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Algebraic properties of edge ideals via combinatorial topology
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
Techn. Universität , Berlin.
2009 (English)In: The Electronic Journal of Combinatorics, ISSN 1077-8926, Vol. 16, no 2Article in journal (Refereed) Published
Abstract [en]

We apply some basic notions from combinatorial topology to establish various algebraic properties of edge ideals of graphs and more general Stanley-Reisner rings. In this way we provide new short proofs of some theorems from the literature regarding linearity, Betti numbers, and (sequentially) Cohen-Macaulay properties of edge ideals associated to chordal, complements of chordal, and Ferrers graphs, as well as trees and forests. Our approach unifies (and in many cases strengthens) these results and also provides combinatorial/enumerative interpretations of certain algebraic properties. We apply our setup to obtain new results regarding algebraic properties of edge ideals in the context of local changes to a graph (adding whiskers and ears) as well as bounded vertex degree. These methods also lead to recursive relations among certain generating functions of Betti numbers which we use to establish new formulas for the projective dimension of edge ideals. We use only well-known tools from combinatorial topology along the lines of independence complexes of graphs, (not necessarily pure) vertex decomposability, shellability, etc.

Place, publisher, year, edition, pages
2009. Vol. 16, no 2
Keyword [en]
COHEN-MACAULAY GRAPHS; CLAW-FREE GRAPHS; MONOMIAL IDEALS; COMPLEXES; HYPERGRAPHS; RESOLUTIONS; THEOREM; RINGS
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-10375ISI: 000263259900002OAI: oai:DiVA.org:kth-10375DiVA: diva2:216360
Note
QC 20100712Available from: 2009-05-08 Created: 2009-05-08 Last updated: 2010-12-03Bibliographically approved
In thesis
1. Topological Combinatorics
Open this publication in new window or tab >>Topological Combinatorics
2009 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis on Topological Combinatorics contains 7 papers. All of them but paper Bare published before.In paper A we prove that!i dim ˜Hi(Ind(G);Q) ! |Ind(G[D])| for any graph G andits independence complex Ind(G), under the condition that G\D is a forest. We then use acorrespondence between the ground states with i+1 fermions of a supersymmetric latticemodel on G and ˜Hi(Ind(G);Q) to deal with some questions from theoretical physics.In paper B we generalize the topological Tverberg theorem. Call a graph on the samevertex set as a (d + 1)(q − 1)-simplex a (d, q)-Tverberg graph if for any map from thesimplex to Rd there are disjoint faces F1, F2, . . . , Fq whose images intersect and no twoadjacent vertices of the graph are in the same face. We prove that if d # 1, q # 2 is aprime power, and G is a graph on (d+1)(q −1)+1 vertices such that its maximal degreeD satisfy D(D + 1) < q, then G is a (d, q)–Tverberg graph. It was earlier known that thedisjoint unions of small complete graphs, paths, and cycles are Tverberg graphs.In paper C we study the connectivity of independence complexes. If G is a graphon n vertices with maximal degree d, then it is known that its independence complex is(cn/d + !)–connected with c = 1/2. We prove that if G is claw-free then c # 2/3.In paper D we study when complexes of directed trees are shellable and how one canglue together independence complexes for finding their homotopy type.In paper E we prove a conjecture by Björner arising in the study of simplicial polytopes.The face vector and the g–vector are related by a linear transformation. We prove thatthis matrix is totaly nonnegative. This is joint work with Michael Björklund.In paper F we introduce a generalization of Hom–complexes, called set partition complexes,and prove a connectivity theorem for them. This generalizes previous results ofBabson, Cukic, and Kozlov, and questions from Ramsey theory can be described with it.In paper G we use combinatorial topology to prove algebraic properties of edge ideals.The edge ideal of G is the Stanley-Reisner ideal of the independence complex of G. Thisis joint work with Anton Dochtermann.

Place, publisher, year, edition, pages
Stockholm: KTH, 2009. vii, 16 p.
Series
Trita-MAT. MA, ISSN 1401-2278
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-10383 (URN)978-91-7415-256-2 (ISBN)
Public defence
2009-05-08, Sal E2, KTH, Lindstedtsvägen 5, Stockholm, 13:00 (English)
Opponent
Supervisors
Note
QC 20100712Available from: 2009-05-08 Created: 2009-05-08 Last updated: 2010-07-12Bibliographically approved

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