Regularity and projective dimension of graphs

Stanley-Reisner rings provides a connection between commutative algebra and combinatorial mathematics so that many problems related to commutative algebra can be solved in terms of combinatorial structures. The central problem of combinatorial commutative algebra is to study the Cohen-Macaulayness, projective dimension and Castelnuovo-Mumford regularity of combinatorial structures such as graphs or simplicial complexes. The study of how local changes in combinatorial structures apply to the algebraic results of Stanley-Reisner rings is attractive. So, the Stanley-Reisner rings are related to the commutative algebra, algebraic geometry, homological algebra and algebraic topology. In this study, Stanley-Reisner rings will be introduced. The results in terms of projective dimension and Castelnuovo-Mumford reularity of a graphs will be given.