On the Depth of Independence Complexes

Year 2017, Volume: 3 Issue: 1, 42 - 44, 30.04.2017

Alper Ülker

Abstract

Let G be a graph and I G be its edge ideal so we call Stanley-Reisner ring of G . The depth of a ring is a well-studied and important algebraic invariant in commutative algebra. In this paper we give some results on the depth of Stanley- Reisner rings of graphs and simplicial complexes. By depth Lemma we reduce the computing depth of a codismantlable graph into its induced subgraphs.

Keywords

Complexes

References

• AUSLANDER M., BUCHSBAUM D. A., (1957). Homological dimension in local rings. Trans. Amer. Math Soc.; 85: no. 2, 390-405. BIYIKOĞLU T., CİVAN Y., (2014). Vertex decomposable graphs, codismantlability, Cohen- Macaulayness and Castelnuovo-Mumford regularity. Electronic J. Combin.; 16:2: 1-17. CHARTRAND G., ZHANG P., (2008). Chromatic graph theory. Chapman and Hall/CRC Press. DAO H., SCHWEIG J., (2013). Projective dimension, graph domination parameters, and independence complex homology, J. Combin. Theory. Ser. A; 120: 453-469. FRÖBERG R., (1990). On Stanley-Reisner rings, Topics in Algebra, Banach Center Publications, Polish Scientific Publishers; 26:2: 57-69. GITLER I., VALENCIA C.E., (2005). Bounds for invariants of edge-rings. Comm. Algebra; 33: 1603- 1616. KHOS-AHANG F., MORADI S., (2014). Rregularity and projective dimension of the edge ideal of 5 C -free vertex-decomposable graphs. Proc. AMS; 142:5: 1567- 1576. KUMMINI M., (2009). Regularity, depth and arithmetic rank of bipartite edge ideals. J Algebra Comb; 30: 4429-445. MOREY S., (2010). Depths of powers of the edge ideal of a tree, Comm. Algebra; 38: 4042-4055. REISNER G. A., (1976). Cohen-macaulay quotients of polynomial rings. Adv. in Maths.; 21: 30-49. VİLLARREAL R.H., (1990). Cohen Macaulay graphs. Manuscripta Maths.; 66: 3, 277-293. VİLLARREAL R.H., (2015). Monomial algebras, 2nd edition. Chapman and Hall/CRC Press.