SOME MONOMIAL SEQUENCES ARISING FROM GRAPHS

s-sequences and d-sequences are fundamental sequences in- tensively studied in many fields of algebra. In this paper we are interested in dealing with monomial sequences associated to graphs in order to es- tablish conditions for which they are s-sequences and/or d-sequences. In this work we consider monomial sequences establishing conditions for which they are s-sequences and d-sequences in order to deduce some properties of the symmetric algebra of monomial ideals, in particular of some monomial ideals arising from graphs. In (2) the notion of d-sequence was firstly given by Huneke for the study of Rees rings. In (1) the notion of s-sequence is employed to compute the invariants of the symmetric algebra of finitely generated modules and it is proved that any d-sequence is a strong s-sequence. Some properties about monomial s-sequences are studied. In (1) it is given a necessary and sufficient condition for monomial sequences of length three to be s-sequences. Afterwards, in (6) it is shown a necessary and sufficient condition for monomial squarefree sequences of length four, and in (5) a more general statement for monomial sequences of any length related to forests. Our aim is to find necessary and sufficient conditions for monomial sequences of length greater than four to be s-sequences. Some results are obtained for edge sequences associated to very important classes of graphs and components of them. The d-sequences have been intensively studied by many algebraists, the monomial d-sequences are characterized in (6). Our aim is to investigate the notion of d-sequence for the monomial ideals arising from simple graphs in order to compute standard algebraic invariants of their symmetric algebra in terms of the corresponding invariants of special quotients of the polynomial ring related to the graphs.