  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 edges 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.
