Beyond Graph Theory

Graph theory offers many useful characterizations of molecules. If the molecular property is bond additive, the modeling by graphs is quite adequate. Graph theory, even when not explicitly mentioned, has been behind many successful mathematical or quantum-chemical models. For example, Hameka studied the magnetic susceptibility of alkanes from a quantum-chemical point of view. However, the very same quantum-chemical model can be translated without difficulties in the graph theoretical terms when it leads to even simpler expressions for the same magnetic susceptibilities.  

Methane as a 3D structure is defined by the coordinates of all of its atoms. The precise form of the coordinates will depend on the orientation of the molecule relative to the coordinate system, just as the form of the adjacency matrix depends on the choice 

Sometimes it is overlooked that characterization of molecules as 3D objects is deficient,] x% as is the characterization of molecules based on the molecular adjacency or the graph-distance matrix. In the former there is no information on connectivity, in the latter there is no information on spatial geometry. In a way the two approaches complement each other, one having the information that is absent in the other. Therefore, it should not be surprising that combining such matrices will yield a matrix that has information on adjacency as well as on interatomic separations. 

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Thus, we have established that molecules can be represented by means of graphs. To proceed beyond this rather trivial statement, let us resort again to the mathematical graph theory and ask the following question is it possible to represent a graph without resort to a drawing, in other words, is it possible to show the relationship of the graph s vertices not by means of points and dashes but differently, by means of other mathematical objects ... 

More advanced mathematical aspects of the graph-theoretical models for aromaticity are given in other references [36, 48, 49]. Some alternative methods, beyond the scope of this chapter, for the study of aromaticity in deltahedral molecules include tensor surface harmonic theory [51-53] and the topological solutions of non-linear field theory related to the Skyrmions of nuclear physics [54]. 

The transformation from one matrix to any other, within the supermatrix, was to be accomplished with advanced mathematics such as group theory. Knowing that such transformations were beyond his capability at the time, Shchukarev and his colleagues collected data in preparation for the day when the capability would materialize. The data are represented in scores of graphs such as Fig. 3. 

Entropy versus temperature graphs such as Figure 19.12 can be obtained by carefully measuring how the heat capacity of a substance aoo (Section 5.5) varies with temperature, and we can use the data to obtain the absolute entropies at different temperatures. (The theory and methods used for these measurements and calculations are beyond the scope of this text.) Entropies are usually tabulated as molar quantities, in units of joules per mole-kelvin (J/mol-K). 

In Section V it was shown how Wertheim s multi-density approach could be used to develop an equation for associating fluids with an arbitrary number of association sites provided a number of assumptions were satisfied. The simplicity of the TPTl solution results from the fact that the solution is that of an effective two-body problem. Only one bond at a time is considered. This allows the theory to be written in terms of pair correlation functions only, as well as obtain analytical solutions for the bond volume. Moving beyond TPTl is defined as considering irreducible graphs that contain more than one association bond. 

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