Tree graphs

Conversion of Networks Containing Rings to Noncyclic Structures (Tree Graphs)... 

If the first-encountered ring atom of the ligand is the quaternary common atom of a bicyclic system, one of the three ring bonds is kept intact and the other two are broken. Three branches of the tree graph, which are provided at their ends with requisite duplicate atoms, are therefore produced (Example 2, Table 4). 

It is important to note (see Example 3) that a tree graph is not an equivalent of a molecule. It is, rather, a device to define ligands, via pathways in the network of bonds, for individual stereogenic centers. Accordingly, each center requires its own tree graph and the various tree graphs can be identical or different. 

Tabic 4. Construction of Tree Graphs of Molecules Containing Rings... 

The sequence rules do not suffice to deal with constitutional properties, as these are not fully given by the tree graph. Additional rales are required to define the relative importance of atoms or groups within the tree graph. The corresponding hierarchical tree graph is defined as follows ... 

Step 1 Open the rings to get the tree graph 4a of Table 4 (not essential for the argument)... 

The sextet rotation to the set of the Kekule patterns k, gives a directed tree graph with a root, or the root Kekule pattern, representing the hierarchical structure of kj, where each point corresponds to a Kekule pattern. 

Geometrical interpretations of MEISs. Kinetic and thermodynamic surfaces. Representation of kinetics in the space of thermodynamic variables. Thermodynamic tree. Graphs of chemical reactions, hydraulic flows, and electric currents. 

Activity Class Characteristic Substructures substructure descriptors ( structural keys) acyclic graph = tree graph... 

In a tree graph, the sum of all path-counted vertex proximities MS(UCJDp) is tvrice the sum of all distances in the graph or twice the Wiener index W. Moreover, in trees, the PI index. 

Gutman, L, Estrada, E. and Ivanduc, O. (1999) Some properties of the Wiener polynomial trees. Graph Theory Notes, New York, 36, 7-13. 

Hosoya, H. (2007) Important mathematical structures of the topological index Z for tree graphs. /. Chem. Inf. Model., 47, 744-750. 

Young Diagram lattice (of Ruch) and the ordering scheme of tree graphs (of Gutman and Randic) are described and it is shown, how the two schemes coincide with each other, i.e. generate identical orders. 

Fig. 10. Ordering of the set of Young diagrams containing 6 boxes. The corresponding tree graphs are shown. Underlined graphs are non-caterpillar trees...

A remarkable type of tree graph is called a Caterpillar El-Basil (1987) (or a caterpillar tree) P, (m1 m2,. .., m,) which is obtained by the addition of mi monovalent vertices to the first vertex iq of path P,, m2 monovalent vertices to n2 of Pj and so on. The three tree graphs shown in Fig. 10 are all caterpillar trees and may be designated respectively as ... 

An observation regarding Young diagrams and tree graphs... 

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