The Edge Ideals Of Graphs And Cohen-Macaulay Property

Edge ideals of graphs are bridges between commutative algebras,combina-tions and graph theory.The classification of Cohen-Macaulay graphs is one of the central problems in combinatorial commutative algebra.Only a few classes of Cohen-Macaulay graphs were completely characterized,for example,trees,bipar-tite graphs and chordal graphs.T.Hibi et al.[56]proved that a Cameron-Walker graph is Cohen-Macaulay if and only if it is unmixed.D.Cook and N.Nagel[19]give a construction of Cohen-Macualay graph.The complete classification of Cohen-Macaulay graphs is far from success,and it is a challenging problem.There-fore,the constructions of new class of Cohen-Macaulay graphs is of great theoretical significance.In this paper,we give two classes of Cohen-Macaulay graphs and,we give a new construction of Cohen-Macaulay graphs.In fact,according to our construction,we can get many vertex decomposable graphs.In the last chapter,we give some classes of graphs whose Castelnuovo-Mumford regularity is no more than 3The detailed contents are summarized as follows·In chapter 1,we introduce some concepts and notations.We also give a brief survey on related background·In chapter 2,we give the construction of vertex decomposable graphs.Starting from any simple graph,we construct a series of vertex decomposable graphs by adding vertices and edges in different ways,and many Cohen-Macaulay graphs can be constructed in this way·In chapter 3,we study Boolean graphs,which are constructed from zero-divisor graphs of Boolean rings.It is proved by mathematical induction that Boolean graphs are vertex decomposable.We also prove that they are unmixed graph-s,hence Cohen-Macaulay.Furthermore,the complement graphs of Boolean graphs are vertex decomposable graphs but not unmixed.·In chapter four,we study interlacing graphs.The construction of such graphs is derived from the triangulation of n-convex polygons.We prove that such graphs are unmixed and vertex decomposable,hence Cohen-Macaulay.·In chapter five,we study two subclasses of gap-free graphs.We prove that the regularity of the chair-free and gap-free graphs is no more than 3;we also prove that the regularity of the kite-free and gap-free graphs is no more than 3.