Groebner bases associated to simple graphs

Classes of graphs are studied using computational and algebraic methods in order to give some models in real connection problems. Let G be a graph. An algebraic object attached to G is the edge ideal I(G), a monomial ideal of the polynomial ring in nvariables R = K[X_1,...,X_n], K a field. When G is a simple (or loopless) graph, I(G) is generated by squarefree monomials of degree 2 in R, but when G is a graph having loops, among the generators of I(G) there are also non-squarefree monomials. We investigate algebraic properties of edge ideals via Groebner bases. More precisely, we use the theory of Groebner bases to characterize monomial s-sequences that arise from G. We introduce some classes of graphs for which, using the Gröbner bases theory, we give necessary and sufficient conditions in order that their edge ideals are generated by s-sequences. The notion of s-sequence is employed to compute algebraic invariants of the symmetric algebra associated to I(G). Our proposal is to compute standard invariants of the symmetric algebra of monomial ideals of graphs in terms of the corresponding invariants of special quotients of the polynomial ring related to such graphs. This computation can be obtained for graph ideals generated by an s-sequence.