# The Zero Divisor Graph of the Ring Zqp.

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 Z_{qp}. The star zero divisor graph is obtained by removing some vertices from the zero divisor graph Γ(Z_{qp}), in different ways , but the best way to get star zero divisor graph SΓ(Z_{qp}) is by removing an odd number of zero divisors. Finally the crossing number, the girth and the diameter of this graph is also determined.

### References

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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.

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

*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

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