In this paper we introduce the concept of generalized trees and compute the Hilbert series of their BINOMIAL EDGE IDEALs

We give upper bounds for the regularity of EDGE IDEAL of some classes of graphs in terms of invariants of graph. We introduce two numbers a’ (G) and n (G) depending on graph G and show that for a vertex decomposable graphG, reg (R/I (G)) £ min{a’ (G), n (G)} and for a shellable graph G, reg (R/I (G))£n (G). Moreover, it is shown that for a graphG, where Gc is a d-tree, we have pd (R/I (G)) =maxvÎV (G) {degG (v)}.

Let G be a simple, oriented connected graph with n vertices and m EDGEs. Let I (B) be the BINOMIAL IDEAL associated to the incidence matrix B of the graph G. Assume that IL is the lattice IDEAL associated to the rows of the matrix B. Also let Bi be a sub matrix of B after removing the i -th row. We introduce a graph theoretical criterion for G which is a sufficient and necessary condition for I (B) =I (Bi) and I (Bi) =IL. After that we introduce another graph theoretical criterion for G which is a sufficient and necessary condition for I (B) =IL. It is shown that the heights of I (B) and I (Bi) are equal to n - 1 and the dimensions of I (B) and I (Bi) are equal to m - n+1; then I (Bi) is a complete intersection IDEAL.

Aim: This study was designed to determine any profile changes and indentify the role of the most significant factors responsible for these changes.Materials and Methods: Cephalograms of 56 class II Division I patients (36 female and 18 male with mean age of 10 years) were used to analysis any changes in the perioral soft tissue profile before and after the treatment. A correlation analysis and multiple regression analysis (forward) revealed complex interaction between dentition, bony structures and soft tissues of the perioral area.Results: Retraction at the upper lip and increase in lower lip length in boys were more predictable than that of others. Significant soft tissue changes occured in response to orthodontic treatment, Upper lip retraction was shown to be related to the following factors: a- Retraction of upper incisal EDGEs during treatment, b-Retraction of Point A, c- Pretreatment upper sulcus thickness and d- Retraction of sulcus superior.Conclusion: A greater increase in lower lip length takes place with greater amount of maxillary incisa1 EDGE retraction during treatment and a greater amount of sulcus superior retraction.