Journal of Commutative Algebra

Finite commutative rings whose unitary Cayley graphs have positive genus

Huadong Su and Yiqiang Zhou

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Abstract

The unitary Cayley graph of a ring $R$ is the simple graph whose vertices are the elements of $R$, and where two distinct vertices $x$ and $y$ are linked by an edge if and only if $x-y$ is a unit in $R$. The genus of a simple graph $G$ is the smallest nonnegative integer $g$ such that $G$ can be embedded into an orientable surface $\mathbb {S}_{g}$. It is proven that, for a given positive integer $g$, there are at most finitely many finite commutative rings whose unitary Cayley graphs have genus $g$. We determine all finite commutative rings whose unitary Cayley graphs have genus 1, 2 and 3, respectively.

Article information

Source
J. Commut. Algebra, Volume 10, Number 2 (2018), 275-293.

Dates
First available in Project Euclid: 13 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.jca/1534125829

Digital Object Identifier
doi:10.1216/JCA-2018-10-2-275

Zentralblatt MATH identifier
06917497

Subjects
Primary: 05C25: Graphs and abstract algebra (groups, rings, fields, etc.) [See also 20F65] 13A99: None of the above, but in this section
Secondary: 05C75: Structural characterization of families of graphs 13M05: Structure

Keywords
Unitary Cayley graph complete graph complete bipartite graph genus finite commutative ring

Citation

Su, Huadong; Zhou, Yiqiang. Finite commutative rings whose unitary Cayley graphs have positive genus. J. Commut. Algebra 10 (2018), no. 2, 275--293. doi:10.1216/JCA-2018-10-2-275. https://projecteuclid.org/euclid.jca/1534125829


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References

  • S. Akbari, H.R. Maimani and S. Yassemi, When a zero-divisor graph is planar or a complete $r$-partite graph, J. Algebra 270 (2003), 169–180.
  • R. Akhtar, T. Jackson-Henderson, R. Karpman, M. Boggess,I. Jiménez, A. Kinzel and D. Pritikin, On the unitary Cayley graph of a finite ring, Electr. J. Combin. 16 (2009), 13 pages.
  • T. Asir and T.T. Chelvam, On the genus of generalized unit and unitary Cayley graphs of a commutative ring, Acta Math. Hungar. 142 (2014), 444–458.
  • R. Belshoff and J. Chapman, Planar zero-divisor graphs, J. Algebra 316 (2007), 471–480.
  • P. Berrizbeitia and R.E. Giudici, Counting pure $k$-cycles in sequences of Cayley graphs, Discr. Math. 149 (1996), 11–18.
  • P. Berrizbeitia and R.E. Giudici, On cycles in the sequence of unitary Cayley graphs, Discr. Math. 282 (2004), 239–243.
  • N. Bloomfield and C. Wickham, Local rings with genus two zero divisor graph, Comm. Algebra 38 (2010), 2965–2980.
  • B. Corbas and G.D. Williams, Rings of order $p^5$, Part I, Nonlocal rings, J. Algebra 231 (2000), 677–690.
  • A.K. Das, H.R. Maimani, M.R. Pournaki and S. Yassemi, Nonplanarity of unit graphs and classification of the toroidal ones, Pacific J. Math. 268 (2014), 371–387.
  • I.J. Dejter and R.E. Giudici, On unitary Cayley graphs, J. Combin. Math. Comp. 18 (1995), 121–124.
  • E.D. Fuchs, Longest induced cycles in circulant graphs, Electr. J. Combin. 12 (2005), 12 pages.
  • N. Ganesan, Properties of rings with a finite number of zero divisors, Math. Ann. 157 (1964), 215–218.
  • A. Ilić, The energy of unitary Cayley graphs, Linear Alg. Appl. 431 (2009), 1881–1889.
  • K. Khashyarmanesh and M.R. Khorsandi, A generalization of unit and unitary cayley graphs of a commutative ring, Acta Math. Hungar. 137 (2012), 242–253.
  • D. Kiani and M.M.H. Aghaei, On the unitary Cayley graph of a ring, Electr. J. Combin. 19 (2012), 10 pages.
  • D. Kiani, M.M.H. Aghaei, Y. Meemark and B. Suntornpoch, Energy of unitary Cayley graphs and gcd-graphs, Linear Alg. Appl. 435 (2011), 1336–1343.
  • W. Klotz and T. Sander, Some properties of unitary Cayley graphs, Electr. J. Combin. 14 (2007), 12 pages.
  • C. Lanski and A. Maróti, Ring elements as sums of units, Central Europ. J. Math. 7 (2009), 395–399.
  • X. Liu and S. Zhou, Spectral properties of unitary Cayley graphs of finite commutative rings, Electr. J. Combin. 19 (2012), 19 pages.
  • H.R. Maimani, C. Wickham and S. Yassemi, Rings whose total graphs have genus at most one, Rocky Mountain J. Math. 42 (2012), 1551–1560.
  • N.O. Smith, Planar zero-divisor graphs, Inter. J. Commutative Rings 2 (2003), 177–188.
  • T. Tamizh Chelvam and T. Asir, On the genus of the total graph of a commutative ring, Comm. Algebra 41 (2013), 142–153.
  • C. Thomassen, The graph genus problem is NP-complete, J. Algorithms 10 (1989), 568–576.
  • Hsin-Ju Wang, Zero-divisor graphs of genus one, J. Algebra 304 (2006), 666–678.
  • A.T. White, Graphs, groups and surfaces, North-Holland Math. Stud. 8, North-Holland Publishing Company, Amsterdam, 1984.
  • C. Wickham, Rings whose zero-divisor graphs have positive genus, J. Algebra 321 (2009), 377–383.
  • ––––, Classification of rings with genus one zero-divisor graphs, Comm. Algebra 36 (2008), 325–345.