AIMS Mathematics, 2019, 4(3): 686-698. doi: 10.3934/math.2019.3.686.

Research article

Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

Depth and Stanley depth of edge ideals associated to some line graphs

School of Natural Sciences, National University of Sciences and Technology Islamabad, Sector H-12, Islamabad, Pakistan

In this paper, we compute some upper and lower bounds for depth and Stanley depth of edge ideals associated to line graphs of the ladder and circular ladder graphs. Furthermore, we determine some bounds for the dimension of the quotient rings of the edge ideals associated to these graphs.
  Figure/Table
  Supplementary
  Article Metrics

Keywords depth; Stanley depth; dimension; monomial ideal; line graph

Citation: Zahid Iqbal, Muhammad Ishaq. Depth and Stanley depth of edge ideals associated to some line graphs. AIMS Mathematics, 2019, 4(3): 686-698. doi: 10.3934/math.2019.3.686

References

  • 1.R. P. Stanley, Linear Diophantine equations and local cohomology, Invent. Math., 68 (1982), 175-193.    
  • 2.B. Ichim, L. Katthän, J. J. Moyano-Fernández, The behavior of Stanley depth under polarization, J. Combin. Theory, Ser. A, 135 (2015), 332-347.    
  • 3.A. M. Duval, B. Goeckneker, C. J. Klivans, et al. A non-partitionable Cohen-Macaulay simplicial complex, Adv. Math., 299 (2016), 381-395.    
  • 4.J. Herzog, T. Hibi, The depth of powers of an ideal, J. Algebra, 291 (2005), 534-550.    
  • 5.J. Herzog, M. Vladoiu, X. Zheng, How to compute the Stanley depth of a monomial ideal, J. Algebra, 322 (2009), 3151-3169.    
  • 6.C. Biro, D. M. Howard, M. T. Keller, et al. Interval partitions and Stanley depth, J. Combin. Theory, Ser. A, 117 (2010), 475-482.    
  • 7.M. Cimpoeas, Stanley depth of squarefree Veronese ideals, An. St. Univ. Ovidius Constanta, 21 (2013), 67-71.
  • 8.M. Ishaq, Upper bounds for the Stanley depth, Comm. Algebra, 40 (2012), 87-97.    
  • 9.M. Ishaq, M. I. Qureshi, Upper and lower bounds for the Stanley depth of certain classes of monomial ideals and their residue class rings, Comm. Algebra, 41 (2013), 1107-1116.    
  • 10.M. T. Keller, Y. Shen, N. Streib, et al. On the Stanley Depth of Squarefree Veronese Ideals, J. Algebr. Comb., 33 (2011), 313-324.    
  • 11.B. Ichim, L. Katthän, J. J. Moyano-Fernández, How to compute the Stanley depth of a module, Math. Comput., 86 (2016), 455-472.    
  • 12.B. Ichim, L. Katthän, J. J. Moyano-Fernández, LCM lattices and Stanley depth: a first computational approach, Exp. Math., 25 (2016), 46-53.    
  • 13.B. Ichim, L. Katthän, J. J. Moyano-Fernández, Stanley depth and the lcm-lattice, J. Combin. Theory, Ser. A, 150 (2017), 295-322.    
  • 14.Z. Iqbal, M. Ishaq, M. Aamir, Depth and Stanley depth of the edge ideals of square paths and square cycles, Comm. Algebra, 46 (2018), 1188-1198.    
  • 15.Z. Iqbal, M. Ishaq, Depth and Stanley depth of the edge ideals of the powers of paths and cycles, An. St. Univ. Ovidius Constanta, In press. Available from: http://arxiv.org/pdf/1710.05996.pdf.
  • 16.M. R. Pournaki, S. A. Seyed Fakhari, S. Yassemi, Stanley depth of powers of the edge ideals of a forest, P. Am. Math. Soc., 141 (2013), 3327-3336.    
  • 17.J. Herzog, T. Hibi, Monomial Ideals, Springer-Verlag London Limited, 2011.
  • 18.R. H. Villarreal, Monomial algebras, Monographs and Textbooks in Pure and Applied Mathematics 238, Marcel Dekker, New York, 2001.
  • 19.J. A. Bondy, U. S. R. Murty, Graph Theory with applications, Springer, 2008.
  • 20.F. Harary, Graph theory, Addison-Wesley, Reading, MA, 1969.
  • 21.A. Rauf, Depth and Stanley depth of multigraded modules, Comm. Algebra, 38 (2010), 773-784.    
  • 22.M. Cimpoeas, Several inequalities regarding Stanley depth, Romanian Journal of Mathematics and Computer Science, 2 (2012), 28-40.
  • 23.S. Morey, Depths of powers of the edge ideal of a tree, Comm. Algebra, 38 (2010), 4042-4055.    
  • 24.A. Stefan, Stanley depth of powers of path ideal}. Available from: http://arxiv.org/pdf/1409.6072.pdf.
  • 25.M. Cimpoeas, On the Stanley depth of edge ideals of line and cyclic graphs, Romanian Journal of Mathematics and Computer Science, 5 (2015), 70-75.
  • 26.L. Fouli, S. Morey, A lower bound for depths of powers of edge ideals, J. Algebr. Comb., 42 (2015), 829-848.    
  • 27.G. Rinaldo, An algorithm to compute the Stanley depth of monomial ideals, Le Matematiche, 63 (2008), 243-256.

 

Reader Comments

your name: *   your email: *  

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Export Citation

Copyright © AIMS Press All Rights Reserved