On the Line Graph of the Complement Graph for the Ring of Gaussian Integers Modulo n

ABSTRACT

The line graph for the complement of the zero divisor graph for the ring of Gaussian integers modulo n is studied. The diameter, the radius and degree of each vertex are determined. Complete characterization of Hamiltonian, Eulerian, planer, regular, locally and locally connected is given. The chromatic number when is a power of a prime is computed. Further properties for and are also discussed.

The line graph for the complement of the zero divisor graph for the ring of Gaussian integers modulo n is studied. The diameter, the radius and degree of each vertex are determined. Complete characterization of Hamiltonian, Eulerian, planer, regular, locally and locally connected is given. The chromatic number when is a power of a prime is computed. Further properties for and are also discussed.

Cite this paper

M. Ghanem and K. Nazzal, "On the Line Graph of the Complement Graph for the Ring of Gaussian Integers Modulo n,"*Open Journal of Discrete Mathematics*, Vol. 2 No. 1, 2012, pp. 24-34. doi: 10.4236/ojdm.2012.21006.

M. Ghanem and K. Nazzal, "On the Line Graph of the Complement Graph for the Ring of Gaussian Integers Modulo n,"

References

[1] E. Abu Osba, S. Al-Addasi and N. Abu Jaradeh, “Zero Divisor Graph for the Ring of Gaussian Integers Modulo n,” Communication in Algebra, Vol. 36, No. 10, 2008, pp. 3865-3877. doi:10.1080/00927870802160859

[2] E. Abu Osba, S. Al-Addasi and B. Al-Khamaiseh, “Some Properties of the Zero Divisor Graph for the Ring of Gaussian Integers Modulo n,” Glasgow Journal of Math, Vol. 53, No. 1, 2011, pp. 391-399. doi:10.1017/S0017089511000024

[3] E. Abu Osba, “The Complement Graph for Gaussian Integers Modulo n,” Communication in Algebra, accepted.

[4] K. Nazzal and M. Ghanem, “On the Line Graph of the Zero Divisor Graph for the Ring of Gaussian Integers Modulo n,” International Journal of Combinatorics, Vol. 2012, Article ID 957284. doi:10.1155/2012/957284

[5] P. F. Lee, “Line Graph of Zero Divisor Graph in Commutative Rings,” Master’s Thesis, Colorado Christian University, 2007.

[6] J. Sedlà?ek, “Some Properties of Interchange Graphs, Theory of Graphs and Its Applications,” Academic Press, New York, 1962, pp. 145-150.

[7] G. Dersden and W. M. Dymcek, “Finding Factors of Factor Rings over the Gaussian Integers,” American Mathematical Monthly, Vol. 112, No. 7, 2005, pp. 602- 611. doi:10.2307/30037545

[8] V. G. Vising, “The Number of Edges in a Graph of a Given Radius,” Soviet Mathematics—Doklady, Vol. 8, 1967, pp. 535-536.

[9] C. Berg, “Graphs and Hypergraphs,” American Elsevier Publishing Co, Inc., New York, 1976.

[10] H. J. Veldman, “A Result on Hamiltonian Line Graphs Involving Restrictions on Induced Subgraphs,” Journal of Graph Theory, Vol. 12, No. 3, 1988, pp. 413-420. doi:10.1002/jgt.3190120312

[11] D. J. Oberly and D. P. Sumner, “Every Connected, Locally Connected Nontrivial Graph with No Induced Claw Is Hamiltonian,” Journal Graph Theory, Vol. 3, No. 4, 1979, pp. 351-356. doi:10.1002/jgt.3190030405

[12] M. J. Plantholt, “The Chromatic Index of Graphs with Large Maximum Degree,” Discrete Mathematics, Vol. 47, 1983, pp.91-96. doi:10.1016/0012-365X(83)90074-2

[13] B.-L. Chen and H.-L. Fu, “Total Chromatic Number and Chromatic Index of Split Graphs,” The Journal of Combinatorial Mathematics and Combinatorial Computing, Vol. 17, 1995, pp. 137-146.

[14] S. Akbari and A. Mohamamadaian, “On the Zero Divisor Graph of a Commutative Ring,” Journal of Algebra, Vol. 274, No. 2, 2004, pp. 847-855. doi:10.1016/S0021-8693(03)00435-6

[15] S. Arumugam and S. Velammal, “Edge Domination in Graphs,” Taiwanese Journal of Mathematics, Vol. 2, No. 2, 1998, pp. 173-179.

[1] E. Abu Osba, S. Al-Addasi and N. Abu Jaradeh, “Zero Divisor Graph for the Ring of Gaussian Integers Modulo n,” Communication in Algebra, Vol. 36, No. 10, 2008, pp. 3865-3877. doi:10.1080/00927870802160859

[2] E. Abu Osba, S. Al-Addasi and B. Al-Khamaiseh, “Some Properties of the Zero Divisor Graph for the Ring of Gaussian Integers Modulo n,” Glasgow Journal of Math, Vol. 53, No. 1, 2011, pp. 391-399. doi:10.1017/S0017089511000024

[3] E. Abu Osba, “The Complement Graph for Gaussian Integers Modulo n,” Communication in Algebra, accepted.

[4] K. Nazzal and M. Ghanem, “On the Line Graph of the Zero Divisor Graph for the Ring of Gaussian Integers Modulo n,” International Journal of Combinatorics, Vol. 2012, Article ID 957284. doi:10.1155/2012/957284

[5] P. F. Lee, “Line Graph of Zero Divisor Graph in Commutative Rings,” Master’s Thesis, Colorado Christian University, 2007.

[6] J. Sedlà?ek, “Some Properties of Interchange Graphs, Theory of Graphs and Its Applications,” Academic Press, New York, 1962, pp. 145-150.

[7] G. Dersden and W. M. Dymcek, “Finding Factors of Factor Rings over the Gaussian Integers,” American Mathematical Monthly, Vol. 112, No. 7, 2005, pp. 602- 611. doi:10.2307/30037545

[8] V. G. Vising, “The Number of Edges in a Graph of a Given Radius,” Soviet Mathematics—Doklady, Vol. 8, 1967, pp. 535-536.

[9] C. Berg, “Graphs and Hypergraphs,” American Elsevier Publishing Co, Inc., New York, 1976.

[10] H. J. Veldman, “A Result on Hamiltonian Line Graphs Involving Restrictions on Induced Subgraphs,” Journal of Graph Theory, Vol. 12, No. 3, 1988, pp. 413-420. doi:10.1002/jgt.3190120312

[11] D. J. Oberly and D. P. Sumner, “Every Connected, Locally Connected Nontrivial Graph with No Induced Claw Is Hamiltonian,” Journal Graph Theory, Vol. 3, No. 4, 1979, pp. 351-356. doi:10.1002/jgt.3190030405

[12] M. J. Plantholt, “The Chromatic Index of Graphs with Large Maximum Degree,” Discrete Mathematics, Vol. 47, 1983, pp.91-96. doi:10.1016/0012-365X(83)90074-2

[13] B.-L. Chen and H.-L. Fu, “Total Chromatic Number and Chromatic Index of Split Graphs,” The Journal of Combinatorial Mathematics and Combinatorial Computing, Vol. 17, 1995, pp. 137-146.

[14] S. Akbari and A. Mohamamadaian, “On the Zero Divisor Graph of a Commutative Ring,” Journal of Algebra, Vol. 274, No. 2, 2004, pp. 847-855. doi:10.1016/S0021-8693(03)00435-6

[15] S. Arumugam and S. Velammal, “Edge Domination in Graphs,” Taiwanese Journal of Mathematics, Vol. 2, No. 2, 1998, pp. 173-179.