Generic Structures Graphs

While the rows and columns of A obviously depend on a particular choice of vertex labels, the generic structural j)roperties of G must remain invariant under a permutation of rows and columns. Much of this structural information can in fact be extracted from the spectrum of G the spectrum of a graph G,... 

Figure 4, A generic structure and the two forms of reduced chemical graphs describing it.

The complexity of the generic structures in the database, and the comparative strengths of each of the methods of chemical graph reduction are illustrated in Figure 10. 

The size and complexity of generic structures leads to the requirement for more elaborate screening systems than for specific structures. The different approaches which have been implemented at Sheffield include fragment and ring screens and also reduced graphs of var3dng discriminatory power. 

Figure 4. The GENSAL, ECTR and ring/non-ring reduced graph representations of a generic structure...

In the previous chapter we have defined the generic rank as a property of structural matrices. Let us now introduce some new concepts in connection with structural systems and their associated graphs. 

Gillet, V.J., Downs, G.M., Holliday, J.D., Lynch, M.F., Dethlefsen, W. Computer storage and retrieval of generic chemical structures in patents. 13. Reduced graph generation. Journal of Chemical Information and Computer Science 1991, 31, 260-270. 

Proof. If ip0 had 2 extensions ip" Spec (R) X, then apply the definition of scheme to ip1 and ip" and it follows that ip1 = ip". This takes care of uniqueness. Next apply the definition of proper to cp Spec (R) -> Y and to the closed subset of X xy Spec (R) obtained by taking the closure Z of (ipo,i) [Spec K]. Give Z the structure of reduced closed subschemes. Then p2(Z) is a closed subset of Spec (R) containing its generic point, hence equal to Spec (R). 1 claim that p2 Z - Spec (R) is an isomorphism, hence Z is the graph of the ip we are looking for. 

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