iraq.

Payman A. Rashed, University of Salahaddin

College of Basic Education, University of Salahaddin, Kurdistan region – Iraq.

References

S. Akbari , H. R. Maimani and S. Yassemi, (2003) "When zero divisor graph is planner or a complete r-partite graph", J. of Algebra ,Vol.270, No.1 , pp. 169-180).

D.D. Anderson and P.S. Livingston, (1999)"The zero divisor graph of a commutative ring" . J. Algebra 217, pp. 434-447.

V. K. Bhat and R. Raina, (2007)"A note on zero divisor graph over rings", International Journal Contemp. Math. Sci. ,Vol. 2, No.14, pp.667-671.

I. Beck, (1988) "Coloring of commutative ring". J. Algebra 116, pp. 208 -226.

J. A. Bondy and U. S. R. Murty, (2013) " Graph Theory and Application" north – Holland, New York. Amsterdam, Oxford.

F. Buckley and F. Harary. (1990) "Distance in Graphs". Addison-Wesley Publishing Company, CA.

J. Coykendall, S.S. Wagstaff, L. Sheppardon and S. Spiroff, (2012) "On zero divisor graph". J. Commutative Algebra 2, pp.241-299.

A. Duane, (2006) "Proper coloring and p- partite structures of the zero divisor graph" Rose Holman Undergraduate Math Journal 7, Issue 2, pp.223-230.

R. S. Gupta, (2013) "The graph Γ2(R) over a ring R" I. J.of Pure and Applied Mathematics,Volume 86 No. 6,pp. 893-904.

F. Harary, (1969) "Graph Theory" Addison-Wesley Publishing Company NY.

A. Ramin, (2013) " The total graph of a finite commutative ring" Turk. J. Math 37, pp. 391-397.

N. H. Shuker, H.Q. Mohammad and A.M. Ali, (2012) "The zero divisor graph of " J. Algebra 6 pp. 1049-1055.
Published
2015-09-30
How to Cite
Shuker, N., & Rashed, P. (2015). The Zero Divisor Graph of the Ring Zqp. Science Journal of University of Zakho, 3(2), 281-290. Retrieved from https://sjuoz.uoz.edu.krd/index.php/sjuoz/article/view/170
Section
Science Journal of University of Zakho