Cyclomatic graph number

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 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 ... 

Noting that faces are exactly what in graph theory are called cycles, the previous Euler formula provides the definition of —> cyclomatic number C ... 

Then, starting from the —> Eider sformula for a graph, corresponding to the usual expression for the calculation of the number of rings, that is, the CYclomatic number C ... 

The cyclomatic number is the simplest descriptor that discriminates cyclic compounds from acyclic ones and is related, via the —> Euler formula applied to a graph, to the number of bonds B and atoms A in a molecule as... 

Dobrynin, A.A. and Mel nikov, L.S. (2005a) Wiener index for graphs and their line graphs with arbitrary large cyclomatic numbers. Appl. Math. Lett., 18, 307-312. 

The summation in Eq. [12] is over the number of edges, P. The quantity R is also known as the cyclomatic number of the graph. 

First, for a graph G a simple invariant is the difference tV(G) - E G) between the vertex and edge counts for G. Clearly this (which is very nearly the cyclomatic number of G) is additive in terms of any disconnected components G might have. It takes the value 0 for a simple cycle and the value 1 for a double cycle such as naphthalene. 

International Union of Pure and Applied Chemists (lUPAC) nomenclature rules and the rings and unsaturations formula are based on graph theory. The number of rings and unsaturations in any chemical structure is also known as the cyclomatic number. 

Remark (Degree sequences of graphs by cyclomatic number) A partition a h 2m of an even number... 

M. Schocker. On degree sequences of graphs with given cyclomatic number. Publ. Inst Math. (Beograd) (N.S.), 69(83) 34-40,2001. 

The cyclomatic number (C) descriptor is simply defined to be L-N+1, where L is the number of chemical bonds and N is the number of atoms. In calculation of the cyclomatic number, the molecular graph can be used with or without hydrogen atoms. 

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