Fuzzy graph theory

These two themes in chemistry have a great deal in common, however, and many of the algorithms developed for one can be applied with little modification to the other. Intrinsically, both CASE and CASD can be based upon fuzzy graph theory.  

A crisp system is a fundamental system in the sense that it is used to make a decision. The basic rule in such a system may be represented as  

Unfortunately, these assumptions do not always hold in practical structure elucidation, because observed spectral data are usually ambiguous, incomplete, and may contain artifacts or impurities. To take these considerations into account, Eq. (1) has to be modified to 

Systems that obey Eq. (1) are called crisp systems and systems that obey Eq. (2) are called fuzzy systems. The real world is actually a mbcture of crisp and fuzzy systems. 

To arrive at a conclusion, usually an attempt is made to use absolutely explicit evidence. If such evidence is not available, Eq. (2) may be used to rank the possible conclusions. The best ranked possible conclusion is not necessarily the correct conclusion. Therefore, to reach a reliable conclusion, additional evidence is needed. Fuzzinesss is a description or outcome of the real world, rather than a criterion used in making a decision. The fuzzy degree of a measurement is used to reduce a search space so that the correct answer may be reached more efficiently. For example, from the H NMR spectrum of a peptide, a set of peaks (PS parts per million) is observed as follows  

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III. FUZZY GRAPH THEORY APPLICATIONS IN COMPUTER-ASSISTED BIOPOLYMER NMR ASSIGNMENT... 

Most proton NMR resonance assignments of proteins are based upon Wiithrich s strategy. In fuzzy logic and fuzzy graph theory, the strategy... 

The definition of a graph given in Eq. (3) shows that it is formed of two sets a set of vertices V and a set of edges E. This definition can be formulated in the context of fuzzy set theory with both sets V and E represented by fuzzy sets (respectively and E to constitute the fuzzy topological graph G = (V E ). This means that each vertex e E and each edge e E may be associated with the membership functions and which map these two sets on the range of real values... 

The class of the empirical descriptors is a fuzzy, not well-defined class. In principle, empirical descriptors are those not defined on the basis of a general theory such as, for example, quantum chemistry or graph theory. Rather they are defined by practical rules derived from chemical experience, e.g. considering specific or local structural factors present in the molecules, often sets of congeneric compounds. As a consequence, in most cases, empirical descriptors represent limited subsets of compounds and cannot be extended to classes of compounds different from those for which they were defined. Empirical descriptors have not to be confused with experimentally derived descriptors even if it is well known that several of them are empirically derived. 

Automation of Protein 2D Proton NMR Assignment by Means of Fuzzy Mathematics and Graph Theory. 

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