Topology branching

Dendrimer Topology, Branch-Cell Differentiation and Surfaces. . 279... 

A. The star topology is the easiest to modify. A physical star topology branches each network device off a central device called a hub, making it easy to add a new workstation. 

Topological indices Topology, branching, general shape 0.1 to 10 s... 

If the principal node K is an endpoint, the planted tree consists of a , the root and the stem, which connects these two points. There are no vertices of degree 4, there are no principal branches. There are no two noncongruent planted trees of this type, whether we deal with topological, spatial, or planar congruence. Hence... 

Now we get to the determination of the number of topologically different free centric trees with n vertices. Select those among the trees whose center carries s branches. The number of such trees is, according to the preceding discussion, equal to the number of nonequivalent configurations of s planted trees with respect to 3. It is the coefficient of in the expression we obtain by replacing f(x,y,z) in (1.12) by... 

From the topology of branched systems with trifunctional branchpoints, for any given molecule the number of branched units is equal to the number of terminal unit minus one. Thus, Eq. 1 can be further simplified to... 

Dendrimers are complex but well-defined chemical compounds, with a treelike structure, a high degree of order, and the possibility of containing selected chemical units in predetermined sites of their structure [4]. Dendrimer chemistry is a rapidly expanding field for both basic and applicative reasons [5]. From a topological viewpoint, dendrimers contain three different regions core, branches, and surface. Luminescent units can be incorporated in different regions of a dendritic structure and can also be noncovalently hosted in the cavities of a dendrimer or associated at the dendrimer surface as schematically shown in Fig. 1 [6]. 

Dendrimers, a relatively new class of macromolecules, differ from traditional Hnear, cross-Hnked, and branched polymers. The conventional way of introducing an active moiety into polymers is to Hnk it chemically into the polymeric backbone or a polymer branch. This synthetic approach results in a topologically complex material. Therefore, a significant effort has to be devoted to improve the structural complexities and functions of the polymers. 

In the case of n-butene isomerization it was demonstrated (Figure 2) that the ideal micro-pore topology led to retardation of the C8 dimer intermediate and that the catalyst based on the ferrierite structure was close to optimal in this respect [1). For selective isodewaxing a one-dimensional pore structure which constrained the skeletal isomerization transition state and thereby minimized multiple branching such as the SAPO-11 structure was found to meet these criteria. Clearly, these are ideal systems in which to apply computational chemistry where the reactant and product molecules are relatively simple and the micro-porous structures are ordered and known in detail. 

To conclude this section, it could be helpful to make a connection between the pictorial discussion we have just given and the type of computation that one can carry out in quantum chemistry. The double cone topology shown in Figure 9.3 can be represented mathematically by Eqs 9.3a and 9.3b. Qx, Qx, are the branching space coordinates. This equation is valid close to the apex of the cone. (A full discussion of the analytical representation of conical intersections can be found in references 9 and 10.)... 

---