Cyclomatic

Polyphosphazenes are the most important and the most thermally characterized of the phosphorus-containing inorganic polymers. Linear, cycloli-near, and cross-linked cyclomatic polymers based on phosphazene structures have been produced. The repeating units of some polyphosphazenes are as follows ... 

For the two examples above, i.e., G, and G2, one obtains A = 7 and 10, respectively. The total adjacency index, however, seems to be of little use for molecular studies since it can only distinguish between molecules having different numbers of cycles (i.e., cyclomatic numbers) p ... 

The number of bonds is considered in the - cyclomatic number and appears in several - molecular descriptors such as the Balaban distance connectivity index, the - mean Randic branching index, the -> information bond index, and several - topological information indices. 

These are simple molecular descriptors based on counting the defined elements of a compound. The most common chemical count descriptors are atom number A, bond number B, cyclomatic number C, -> hydrogen-bond acceptor number and -> hydrogen-bond donor number, -> distance-counting descriptors, -> path counts, walk counts. 

It is the number of independent cycles C (or rings) in a molecule, and, more exactly, the number of non-overlapping cycles. The cyclomatic number of a polycyclic graph is... 

The cyclomatic number is the simplest descriptor which discriminates cyclic compounds from acyclic ones and is related to the number of bonds B and atoms A in a molecule as follows ... 

The cyclomatic number must not be confused with the graph - cyclicity CA. Thus, for example, naphthalene has a cyclomatic number equal to two (the two benzene rings) and a cyclicity equal to three (the two benzene rings plus the more external 10-atom ring). 

A self-returning path is called a cyclic path (or cycle or circuit), i.e. a cycle is a walk with no repeated vertices (i.e. a path) other than its first and last ones (vj = v ). The number of independent cycles (or rings) in a graph is the -> cyclomatic number. Cyclicity C is the number of all possible cycles in a graph. 

The cycle is a chain of the graph consisting of arcs and vertices which begins and finishes on the same vertex. One of most important characteristics of any graph is its cyclomatic number 0 which gives the number of independent cycles it contains. This number can be determined from the following formula ... 

The Horiuti Rule for linear mechanisms is the chemical equivalent of the cyclomatic number equation, which expresses general topological characteristics of a graph (mechanism). 

For greater clarity we illustrate our results by the model graph R, as shown in Fig. 1.7. The cyclomatic number of this graph is ... 

Method for finding spanning trees based on the cyclomatic matrix... 

To determine the set of sections we use the cyclomatic matrix A/fOf the graph R in which the rows correspond to the independent cycles, and the columns to the arcs of the graph. If an arc participates in a given cycle, the corresponding matrix element is equal to the arc otherwise it is equal to zero. We obtain the following matrix for the graph R ... 

Methods based on cyclomatic and section matrices After shrinking the given cycle to the vertex, the subgraph obtained is treated according to the proposed methods. For every cycle this operation should be performed separately. 

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