International Journal of Combinatorics
Volume 2012 (2012), Article ID 957284, 13 pages
http://dx.doi.org/10.1155/2012/957284
Research Article

On the Line Graph of the Zero Divisor Graph for the Ring of Gaussian Integers Modulo 𝑛

Khalida Nazzal1 and Manal Ghanem2

1Department of Mathematics, Palestine Technical University-Kadoorie, P.O. Box 7 Tulkarm, West Bank, Palestine
2Department of Mathematics, Irbid National University, P.O. Box 2600 Irbid, 21110, Jordan

Received 9 September 2011; Accepted 26 December 2011

Academic Editor: Gelasio Salazar

Copyright © 2012 Khalida Nazzal and Manal Ghanem. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let Γ ( ℤ 𝑛 [ 𝑖 ] ) be the zero divisor graph for the ring of the Gaussian integers modulo 𝑛 . Several properties of the line graph of Γ ( ℤ 𝑛 [ 𝑖 ] ) , 𝐿 ( Γ ( ℤ 𝑛 [ 𝑖 ] ) ) are studied. It is determined when 𝐿 ( Γ ( ℤ 𝑛 [ 𝑖 ] ) ) is Eulerian, Hamiltonian, or planer. The girth, the diameter, the radius, and the chromatic and clique numbers of this graph are found. In addition, the domination number of 𝐿 ( Γ ( ℤ 𝑛 [ 𝑖 ] ) ) is given when 𝑛 is a power of a prime. On the other hand, several graph invariants for Γ ( ℤ 𝑛 [ 𝑖 ] ) are also determined.