Some algebraic properties of hypergraphs

Eric Emtander; Fatemeh Mohammadi; Somayeh Moradi

Czechoslovak Mathematical Journal (2011)

  • Volume: 61, Issue: 3, page 577-607
  • ISSN: 0011-4642

Abstract

top
We consider Stanley-Reisner rings k [ x 1 , ... , x n ] / I ( ℋ ) where I ( ℋ ) is the edge ideal associated to some particular classes of hypergraphs. For instance, we consider hypergraphs that are natural generalizations of graphs that are lines and cycles, and for these we compute the Betti numbers. We also generalize some known results about chordal graphs and study a weak form of shellability.

How to cite

top

Emtander, Eric, Mohammadi, Fatemeh, and Moradi, Somayeh. "Some algebraic properties of hypergraphs." Czechoslovak Mathematical Journal 61.3 (2011): 577-607. <http://eudml.org/doc/196326>.

@article{Emtander2011,
abstract = {We consider Stanley-Reisner rings $k[x_1,\ldots ,x_n]/I(\mathcal \{H\})$ where $I(\mathcal \{H\})$ is the edge ideal associated to some particular classes of hypergraphs. For instance, we consider hypergraphs that are natural generalizations of graphs that are lines and cycles, and for these we compute the Betti numbers. We also generalize some known results about chordal graphs and study a weak form of shellability.},
author = {Emtander, Eric, Mohammadi, Fatemeh, Moradi, Somayeh},
journal = {Czechoslovak Mathematical Journal},
keywords = {Betti numbers; chordal hypergraphs; connectivity; homologically connected hypergraphs; hypercycles; line hypergraphs; shellability; Betti numbers; chordal hypergraph; connectivity; hypercycle; line hypergraph; shellability},
language = {eng},
number = {3},
pages = {577-607},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Some algebraic properties of hypergraphs},
url = {http://eudml.org/doc/196326},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Emtander, Eric
AU - Mohammadi, Fatemeh
AU - Moradi, Somayeh
TI - Some algebraic properties of hypergraphs
JO - Czechoslovak Mathematical Journal
PY - 2011
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 61
IS - 3
SP - 577
EP - 607
AB - We consider Stanley-Reisner rings $k[x_1,\ldots ,x_n]/I(\mathcal {H})$ where $I(\mathcal {H})$ is the edge ideal associated to some particular classes of hypergraphs. For instance, we consider hypergraphs that are natural generalizations of graphs that are lines and cycles, and for these we compute the Betti numbers. We also generalize some known results about chordal graphs and study a weak form of shellability.
LA - eng
KW - Betti numbers; chordal hypergraphs; connectivity; homologically connected hypergraphs; hypercycles; line hypergraphs; shellability; Betti numbers; chordal hypergraph; connectivity; hypercycle; line hypergraph; shellability
UR - http://eudml.org/doc/196326
ER -

References

top
  1. Berge, C., Hypergraphs: Combinatorics of finite sets, {North-Holland} (1989). (1989) Zbl0674.05001MR1013569
  2. Bruns, W., Herzog, J., Cohen-Macaulay rings, revised ed, Cambridge University Press (1998). (1998) MR1251956
  3. Dochstermann, A., Engström, A., Algebraic properties of edge ideals via combinatorial topology, arXiv:0810.4120 (2008). (2008) MR2992613
  4. Eagon, J. A., Reiner, V., 10.1016/S0022-4049(97)00097-2, J. Pure Appl. Algebra 130 (1998), 265-275. (1998) Zbl0941.13016MR1633767DOI10.1016/S0022-4049(97)00097-2
  5. Eisenbud, D., Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics 150, Springer (1995). (1995) Zbl0819.13001MR1322960
  6. Eliahou, S., Kervaire, M., 10.1016/0021-8693(90)90237-I, J. Algebra 129 (1990), 1-25. (1990) Zbl0701.13006MR1037391DOI10.1016/0021-8693(90)90237-I
  7. Emtander, E., 10.1080/00927870802098158, Communications in algebra 37 (2009), 1545-1571. (2009) Zbl1191.13015MR2526320DOI10.1080/00927870802098158
  8. Emtander, E., A class of hypergraphs that generalizes chordal graphs, arXiv:0803.2150 (2008). (2008) MR2603461
  9. Faridi, S., The facet ideal of a simplicial complex, Manuscripta Math. 242 (2002), 92-108. (2002) Zbl1005.13006MR1935027
  10. Faridi, S., 10.1016/j.jcta.2004.09.005, J. Combin. Theory Ser. A 109 (2005), 299-329. (2005) Zbl1101.13015MR2121028DOI10.1016/j.jcta.2004.09.005
  11. Francisco, C. A., Tuyl, A. Van, Sequentially cohen-macaulay edge ideals, Proc. Amer. Math. Soc. (2007), 2327-2337. (2007) MR2302553
  12. Francisco, Christopher A., Hà, H. T., Tuyl, A. Van, Splittings of monomial ideals, arXiv:0807.2185 (2008). (2008) MR2515396
  13. Fröberg, R., 10.1016/0022-4049(85)90011-8, J. Pure Appl. Algebra 38 (1985), 235-241. (1985) MR0814179DOI10.1016/0022-4049(85)90011-8
  14. Fröberg, R., On Stanley-Reisner rings, Topics in Algebra 26 (1990). (1990) MR1171260
  15. Hà, H. T., Tuyl, A. Van, Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers, arXiv:math/0606539 (2006). (2006) MR2375493
  16. Hà, H. T., Tuyl, A. Van, 10.1016/j.jalgebra.2006.08.022, J. Algebra 309 (2007), 405-425. (2007) MR2301246DOI10.1016/j.jalgebra.2006.08.022
  17. Herzog, J., Hibi, T., Zheng, X., Cohen-Macaulay chordal graphs, arXiv:math/0407375v1 (2004). (2004) MR2231097
  18. Herzog, J., Hibi, T., Zheng, X., 10.1016/j.ejc.2003.12.008, European Journal of Combinatorics 25 (2004). (2004) Zbl1062.05075MR2083448DOI10.1016/j.ejc.2003.12.008
  19. Jacques, S., Betti Numbers of Graph Ideals, Ph.D. thesis, University of Sheffield, 2004, arXiv:math/0410107. 
  20. Miller, E., Sturmfels, B., Combinatorial Commutative Algebra, Springer (2005). (2005) Zbl1090.13001MR2110098
  21. Morey, S., Reyes, E., Villarreal, R. H., Cohen-Macaulay, Shellable and unmixed clutters with a perfect matching of König type, arXiv:0708.3111v3 (2007). (2007) MR2400742
  22. Villarreal, R. H., 10.1007/BF02568497, Manuscripta Math. 66 (1990), 277-293. (1990) Zbl0737.13003MR1031197DOI10.1007/BF02568497
  23. Zheng, X., 10.1081/AGB-120037222, Comm. Algebra 32 (2004), 2301-2324. (2004) Zbl1089.13014MR2100472DOI10.1081/AGB-120037222

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.