Connected graphs, circuits, trees

When analyzing a graph, the first step is clearly its decomposition into connected components. This done, we can focus our attention upon the properties of connected graphs. For convenience, let us designate S the number of 

The inequality can be proved by induction starting from [ N = 2 and going from I NI to I N I +1. 

Let Nc be the set of nodes we do not require that the nodes n, —, % are two-by-two distinct, hence we have only the inequality N, C. The order of the arcs in (i) is not necessarily unique. Imagine a circuit like 

A simple circuit is such circuit C that = C (thus the nodes n, —, % 

The idea of the proof is straightforward. Starting from node n, and going through the sequence, if the path returns to some nj ( ,) we separate the circuit going from back to 2 and so on finally, we have a decomposition into simple circuits. 

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Minimal spanning tree Least sum-of-link path that connects all nodes in a graph with no cycles (closed circuits). 

Figure 2. Open chains (spanning trees ) that could be used for an application of McWeeny s method via eq 16 (section V) in order to perform a ring-current calculation on the conjugated hydrocarbon pyrene (14) (a) the carbon-carbon cr-bond connectivity of pyrene (b) a continuous open chain (from this semi Hamiltonian molecular-graph ) for pyrene (c) circuit completing bonds associated with the open chain in part b. (Adapted and reprinted with permission from ref 3. Copyright 1997 Research Studies Press Ltd.)...

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