Isomorphic molecular graphs

How can one simply interpret the notion of isomorphism for molecular graphs Loosely speaking, and with some simplification, the isomorphic molecular graphs and 0, 2 only in the indexing of their... 

The study of chemical reactions requires the definition of simple concepts associated with the properties ofthe system. Topological approaches of bonding, based on the analysis of the gradient field of well-defined local functions, evaluated from any quantum mechanical method are close to chemists intuition and experience and provide method-independent techniques [4-7]. In this work, we have used the concepts developed in the Bonding Evolution Theory [8] (BET, see Appendix B), applied to the Electron Localization Function (ELF, see Appendix A) [9]. This method has been applied successfully to proton transfer mechanism [10,11] as well as isomerization reaction [12]. The latter approach focuses on the evolution of chemical properties by assuming an isomorphism between chemical structures and the molecular graph defined in Appendix C. 

Faulon, J.-L. (1998). Isomorphism, Automorphism Partitioning, and Canonical Labeling Can Be Solved in Polynomial-Time for Molecular Graphs. J.Chem.lnf.Comput.Scl, 38,432-444. 

However, a unique enumeration does not solve the problem of substructure search. Superimposition of substructures on structures would require the mapping of any combination of molecular graphs to find a graph isomorphism this is a tedious and time-consuming process. Because the rate of search is always one of the most important limitations for database applications, substructure search should incorporate additional preprocessing steps that restrict the number of molecules to be compared in an atom-by-atom matching algorithm. 

Faulon, J.-L. (1998) Isomorphism, automorphism partitioning, and canonical labeling can be solved in polynomial-time for molecular graphs. J. Chem. Inf. Comput. Sci., 38, 432-444. 

Nicholson V, Tsai C-C, Johnson M, Naim M. A subgraph isomorphism theorem for molecular graphs. Stud Phys Theor Chem 1987 51 226-230. 

The Hiickel Hamiltonian-matrix for an n-annulene (Fig. 5-1) is the matrix in the determinant found in equation (5-1), with the quantities ( — x), along the diagonal, replaced by zero (i.e. a). As pointed out in Appendix A, this matrix is isomorphic with the adjacency matrix, A(C ), of the correspondingly-labelled molecular graph furthermore, the eigenvalues of the matrix, A(C ), are the annulene energy-levels we require (expressed in units of fi and... 

Two molecular graphs M, M )vt are equivalent molecular graphs or isomorphic molecular prap/isiftheybelongtothesameorbitunderthis action. The elements of = S )vt , i.e. the orbits, are called equivalence classes of molecular graphs. The class of M will be indicated as M, and the set of all these classes is denoted by... 

Speed plays an important role in structure enumeration, but only few theoretical results about the computational complexity are known. Goldberg s work [91] proves that the results in orderly enumeration can be computed with polynomial delay, and a paper of Luks [188] shows that isomorphism testing of molecular graphs can be done in polynomial time. 

Finally we have to decompose the product graphs into connected components and to eliminate the isomorphic duplicates. For this reason we introduce the set of connected labeled graphs for an arbitrary set of molecular graphs... 

The concept of isomerism and isomorphism determined over a universe of molecular graphs can be formally considered as a realization of an "equivalence between abstract elements. For instance, one can easy verify that the following three laws are satisfied ... 

The problem of canonical coding, graph isomorphism, and graph automorphism has both mathematical and chemical significance. The mathematical formulation of the problem is briefly set out below, and some cormections with the chemical counterpart are presented. In the subsequent sections, the main algorithms used in chemistry for canonical coding of molecular graphs and constitutional symmetry perception are presented and compared. 

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