Diameter of Zero Divisor Graphs of Finite Direct Product of Lattices

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





References:
[1] M. Alizadeh, A. K. Das, H. R. Maimani, M. R. Pournaki, AND S.
Yassemi, On the diameter and girth of zero-divisor graphs of posets,
Discrete Appl. Math. 160 (2012), 1319-1324.
[2] M. Alizadeh, H. R. Maimani, M. R. Pournaki, AND S. Yassemi, An ideal
theoretic approach to complete partite zero-divisor graphs of posets, J.
Algebra Appl 12 (2013), 1250148-1250159.
[3] D. F. Anderson and P. S. Livingston, The zero-divisor graph of a
commutative ring, J. Algebra 217(1999), 434-447.
[4] S. E. Atani and M. S. Kohan, The diameter of a zero-divisor graph
for finite direct product of commutative rings, Sarajevo Journal of
Mathematics, 16 (2007), 149-156.
[5] I. Beck, Coloring of a commutative ring, J. Algebra 116 (1988), 208-
226.
[6] F. R. DeMeyer, T. McKenzie and K. Schneider, The zero-divisor graph
of a commutative semigroup, Semigroup Forum 65 (2002), 206-214.
[7] E. Estaji and K. Khashyarmanesh, The zero-divisor graph of a lattice,
Results Math. 61 (2012), 1-11.
[8] R. Halaˇs and M. Jukl, On Beck’s coloring of posets, Discrete Math. 309
(2009), 4584-4589.
[9] R. Halaˇs and H. L¨anger, The zero divisor graph of a qoset, Order 27
(2010), 343-351.
[10] Vinayak Joshi, Zero divisor graph of a poset with respect to an ideal,
Order 29 (2012), 499-506.
[11] Vinayak Joshi and A. Khiste, Complement of the zero divisor graph of
a lattice, Bull. Aust. Math. Soc. 89 (2014), 177-190.
[12] Vinayak Joshi and Nilesh Mundlik, Prime ideals in 0-distributive posets,
Cen. Eur. J. Math. 11 (2013), 940-955.
[13] Vinayak Joshi, B. N. Waphare, and H. Y. Pourali, Zero divisor graphs
of lattices and primal ideals, Asian-Eur. J. Math. 5 (2012), 1250037-
1250046.
[14] Vinayak Joshi, B. N. Waphare, and H. Y. Pourali, On generalized zero
divisor graph of a poset, Discrete Appl. Math. 161 (2013), 1490-1495.
[15] Vinayak Joshi, B. N. Waphare, and H. Y. Pourali, The graph of
equivalence classes of zero divisors , ISRN Discrete Math. (2014),
Article ID 896270, 7 pages. http://dx.doi.org/101155/2014/896270.
[16] D. Lu and T. Wu, The zero-divisor graphs of posets and an application
to semigroups, Graphs Combin. 26 (2010), 793-804.
[17] S. K. Nimbhorkar, M. P. Wasadikar and Lisa DeMeyer, Coloring of
semilattices, Ars Comb. 12 (2007), 97-104.
[18] S.P. Redmond, The zero-divisor graph of a non-commutative ring, Int.
J. Comm. Rings 4 (2002), 203-211.
[19] J. Varlet, A generalization of notion of pseudo-complementness, Bull.
Soc. Roy. Sci. Li´ege 36 (1968), 149-158.
[20] D. B. West, Introduction to Graph Theory, Practice Hall, New Delhi,
2009.