General Theorematic Results

One of the most fundamental results involves an area integral over all (Gaussian) curvatures on a surface S  

Gauss-Bonnet Theorem. For an orientable dosed smooth surface Sin E, the total Gaussian curvature integrated over S is a topological invariant 

That is, for the surfaces S around the outside of a basketball, or of an (American) football, or of a discus, the total Gaussian curvature integrated over all of S has the same value 4n, whereas for the surfaces around a doughnut or around a coffee mug, the result is another characteristic value 0. 

In fact, this result may be viewed to be the same as that of the Gauss-Bonnet theorem if we recall that the angle defects are essentially the net (i.e., integrated) Gaussian curvatures associated with each vertex. 

A topohedron S (or also the associated graph embedding) is also characterized by counts N, E, and F of its vertices, edges, and faces. These counts are related by  

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Here then it is intended to note some inter-relations between VB and MO theories, particularly as regards general predictive correspondences. With the focus on generality of correspondences the emphasis naturally shifts to the simpler (typically semiempirical) models for which it is easier to obtain general (e.g. theorematic) results. Thence here attention is focused on the rather well studied such models for n-electron networks in neutral (i.e. non-ionic) organic molecules. 

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