Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 15 (2020), 371 -- 397

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This work is licensed under a Creative Commons Attribution 4.0 International License.

ZERO-DIVISOR GRAPHS OF FINITE COMMUTATIVE RINGS: A SURVEY

Pradeep Singh and Vijay Kumar Bhat

Abstract. This article gives a comprehensive survey on zero-divisor graphs of finite commutative rings. We investigate the results on structural properties of these graphs.

2020 Mathematics Subject Classification: 05C25, 13A99, 13M99, 13P25
Keywords: commutative ring; zero-divisor graph; structural properties; Wiener index

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References

  1. E. E. AbdAlJawad, H. Al-Ezeh, Domination and Independence Numbers of Γ(ℤn), Int. Math. Forum (3)(2008), 503-511. MR2386203. zbl 1151.13300.

  2. E. AbuHijleh, M. Abudayah, O. Alomari, H. Al-Ezeh, Matching Number, Independence Number, and Covering Vertex Number of Γ(ℤn), Mathematics 7 (2019), 49, https://doi.org/10.3390/math7010049.

  3. M. R. Ahmadi, R. Jahani-Nezhad, Energy and Wiener index of zero-divisor graphs, Iranian J. Math. Chem 2 (2011), 45-51.

  4. S. Akbari, H. R. Maimani, S. Yassemi, When a zero-divisor graph is planar or a complete r-partite graph, J. Algebra 270 (2003) 169-180. MR2016655. Zbl 1032.13014.

  5. S. Akbari, A. Mohammadian, Zero-divisor graphs of non-commutative rings, J. Algebra 296 (2006), 462-479. MR2201052. Zbl 1113.16038.

  6. N. Akgunes, Y. Nacaroglu, Some Properties of zero-divisor Graph Obtained by the Ring Zp × Zq × Zr, Asian-European J. Math 12 (2019), https://doi.org/10.1142/S179355712040001X. MR4018166. Zbl 1423.05077.

  7. D. D. Anderson, M. Naseer, Beck's coloring of a commutative ring, J. Algebra 159 (1993), 500-514. MR1231228. Zbl 0798.05067.

  8. D. F. Anderson, P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217 (1999), 434-447. MR1700509. Zbl 0941.05062.

  9. D. F. Anderson, A. Frazier, A. Lauve, P. S. Livingston, The zero-divisor graph of a commutative ring II, Lecture Notes in Pure and Appl. Math. 220 (2001), 61-72. MR1836591. Zbl 1035.13004.

  10. D. F. Anderson, M. C. Axtell, J. A. Stickles, Zero-divisor graphs in commutative rings. In:Commutative algebra, Noetherian and Non-Noetherian perspectives, ed. by M. Fontana et al., New York:Springer, (2010) pp. 23-45. MR2762487. Zbl 1225.13002.

  11. D. F. Anderson, A. Badawi, The zero-divisor graph of a commutative semigroup, In:A survey, Groups, Modules and Model Theory- Surveys and Recent Developments, ed. by M. Droste et al., Springer International Publishing AG (2017) pp. 23-39. MR3675901.

  12. C. I. Aponte, P. S. Johnson, N. A. Mims, Line graphs of zero-divisor graphs, Lect. Note of SUMSRI, Miami University (2005).

  13. M. Axtell, J. Coykendall, J. Stickles, Zero-divisor graphs of polynomial and power series over commutative rings, Commun. Algebra 33 (2005), 2043-2050. MR2150859. Zbl 1088.13006.

  14. M. Axtell, J. Stickles, J. Warfel, Zero-divisor graphs of direct products of commutative rings, Houst. J. Math 32 (2006), 985-994. MR2268463. Zbl 1110.13004.

  15. I. Beck, Coloring of commutative rings, J. Algebra 116 (1988), 208-226. MR0944156. Zbl 0654.13001.

  16. R. Belshoff, J. Chapman, Planar zero-divisor graphs, J. Algebra 316 (2007), 471-480. MR2354873 Zbl 1129.13028

  17. M. I. Bhat, S. Pirzada, On strong metric dimension of zero-divisor graphs of rings, The Korean J. Math 27 (2019), 563-580. MR4020290. Zbl 1427.13003.

  18. L. M. Birch, J. J. Thibodeaux, R. P. Tucci, Zero-divisor graphs of finite direct products of finite rings, Comm. Algebra 42 (2014), 3852-3860. MR3200064. Zbl 1302.13011.

  19. N. Bloomfield, C. Wickham, Local rings with genus two zero-divisor graph, Comm. Algebra 38 (2010), 2965-2980. MR2730289. Zbl 1226.05132.

  20. J. A. Bondy, U. S. R. Murty, Graph Theory with Applications, Elsevier Science Publishing Co., Inc (1982).

  21. S. Chattopadhyay, K. L. Patra, B. K. Sahoo, Laplacian eigenvalues of the zero-divisor graph of the ring ℤn, Linear Algebra And Appl 584 (2020), 267-286. MR4011590. Zbl 1426.05062.

  22. T. T. Chelvam, T. Asir, Distances in zero-divisor and total graphs from commutative rings- A survey, AKCE Int. J. Graphs Comb. 13 (2016), 290-298. MR3616614. Zbl 1354.05037.

  23. N. Cordova, C. Gholston, H. Hauser, The Structure of Zero-Divisor Graphs, SUMSRI, Miami University, 2005.

  24. B. Cote, C. Ewing, M. Huhn, C. M. Plaut, D. Weber, Cut-sets in zero-divisor graphs of finite commutative rings, Commun. Algebra 39 (2011), 2849-2861. MR2834134. Zbl 1228.13011.

  25. J. Coykendall, S. Sather-Wagstaff, L. Sheppardson, S. Spiroff, On zero-divisor graphs, In:Progress in commutative algebra 2, ed. by C. Francisco et al. de Gruyter Berlin, (2012) pp. 241-299. MR2932598. Zbl 1243.13017.

  26. F. DeMeyer, K. Schneider, Automorphisms and zero-divisor graphs of commutative rings, Int. J. Commut. Rings 1 (2002), 93-106. MR2037656.

  27. A. Duane, Proper colorings and p-partite structures of the zero-divisor graph, Rose-Hulman Undergraduate Mathematics Journal 7 (2006), Article 16.

  28. A. Duric, S. Jevdenic, P. Oblak, N. Stopar, The total zero-divisor graph of commutative rings, J. Algebra Appl. 18 (2019), 1950190, 16pp. MR3989386. Zbl 1423.13055.

  29. K. Elahi, A. Ahmad, R. Hasni, Construction Algorithm for zero-divisor Graphs of Finite Commutative Rings and Their Vertex-Based Eccentric Topological Indices, Mathematics 6 (2018), 301. Zbl 1425.05069.

  30. M. Ghanem, K. Nazzal, On the line graph of the complement graph for the ring of gaussian integers modulo n, Open J. Discrete Math. 2 (2012), 24-34. MR2901214. Zbl 1242.05155.

  31. S. Hedetniemi, Homomorphisms of graphs and automata, Technical Report 03105-44-T, University of Michigan (1966). MR2615860.

  32. T. Ju, M. Wu, On iteration digraph and zero-divisor graph of the ring ℤn, Czechoslovak Math. J. 64(3) (2014), 611-628. MR3298550. Zbl 1349.05145.

  33. A. N. Koam, A. Ahmad, A. Haider, On Eccentric Topological Indices Based on Edges of zero-divisor Graphs, Symmetry 11 (2019), 907 doi:10.3390/sym11070907.

  34. D. Lu, T. Wu, On endomorphism-regularity of zero-divisor graphs, Discrete Math. 308 (2008), 4811-4815. MR2438187. Zbl 1156.05025.

  35. P. Mahadevi, J. B. Babujee, On Special Structures of Zero-Divisor Graphs, Int. J. Pure And Appl. Math. 119 (2018), 281-287.

  36. H. R. Maimani, M. R. Pouranki, A. Trhranian, S. Yassemi, Graphs attached to rings revisited, Arab. J. Sci. Eng. 36 (2011), 997-1011. MR2845527.

  37. M. Malathi, S. Sankeetha, J. R. Sankar, S. Meena, Rectilinear crossing number of a zero-divisor graph, Int. Math. Forum 8 (2013), 583-589. MR3040740. Zbl 1283.05129.

  38. M. Malathi, N. Selvi, J. R. Sankar, The crucial point of behaviour of maximum planarity in rectilinear crossing number of Γ(ℤn), Int. J. Pure And Appl. Math. 115 (2017), 321-329.

  39. M. Malathi, N. Selvi, J. R. Sankar, Edge Non-Edge crossing number of bipartite graph of Γ(ℤn), Int. J. Math. Archive 8 (2017), 65-70.

  40. H. Q. Mohammad, M. N. Authman, Hosoya Polynomial and Wiener Index of Zero-Divisor Graph of ℤn, Raf. J. of Comp. & Math's. 12 (2018), 47-59.

  41. K. Monius, Eigenvalues of zero-divisor graphs of finite commutative rings, arXiv preprint arXiv:1910.12567, (2019).

  42. S. B. Mulay, Cycles and symmetries of zero-divisors, Comm. Algebra 30 (2002), 3533-3558. MR1940489. Zbl 1087.13500.

  43. K. Nazzal, M. Ghanem, On the Line Graph of the zero-divisor Graph for the Ring of Gaussian Integers Modulo n, Int. J. Combinatorics (2012), 1-13. MR2874508. Zbl 1236.05105.

  44. K. Nazzal, Total graphs attached to a commutative ring, Palest. J. Math 5(special I) (2016), 108-126. MR3477620. Zbl 1346.13005

  45. M. Nikmehr, L. Heidarzadeh, N. Soleimani, Calculating different topological indices of total graph of ℤn, Stud. Sci. Math. Hung. 51 (2014), 133-140. Zbl 1299.05164.

  46. E. A. Osba, S. Al-Addasi, N. A. Jaradeh, Zero-divisor graph for the ring of Gaussian integers modulo n, Comm. Algebra 36 (2008), 3865-3877. MR2458411. Zbl 1151.05042.

  47. E. A. Osba, S. Al-Addasi, B. Al-Khamaiseh, Some properties of the zero-divisor graph for the ring of Gaussian integers modulo n, Glasgow Math. J. 53 (2011), 391-399. MR2783168. Zbl 1213.13019.

  48. E. A. Osba, The Complement graph for Gaussian integers modulo n, Comm. Algebra 40 (2012), 1886-1892. MR2924491. Zbl 1246.13006.

  49. R. Pandimeenal, K. A. Devi, Odd and Even Number of the zero-divisor Graph, Int. J. Research in Engineering, Science and Management 2 (2019), 334-335.

  50. K. Patra, P. P. Baruah, On the Adjacency Matrix and Neighborhood Associated with Zero-divisor Graph for Direct Product of Finite Commutative Rings, Int. J. Comp. Appl. Tech. And Research 2 (2013), 315-323.

  51. J. Periaswamy, N. Selvi, J. R. Sankar, Pair Sum Labelings of zero-divisor Graphs, Int. J. Pure And Appl. Math. 115 (2017), 387-394.

  52. J. Periaswamy, N. Selvi, Edge Sum Index of a Graph in a Commutative Ring, Int. J. Math. And Appl. 6(1–A) (2018), 133–137.

  53. J. Periaswamy, N. Selvi, Geometric Mean Labeling of Union and Product of Some Standard Graphs with zero-divisor Graphs, Int. J. Math. And Appl. 6(1–A) (2018), 115–120.

  54. J. Periaswamy, N. Selvi, Integral Edge Sum of zero-divisor Graph ℤn, Int. J. Math. Archive 9 (2018), 37-43.

  55. A. Phillips, J. Rogers, K. Tolliver, F. Worek, Uncharted territory of zero-divisor graphs and their complements, SUMSRI, Miami University (2004).

  56. V. Rajakumaran, N. Selvi, Cyclic Path Covers in zero-divisor Graphs, Int. J. Math. Archive 9 (2018), 163-167.

  57. B. S. Reddy, R. Jain, N. Laxmikanth, Eigenvalues and Wiener index of the zero-divisor graph Γ(ℤn), arXiv preprint arXiv:1707.05083 (2017).

  58. B. S. Reddy, R. Jain, N. Laxmikanth, Vertex and Edge connectivity of the zero-divisor graph Γ(ℤn), arXiv preprint arXiv:1807.02703 (2018).

  59. S. P. Redmond, The zero-divisor graph of a non-commutative ring, Int. J. Comm. Ring 1 (2002), 203-211. MR2037657. Zbl 1195.16038.

  60. S. P. Redmond, An Ideal-based zero-divisor graph of a commutative ring, Comm. Algebra 31 (2003), 4425-4443. MR1995544. Zbl 1020.13001.

  61. S. P. Redmond, Structure in the zero-divisor graph of a non-commutative ring, Houst. J. Math 30 (2004), 345-355. MR2084907. Zbl 1064.16033.

  62. S. P. Redmond, Central sets and the radii of zero-divisor graphs of the commutative rings, Comm. Algebra 34 (2006), 2389-2401. MR2240381. Zbl 1105.13007.

  63. S. P. Redmond, On zero-divisor graphs of small finite commutative rings, Discrete Math. 307 (2007), 1155-1166. MR2292543. Zbl 1107.13006.

  64. J. R. Sankar, S. Sankeetha, R. Vasanthakumari, S. Meena, Crossing number of a zero-divisor graph, Int. J. Algebra 6 (2012), 1499-1505. MR3033227. Zbl 1273.05105.

  65. J. R. Sankar, S. Meena, Connected Domination Number of a commutative ring, Int. J. Math. Research 5 (2012), 5-11.

  66. J. R. Sankar, S. Meena, On Weak Domination in a zero-divisor Graph, Int. J. Appl. Math. 26 (2013), 83-91. MR3098194. Zbl 1288.05126.

  67. P. Sharma, A. Sharma, R. K. Vats, A Study on Adjacency Matrix for Zero-Divisor Graphs over Finite Ring of Gaussian Integer, Int. J. Comp. Sci. & Emerging Tech. 1 (2010), 218-223.

  68. P. Sharma, A. Sharma, R. K. Vats, Analysis of adjacency matrix and neighborhood associated with zero-divisor graph of finite commutative rings, Int. J. Comp. Appl. 14 (2011), 38-42.

  69. N. H. Shuker, H. Q. Mohammad, L. A. Khaleel, Hosoya and Wiener Index of Zero-Divisor Graph of Z Pm Q2, ZANCO J. Pure And Appl. Sci. 31 (2019), 45-52.

  70. J. Skowronek-Kaziow, Some digraphs arising from number theory and remarks on the zero-divisor graph of the ring Zn, Information Processing Letters 108 (2008), 165-169. MR2452147. Zbl 1189.05070.

  71. N. O. Smith, Planar zero-divisor graphs, Int. J. Commut. Rings 2 (2003), 177-188.

  72. C. Subramanian, T. T. Chelvam, Sum cordial labelling of zero-divisor graph, Int. J. Math. Archive 9 (2018), 146-151.

  73. S. Suthar, O. Prakash, Covering of line graph of zero-divisor graph over ring ℤn, British J. Math. And Comp. Sci. 5 (2015), 728-734.

  74. S. Suthar, O. Prakash, Energy and Wiener index of Total Graph over Ring ℤn, Electron. Notes Discrete Math. 63 (2017), 485-495. MR3754839. Zbl 1382.05087.

  75. S. Suthar, O. Prakash, Adjacency Matrix and Energy of the Line Graph of Γ(ℤn), arXiv preprint arXiv:1806.08944 (2018).

  76. H. J. Wang, Zero-divisor graphs of genus one, J. Algebra 304 (2006), 666-678. MR2264274. Zbl 1106.13029.

  77. C. Wickham, Classification of rings with genus one zero-divisor graphs, Comm. Algebra 36 (2008), 325–345. MR2387525. Zbl 1137.13015.

  78. C. Wickham, Rings whose zero-divisor graphs have positive genus, J. Algerba 321 (2009), 377-383. MR2483271. Zbl 1163.13003.

  79. S. Wright, lengths of paths and cycles in zero-divisor graphs and digraphs of semigroups, Commun. Algebra 35 (2007), 1987-1991.. MR2324628. Zbl 1183.13009.

  80. T. Wu, On directed zero-divisor graphs of finite rings, Discrete Math. 296 (2005), 73-86. MR2148482. Zbl 1082.05091.

  81. T. Yanzhao, W. Qijiao, Remarks on the zero-divisor graph of a commutative ring, Advances Appl. Math. 00 (2015), 1–3.



Pradeep Singh and Vijay Kumar Bhat
School of mathematics,
Shri Mata Vaishno Devi University, Katra,
J&K, 182320, India.
e-mail: pradeep333singh@gmail.com


Vijay Kumar Bhat (corresponding author)
School of mathematics,
Shri Mata Vaishno Devi University, Katra,
J&K, 182320, India.
e-mail: vijaykumarbhat2000@yahoo.com

http://www.utgjiu.ro/math/sma