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Article . 2014
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Article . 2014
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Diameter Of Zero Divisor Graphs Of Finite Direct Product Of Lattices

Authors: H. Y. Pourali; V. V. Joshi; B. N. Waphare.;

Diameter Of Zero Divisor Graphs Of Finite Direct Product Of Lattices

Abstract

{"references": ["M. Alizadeh, A. K. Das, H. R. Maimani, M. R. Pournaki, AND S.\nYassemi, On the diameter and girth of zero-divisor graphs of posets,\nDiscrete Appl. Math. 160 (2012), 1319-1324.", "M. Alizadeh, H. R. Maimani, M. R. Pournaki, AND S. Yassemi, An ideal\ntheoretic approach to complete partite zero-divisor graphs of posets, J.\nAlgebra Appl 12 (2013), 1250148-1250159.", "D. F. Anderson and P. S. Livingston, The zero-divisor graph of a\ncommutative ring, J. Algebra 217(1999), 434-447.", "S. E. Atani and M. S. Kohan, The diameter of a zero-divisor graph\nfor finite direct product of commutative rings, Sarajevo Journal of\nMathematics, 16 (2007), 149-156.", "I. Beck, Coloring of a commutative ring, J. Algebra 116 (1988), 208-\n226.", "F. R. DeMeyer, T. McKenzie and K. Schneider, The zero-divisor graph\nof a commutative semigroup, Semigroup Forum 65 (2002), 206-214.", "E. Estaji and K. Khashyarmanesh, The zero-divisor graph of a lattice,\nResults Math. 61 (2012), 1-11.", "R. Hala\u02c7s and M. Jukl, On Beck's coloring of posets, Discrete Math. 309\n(2009), 4584-4589.", "R. Hala\u02c7s and H. L\u00a8anger, The zero divisor graph of a qoset, Order 27\n(2010), 343-351.\n[10] Vinayak Joshi, Zero divisor graph of a poset with respect to an ideal,\nOrder 29 (2012), 499-506.\n[11] Vinayak Joshi and A. Khiste, Complement of the zero divisor graph of\na lattice, Bull. Aust. Math. Soc. 89 (2014), 177-190.\n[12] Vinayak Joshi and Nilesh Mundlik, Prime ideals in 0-distributive posets,\nCen. Eur. J. Math. 11 (2013), 940-955.\n[13] Vinayak Joshi, B. N. Waphare, and H. Y. Pourali, Zero divisor graphs\nof lattices and primal ideals, Asian-Eur. J. Math. 5 (2012), 1250037-\n1250046.\n[14] Vinayak Joshi, B. N. Waphare, and H. Y. Pourali, On generalized zero\ndivisor graph of a poset, Discrete Appl. Math. 161 (2013), 1490-1495.\n[15] Vinayak Joshi, B. N. Waphare, and H. Y. Pourali, The graph of\nequivalence classes of zero divisors , ISRN Discrete Math. (2014),\nArticle ID 896270, 7 pages. http://dx.doi.org/101155/2014/896270.\n[16] D. Lu and T. Wu, The zero-divisor graphs of posets and an application\nto semigroups, Graphs Combin. 26 (2010), 793-804.\n[17] S. K. Nimbhorkar, M. P. Wasadikar and Lisa DeMeyer, Coloring of\nsemilattices, Ars Comb. 12 (2007), 97-104.\n[18] S.P. Redmond, The zero-divisor graph of a non-commutative ring, Int.\nJ. Comm. Rings 4 (2002), 203-211.\n[19] J. Varlet, A generalization of notion of pseudo-complementness, Bull.\nSoc. Roy. Sci. Li\u00b4ege 36 (1968), 149-158.\n[20] D. B. West, Introduction to Graph Theory, Practice Hall, New Delhi,\n2009."]}

In this paper, we verify the diameter of zero divisor graphs with respect to direct product.

Keywords

girth, zero divisor graph., product of graphs, complement of graph, prime ideal, diameter, direct product of lattices, Atomic lattice, 0-distributive lattice

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