Abstract:
Let J G denote the binomial edge ideal of a connected undirected graph G on n
vertices. This is the ideal generated by the binomials x i y j −x j y i , 1 ≤ i < j ≤ n, in the
polynomial ring S = K[x 1 , . . . , x n , y 1 , . . . , y n ] where {i, j} is an edge of G. Our aim in
this thesis is to compute certain algebraic invariants like dimension, depth, system of
parameters, regular sequence, Hilbert series and multiplicity of J G of some particular
classes of binomial edge ideals of graphs. A large amount of information of an ideal
is carried by its minimal free resolution. So we give information on the minimal free
resolution on certain binomial edge ideals. We also give a complete description of
the structure of the modules of deficiencies of binomial edge ideals of some classes of
graphs.
A generalization of the concept of a Cohen-Macaulay ring was introduced by S.
Goto [7] under the name approximately Cohen-Macaulay. In this thesis we collect a
few graphs G such that the associated ring S/J G is approximately Cohen-Macaulay.
We also characterize all the trees that are approximately Cohen-Macaulay.
As more generalized notion than approximately Cohen-Macaulay we also study se-
quentially Cohen-Macaulay property for binomial edge ideals. We give a nice con-
struction principle in this topic.
̃ on n vertices has the property that S/J ̃ is a Cohen-Macaulay
The complete graph G
G
domain with a 1-linear resolution. As one of the main results we clarify the structure
of S/J K m,n , where K m,n denotes the complete bipartite graph.
