  Independent sets play a key role into the study of graphs and important
problems arising in graph theory reduce to them. We define the monomial ideal
of independent sets associated to a finite simple graph and describe its
homological and algebraic invariants in terms of the combinatorics of the
graph. We compute the minimal primary decomposition and characterize the
Cohen--Macaulay ideals. Moreover, we provide a formula for computing the Betti
numbers, which depends only on the coefficients of the independence polynomial
of the graph.
