Factor graphs

The following equation is not as accurate as the K-factor graphs of Appendix A24. However, certain solution techniques for gas-liquid equilibrium calculations with equations-of-state require initial trial values of K-factors. Equation B-61 is useful for this purpose. 

Still another application V can represent a set of CC double bonds within a single Kekule structure while B represents "adjacent" CC double bonds The resulting graph is known as factor graph [14]. The relation between the later and a set of Kekule structures is not bijec tive [IS] as illustrated in Fig. 4. 

Fig. 4 Three Kekule structures all modeled by the same factor graph.

A more striking case which illustrates the advantage of the equivalent alternative forms was discovered sometime ago by one of the authors [33] An alternative form of a KekulS valence-bond structure, K, is its factor graph, F(K), c f Fig 4 In Fig 6 all Kekul structures of picene are hown (in the form of their Clar notation) along with the corresponding factor graphs One recalls that an F(K) is... 

The name factor graph was suggested by Randic, these graphs were elaborated by one of the authors S- El-Basil, biternat- J-Quantum Chem-, 21, 771 779 793 (1982) Match-, 11, 183 199 209 (1982) Chem- Phys- Lett-, 8 , 145 (1982) Bull- Chem- Soc-Japan, 16, 3158 (1983) Croat- Chem- Acta, 57, 47 (1984)-... 

Tn the determination of the crystal structure of hexamethylbenzene (Figure 1.9, Chapter 1), the Bragg reflections 730, 340 and 470 form a triplet. It was found that by placing atoms at the intersections of high positive values for the electron-density waves (plots called structure factor graphs), Kathleen Lonsdale (in 1929) was able to solve the crystal structure, shown for the 730 and 340 electron-density waves in Figure 8.9.39... 

El-Basil S, Shalabi AS (1985) On the ordering of Kekule Structures, spectral moments of factor graphs of benzenoid hydrocarbons. Match Commun Math Comput Chem 17 11-43... 

The so-called factor graphs (Figure 4) depict the individual Kekule valence structures by considering C=C bonds as the vertices of a graph that are connected only when separated from other C=C bonds by a single CC bond. " This approach can be viewed as a translation of the familiar Kekule valence structures to an... 

Flow factor graph for conical hopper and effective angle of internal friction, 6 = 40°. 

Flow factor graph (similar to Figure 3.11) showing the value of ff and the hopper half angle. 

C=C bonds adjacency Joella El-Basil factor graphs... 

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