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Zero-divisor graphs of Catalan monoid

Yıl 2021, Cilt: 50 Sayı: 2, 387 - 396, 11.04.2021
https://doi.org/10.15672/hujms.702478

Öz

Let $\mathcal C_{n}$ be the Catalan monoid on $X_{n}=\{1,\ldots ,n\}$ under its natural order. In this paper, we describe the sets of left zero-divisors, right zero-divisors and two sided zero-divisors of $\mathcal C_{n}$; and their numbers. For $n \geq 4$, we define an undirected graph $\Gamma(\mathcal C_{n})$ associated with $\mathcal C_{n}$ whose vertices are the two sided zero-divisors of $\mathcal C_{n}$ excluding the zero element $\theta$ of $\mathcal C_{n}$ with distinct two vertices $\alpha$ and $\beta$ joined by an edge in case $\alpha\beta=\theta=\beta\alpha$. Then we first prove that $\Gamma(\mathcal C_{n})$ is a connected graph, and then we find the diameter, radius, girth, domination number, clique number and chromatic numbers and the degrees of all vertices of $\Gamma(\mathcal C_{n})$. Moreover, we prove that $\Gamma(\mathcal C_{n})$ is a chordal graph, and so a perfect graph.

Teşekkür

My sincere thanks are due to Prof. Dr. Hayrullah Ayık for his helpful suggestions and encouragement.

Kaynakça

  • [1] D.F. Anderson and P.S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217 (2), 434-447, 1999.
  • [2] G. Ayık, H. Ayık and M. Koç, Combinatorial results for order-preserving and order-decreasing transformations, Turkish J. Math. 35 (4), 617-625, 2011.
  • [3] I. Beck, Coloring of commutative rings, J. Algebra 116 (1), 208-226, 1988.
  • [4] K.C. Das, N. Akgüneş and A.S. Çevik, On a graph of monogenic semigroup, J. Inequal. Appl. 44, 2013.
  • [5] F. DeMeyer and L. DeMeyer, Zero divisor graphs of semigroups, J. Algebra 283 (1), 190-198, 2005.
  • [6] F. DeMeyer, T. McKenzie and K. Schneider, The zero-divisor graph of a commutative semigroup, Semigroup Forum 65 (2), 206-214, 2002.
  • [7] G.A. Dirac, On rigid circuit graphs, Abh. Math. Sem. Univ. Hamburg 25 (1-2), 71-76, 1961.
  • [8] W. Feller, An Introduction to Probability Theory and Its Applications Volume I, New York, John Wiley & Sons Inc, 1957.
  • [9] O. Ganyushkin and V. Mazorchuk, Classical Finite Transformation Semigroups: An Introduction, Springer, 2009.
  • [10] P.M. Higgins, Combinatorial results for semigroups of order-preserving mappings, Math. Proc. Camb. Phil. Soc. 113 (2), 281-296, 1993.
  • [11] J.M. Howie, Fundamentals of Semigroup Theory, New York, NY, USA: Oxford University Press, 1995.
  • [12] A. Laradji and A. Umar, On certain finite semigroups of order-decreasing transformations I, Semigroup Forum, 69 (2), 184-200, 2004.
  • [13] S.G. Mohanty, Lattice Path Counting and Applications, Academic Press, New York, 1979.
  • [14] S.P. Redmond, The zero-divisor graph of a non-commutative ring, Int. J. Commut. Rings 1 (4), 203-211, 2002.
  • [15] R. Stanley, Enumerative Combinatorics, Volume 2, Cambridge University Press, 2001.
  • [16] K. Thulasiraman, S. Arumugam, A. Brandstädt, A. and T. Nishizeki, Handbook of Graph Theory, Combinatorial Optimization, and Algorithms, Chapman & Hall/CRC Computer and Information Science Series. CRC Press, Boca Raton, 2015.
  • [17] K. Toker, On the zero-divisor graphs of finite free semilattices, Turkish J. Math 40 (4), 824-831, 2016.
Yıl 2021, Cilt: 50 Sayı: 2, 387 - 396, 11.04.2021
https://doi.org/10.15672/hujms.702478

Öz

Kaynakça

  • [1] D.F. Anderson and P.S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217 (2), 434-447, 1999.
  • [2] G. Ayık, H. Ayık and M. Koç, Combinatorial results for order-preserving and order-decreasing transformations, Turkish J. Math. 35 (4), 617-625, 2011.
  • [3] I. Beck, Coloring of commutative rings, J. Algebra 116 (1), 208-226, 1988.
  • [4] K.C. Das, N. Akgüneş and A.S. Çevik, On a graph of monogenic semigroup, J. Inequal. Appl. 44, 2013.
  • [5] F. DeMeyer and L. DeMeyer, Zero divisor graphs of semigroups, J. Algebra 283 (1), 190-198, 2005.
  • [6] F. DeMeyer, T. McKenzie and K. Schneider, The zero-divisor graph of a commutative semigroup, Semigroup Forum 65 (2), 206-214, 2002.
  • [7] G.A. Dirac, On rigid circuit graphs, Abh. Math. Sem. Univ. Hamburg 25 (1-2), 71-76, 1961.
  • [8] W. Feller, An Introduction to Probability Theory and Its Applications Volume I, New York, John Wiley & Sons Inc, 1957.
  • [9] O. Ganyushkin and V. Mazorchuk, Classical Finite Transformation Semigroups: An Introduction, Springer, 2009.
  • [10] P.M. Higgins, Combinatorial results for semigroups of order-preserving mappings, Math. Proc. Camb. Phil. Soc. 113 (2), 281-296, 1993.
  • [11] J.M. Howie, Fundamentals of Semigroup Theory, New York, NY, USA: Oxford University Press, 1995.
  • [12] A. Laradji and A. Umar, On certain finite semigroups of order-decreasing transformations I, Semigroup Forum, 69 (2), 184-200, 2004.
  • [13] S.G. Mohanty, Lattice Path Counting and Applications, Academic Press, New York, 1979.
  • [14] S.P. Redmond, The zero-divisor graph of a non-commutative ring, Int. J. Commut. Rings 1 (4), 203-211, 2002.
  • [15] R. Stanley, Enumerative Combinatorics, Volume 2, Cambridge University Press, 2001.
  • [16] K. Thulasiraman, S. Arumugam, A. Brandstädt, A. and T. Nishizeki, Handbook of Graph Theory, Combinatorial Optimization, and Algorithms, Chapman & Hall/CRC Computer and Information Science Series. CRC Press, Boca Raton, 2015.
  • [17] K. Toker, On the zero-divisor graphs of finite free semilattices, Turkish J. Math 40 (4), 824-831, 2016.
Toplam 17 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Matematik
Yazarlar

Kemal Toker 0000-0003-3696-1324

Yayımlanma Tarihi 11 Nisan 2021
Yayımlandığı Sayı Yıl 2021 Cilt: 50 Sayı: 2

Kaynak Göster

APA Toker, K. (2021). Zero-divisor graphs of Catalan monoid. Hacettepe Journal of Mathematics and Statistics, 50(2), 387-396. https://doi.org/10.15672/hujms.702478
AMA Toker K. Zero-divisor graphs of Catalan monoid. Hacettepe Journal of Mathematics and Statistics. Nisan 2021;50(2):387-396. doi:10.15672/hujms.702478
Chicago Toker, Kemal. “Zero-Divisor Graphs of Catalan Monoid”. Hacettepe Journal of Mathematics and Statistics 50, sy. 2 (Nisan 2021): 387-96. https://doi.org/10.15672/hujms.702478.
EndNote Toker K (01 Nisan 2021) Zero-divisor graphs of Catalan monoid. Hacettepe Journal of Mathematics and Statistics 50 2 387–396.
IEEE K. Toker, “Zero-divisor graphs of Catalan monoid”, Hacettepe Journal of Mathematics and Statistics, c. 50, sy. 2, ss. 387–396, 2021, doi: 10.15672/hujms.702478.
ISNAD Toker, Kemal. “Zero-Divisor Graphs of Catalan Monoid”. Hacettepe Journal of Mathematics and Statistics 50/2 (Nisan 2021), 387-396. https://doi.org/10.15672/hujms.702478.
JAMA Toker K. Zero-divisor graphs of Catalan monoid. Hacettepe Journal of Mathematics and Statistics. 2021;50:387–396.
MLA Toker, Kemal. “Zero-Divisor Graphs of Catalan Monoid”. Hacettepe Journal of Mathematics and Statistics, c. 50, sy. 2, 2021, ss. 387-96, doi:10.15672/hujms.702478.
Vancouver Toker K. Zero-divisor graphs of Catalan monoid. Hacettepe Journal of Mathematics and Statistics. 2021;50(2):387-96.