Invariants And Properties Of Simplicial Complexes

This dissertation explores various invariants and properties of simplicial complexes.Simplicial complexes ? are classical objects in combinatorial commutative algebra.This area of research captivated many renowned mathematician’s minds and a lot of great work has been done on it.Still there are plenty research problems which are unsolved yet.So I decided to invest my efforts into this topic after realizing the greatness and beauty of the intense interplay between the combinatorial objects and monomial ideals.Starting with the study of monomial ideals and Gr¨obner bases,there is a fascinating discussion topic of“binomial ideals",which satisfies a lot of algebraic properties.We generalized the idea of binomials into polynomials via using combinatorial object simplicial complexes and found the reduced Gr¨obner basis of a homogeneous ideal from ?.Simplicial complexes of dimension “1” are called graphs.In the next section,Betti numbers are evaluated for several classes of graphs whose complements are bipartite graphs;relations are established for the general case,and counting formulae are given in several particular cases,including the union of several mutually disjoint complete graphs.Last but not the least,we studied the vertex decomposable property of graph G when its complement (?) is r-partite,including the unmixed property and the sequentially Cohen-Macaulay property of G.For r = 2,3,some necessary and sufficient conditions are established for G to be vertex decomposable when G is dual to r-partite;some sufficient conditions are given for r ≥ 4 and connectedness of Ind(G)is also discussed.In a nutshell,this thesis has given me an opportunity to learn about the deep connections of combinatorial algebra in commutative algebra.