### The Zero Divisor Graph of the Ring Z_(2^2 p)

#### Abstract

In this paper, we consider the crossing number and the chromatic number of the zero divisor graph Γ(Z_(2^2 p)) to show that this type of zero divisor graphs is bipartite graph, and the smallest cycle in Γ(Z_(2^2 p)) is of length four this implies that the girth is equal four.

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

Beck, I., 1988. Coloring of commutative ring. Journal of Algebra, 116, pp208-226.

Bondy, J.A. and. Murty, U.S.R., 2013. Graph Theory and Application North. Holland, New York. Amsterdam. Oxford.

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

Duane, A., 2006. Proper coloring and p-partite structures of the zero divisor graph. Rose Holman Undergraduate Math Journal, 7(2), pp.1-7.

Harary, F., 1969. Graph Theory. Addison-Wesley Publishing Company, California.

Harju, T., 2005. Graph theory, Department of Mathematics, University of Turku, Finland.

Malathi, M., Sankeetha, S. and Sankar, J.R., 2013. Rectilinear crossing number of a zero divisor graph. International Mathematical Forum, 8(12), pp583-589.

Sankar, J.R. and Sankeetha, S., 2012. Crossing number of a zero divisor graph. International Journal of Algebra, 6(32), pp.1499-1505.

Shuker, N.H., Mohammad, H.Q. and Ali, A.M., 2012. The zero divisor graph of Z_(p^n q) . International Journal of Algebra, 6, pp.1049-1055.

DOI: http://dx.doi.org/10.14500/aro.10058

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