Edge ideals: algebraic and combinatorial properties

Let C be a clutter and let I© be its edge ideal. This is a survey paper on the algebraic and combinatorial properties of R/I© and C, respectively. We give a criterion to estimate the regularity of R/I© and apply this criterion to give new proofs of some formulas for the regularity. If C is a clutter and R/I© is sequentially Cohen-Macaulay, we present a formula for the regularity of the ideal of vertex covers of C and give a formula for the projective dimension of R/I©. We also examine the associated primes of powers of edge ideals, and show that for a graph with a leaf, these sets form an ascending chain.