Algebraic properties of edge ideals via combinatorial topology
2009 (English)In: The Electronic Journal of Combinatorics, ISSN 1077-8926, Vol. 16, no 2Article in journal (Refereed) Published
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
COHEN-MACAULAY GRAPHS; CLAW-FREE GRAPHS; MONOMIAL IDEALS; COMPLEXES; HYPERGRAPHS; RESOLUTIONS; THEOREM; RINGS
IdentifiersURN: urn:nbn:se:kth:diva-10375ISI: 000263259900002OAI: oai:DiVA.org:kth-10375DiVA: diva2:216360
QC 201007122009-05-082009-05-082010-12-03Bibliographically approved