Commutative algebraic methods on graphs

The aim of this thesis is to introduce the connections between combinatorial objects and commutative algebra. Stanley-Reisner rings play important role for this connection. Using combinatorics and combinatorial topology as a tool we give some results about commutative algebra. This thesis consists of seven chapters. After the introductiory chapter in which the general information concerning to the thesis is given, in the Preliminaries, fundamental and basic notions are presented. In chapter 3, Cohen-Macaulay Complexes are introduced and basic notions about Cohen- Macaulay Complexes are given. Stanley-Reisner rings are introduced and some homology groups are computed. Also most important familiy of complexes for this thesis that is called independence complexes are introduced. In chapter 4, the notion of vertex-decomposability is introduced and some results about vertex decomposability of independence complexes of weakly chordal graphs are given. In chapter 5, the notion of shellability is given and some results about shellability are shown. In chapter 6, an algebraic invariant depth is defined for complexes. For weakly chordal and cycle graphs depth is computed. In chapter 7, we final the work with the summary of the thesis.