Betti numbers of edge ideals of some split graphs. (English) Zbl 1451.13047
Let \(G=(V,E)\) be a simple graph. The graph \(G\) is a split graph if \(V=C\cup S\) and
– for any two distinct elements \(x,x'\in C\), we have \(\{x,x'\}\in E\);
– for any two distinct elements \(y,y'\in S\), we have \(\{y,y'\}\notin E\).
The set \(C\) is called clique, and the set \(S\) is called stable set.
A split graph is said to be complete if the following extra condition holds
– for any \(x\in C\) and \(y\in S\) we have \(\{x,y\}\in E\).
In the first part of Section 3, the authors study some homological invariants of the edge ideal of complete split graphs. In particular they compute the graded Betti numbers in Theorem 3.2.
In the last part of Section 3 the focus is on nearly complete split graphs, that are constructed from complete split graphs by removing a certain number of edges. The graded Betti numbers of the edge ideals of nearly complete split graphs are computed in Theorem 3.7.
Both the results are achieved by using Hochster’s formula.
Section 4 is devoted to the study of the projective dimension of edge ideals of split graphs, an upper and a lower bound is provided in Theorem 4.4.
– for any two distinct elements \(x,x'\in C\), we have \(\{x,x'\}\in E\);
– for any two distinct elements \(y,y'\in S\), we have \(\{y,y'\}\notin E\).
The set \(C\) is called clique, and the set \(S\) is called stable set.
A split graph is said to be complete if the following extra condition holds
– for any \(x\in C\) and \(y\in S\) we have \(\{x,y\}\in E\).
In the first part of Section 3, the authors study some homological invariants of the edge ideal of complete split graphs. In particular they compute the graded Betti numbers in Theorem 3.2.
In the last part of Section 3 the focus is on nearly complete split graphs, that are constructed from complete split graphs by removing a certain number of edges. The graded Betti numbers of the edge ideals of nearly complete split graphs are computed in Theorem 3.7.
Both the results are achieved by using Hochster’s formula.
Section 4 is devoted to the study of the projective dimension of edge ideals of split graphs, an upper and a lower bound is provided in Theorem 4.4.
Reviewer: Giuseppe Favacchio (Catania)
MSC:
| 13D02 | Syzygies, resolutions, complexes and commutative rings |
| 13F55 | Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes |
| 05C69 | Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.) |
References:
| [1] | Barile, M., On ideals whose radical is a monomial ideal, Commun. Algebra, 33, 12, 4479-4490 (2005) · Zbl 1098.13023 · doi:10.1080/00927870500274812 |
| [2] | Dao, H.; Schweig, J., Projective dimension, graph domination parameters and independence complex homology, J. Combin. Theory. Ser. A, 120, 2, 453-469 (2013) · Zbl 1257.05114 · doi:10.1016/j.jcta.2012.09.005 |
| [3] | Decker, W.; Greuel, G.-M.; Pfister, G.; Sch Onemann, H. (2015) |
| [4] | Froberg, R., Topics in Algebra, Vol. 26, On Stanley Reisner rings, 57-70 (1990), Warsaw: PWN, Warsaw · Zbl 0741.13006 |
| [5] | Ghorbani, M.; Azimi, N., Characterization of split graphs with at most four distinct eigenvalues, Disc. Appl. Math, 184, 231-236 (2015) · Zbl 1311.05112 · doi:10.1016/j.dam.2014.10.039 |
| [6] | Hà, H. T.; Van Tuyl, A., Monomial ideals, edge ideals of hypergraphs, and their minimal graded Betti free resolutions, J. Algebr. Comb, 27, 2, 215-245 (2008) · Zbl 1147.05051 · doi:10.1007/s10801-007-0079-y |
| [7] | Hà, H. T.; Van Tuyl, A., Algebra, Geometry and Their Intersections, Contemporary Mathematics, 448, Resolutions of square-free monomial ideals via facet ideals: a survey, 91-117 (2007), Providence, RI: Amer. Math. Soc, Providence, RI · Zbl 1146.13007 |
| [8] | Hochster, M., Ring Theory II (Proc. Second Conf., Univ. Oklahoma, Norman, Okla, 26, Cohen-Macaulay rings, combinatorics, and simplicial complexes, 171-223 (1975), New York: Dekker, New York · Zbl 0351.13009 |
| [9] | Horwitz, N., Linear resolutions of quadratic monomial ideals, J. Algebra, 318, 2, 981-1001 (2007) · Zbl 1142.13011 · doi:10.1016/j.jalgebra.2007.06.006 |
| [10] | Jacques, S., Betti numbers of graph ideals (2005), University of Sheffield: University of Sheffield, Great Britain |
| [11] | Jacques, S., Katzman, M. (2005). The Betti numbers of forests. Preprint. math.AC/0401226. |
| [12] | Le, V. B.; Peng, S.-L., Characterizing and recognizing probe block graphs, Theor. Comput. Sci, 568, 97-102 (2015) · Zbl 1315.05088 · doi:10.1016/j.tcs.2014.12.014 |
| [13] | Morey, S.; Villarreal, R. H., Progress in Commutative Algebra 1, Edge ideals: algebraic and combinatorial properties, 85-126 (2012), Berlin: de Gruyter, Berlin · Zbl 1246.13001 |
| [14] | Rather, S. A.; Singh, P., On Betti numbers of edge ideals of crown graphs, Beitr. Algebra Geom, 60, 1, 123-136 (2019) · Zbl 1408.13038 · doi:10.1007/s13366-018-0397-3 |
| [15] | Rather, S. A.; Singh, P., Graded Betti numbers of crown edge ideals, Commun. Algebra, 47, 4, 1690-1698 (2019) · Zbl 1422.13016 · doi:10.1080/00927872.2018.1513018 |
| [16] | Villarreal, R. H., Rees algebras of edge ideals, Commun. Algebra, 23, 9, 3513-3524 (1995) · Zbl 0836.13014 · doi:10.1080/00927879508825412 |
| [17] | Zheng, X., Resolutions of facet ideals, Commun. Algebra, 32, 6, 2301-2324 (2004) · Zbl 1089.13014 · doi:10.1081/AGB-120037222 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
