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{\bf Anton Dochtermann and Alexander Engstr\"{o}m}
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{\bf Algebraic Properties of Edge Ideals via Combinatorial \phantom{XX} Topology}
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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.

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