On the Genus of the Total Graph of a Commutative Ring

REFERENCES

• Akbari , S. , Kiani , D. , Mohammadi , F. , Moradi , S. ( 2009 ). The total graph and regular graph of a commutative ring . J. Pure Appl. Algebra 213 : 2224 – 2228 .
   Web of Science ®Google Scholar
• Akbari , S. , Maimani , H. R. , Yassemi , S. ( 2003 ). When a zero-divisor graph is planar or a complete r-partite graph . J. Algebra 270 : 169 – 180 .
   Web of Science ®Google Scholar
• Anderson , D. F. , Frazier , A. , Lauve , A. , Livingston , P. S. ( 2001 ). The Zero-divisor Graph of a Commutative Ring, II. Lecture Notes in Pure and Appl. Math. 202 . New York : Marcel Dekker , pp. 61 – 72 .
   Google Scholar
• Anderson , D. F. , Badawi , A. ( 2008 ). The total graph of a commutative ring . J. Algebra. 320 : 2706 – 2719 .
   Web of Science ®Google Scholar
• Anderson , D. F. , Livingston , P. S. ( 1999 ). The zero-divisor graph of a commutative ring . J. Algebra. 217 : 434 – 447 .
   Web of Science ®Google Scholar
• Archdeacon , D. ( 1996 ). Topological graph theory: A survey . Congr. Numer. 115 : 5 – 54 .
   Google Scholar
• Atiyah , M. F. , Macdonald , I. G. ( 1969 ). Introduction to Commutative Algebra . London : Addison-Wesley Publishing Company .
   Google Scholar
• Belshoff , R. , Chapman , J. ( 2007 ). Planar zero-divisor graphs . J. Algebra. 316 ( 1 ): 471 – 480 .
   Web of Science ®Google Scholar
• Bloomfield , N. , Wickham , C. ( 2010 ). Local rings with genus two zero divisor graph . Comm. Algebra. 38 : 2965 – 2980 .
   Web of Science ®Google Scholar
• Chartrand , G. , Zhang , P. ( 2006 ). Introduction to Graph Theory . New Delhi : Tata McGraw-Hill .
   Google Scholar
• Ganesan , N. ( 1964 ). Properties of rings with a finite number of zero-divisors . Math. Ann. 157 : 215 – 218 .
   Web of Science ®Google Scholar
• Kaplansky , I. (1974). Commutative Rings, Rev. ed . Chicago : University of Chicago Press.
   Google Scholar
• Maimani , H. R. , Wickham , C. , Yassemi , S. ( 2011 ). Rings whose total graphs have genus at most one. Rocky Mountain J. Math. To appear. arXiv: 1001.5338v1[math.AC] .
   Google Scholar
• Mohar , B. , Thomassen , C. ( 2001 ). Graphs on Surfaces . Baltimore , MD : Johns Hopkins .
   Google Scholar
• Ringel , G. ( 1977 ). On the genus of the graph K n  × K 2 or the n-prism . Discrete Math. 20 : 287 – 294 .
   Web of Science ®Google Scholar
• Smith , N. O. ( 2003 ). Planar zero-divisor graphs . Internat. J. Commutative Rings 2 : 177 – 188 .
   Google Scholar
• Tamizh Chelvam , T. , Asir , T. ( 2011 ). A note on total graph of ℤ n . J. Discrete Math. Sci. Cryptography. 14 ( 1 ): 1 – 7 .
   Google Scholar
• Tamizh Chelvam , T. , Asir , T. ( 2012 ). Intersection graph of gamma sets in the total graph . Discuss. Math. Graph Theory 32 : 339 – 354 .
   Web of Science ®Google Scholar
• Wang , H.-J. ( 2006 ). Zero-divisor graphs of genus one . J. Algebra. 304 ( 2 ): 666 – 678 .
   Web of Science ®Google Scholar
• White , A. T. ( 1973 ). Graphs, Groups and Surfaces . Amsterdam : North-Holland .
   Google Scholar
• White , A. T. ( 1971 ). The genus of the Cartesian product of two graphs . J. Combin. Theory Ser. B. 11 : 89 – 94 .
   Google Scholar
• Wickham , C. ( 2008 ). Classification of rings with genus one zero-divisor graphs . Comm. Algebra. 36 ( 2 ): 325 – 345 .
   Web of Science ®Google Scholar
• Wickham , C. ( 2009 ). Rings whose zero-divisor graphs have positive genus . J. Algebra. 321 ( 2 ): 377 – 383 .
   Web of Science ®Google Scholar
• Communicated by I. Swanson.
   Google Scholar