Graph isomorphic

It is very important to understand what is meant by two graphs being the same or different . For this purpose we introduce the notion of graph isomorphism. Two graphs, G and G2, are isomorphic, which we write as Gi = G2, if there exists a bijection ij) V V2 which preserves adjacency (i.e. such that e i,j) 6 Ej if... 

Structure searching is the chemical equivalent of graph isomorphism, that is, the matching of one graph against another to determine whether they are identical. This can be carried out very rapidly if a unique structure representation is available, because a character-by-character match will then suffice to compare two structures for identity. However, connection tables are not necessarily unique, because very many different tables can be created for the same molecule depending upon the way in which the atoms in the molecule are numbered. Specifically, for a molecule containing N atoms, there are N ... 

Read RC, Cornell DG. The graph isomorphism disease. / Graph Theory 1977 1 339-63. 

The strongest definition of structural similarity is the concept of graph isomorphism which is, of course, tantamount to complete identity. This concept can be defined for subschemes as well as for schemes. 

Graph isomorphism is a completely syntactic concept and is clearly testable. 

B contains all nodes reachable from node m B is graph isomorphic to B... 

Here A is a single entry subgraph with entry node and a self-loop back to n. We now add a single entry subgraph A graph isomorphic to A and connect A to n, , the entry of A. There are two possibilities. If node m in A was connected to, then the corresponding node m in A can be connected either back to the entry node of A or A or to the entry node n of A. ... 

If a(P) = 0, there is nothing to prove. Assume that we have shown the result for any flow diagram G with a(G) d-1 and that a(P) = d > 1. Let n and m be any two nodes in P such that the direct connection from n to m is anomalous. Let G- be a graph isomorphic copy of Sm with entry node m of G mapped by the graph isomorphism f onto m. Now the direct connection from n to... 

The most successful algorithms for general graph isomorphism use the backtrack approach (as a fall-back method) in combination... 

Hoffmann CM (1982) Group-theoretic algorithms and graph isomorphisms, Lecture Notes in Computer Science, vol. 136, Springer, Heidelberg... 

Gi equals the number of edges connecting the respective vertices of Gg, then we call the graphs isomorphic (see Fig. 7). 

Bonchev, D., Mekenyan, O. and Balaban, A.T. (1985). Unique Description of Chemical Structures Based on Hierarchically Ordered Extended Connectivities (HOC Procedures). IV. Recognition of Graph Isomorphism jind Graph Symmetries. MATCH (Comm.Math.Comp. Chem.), 18, 83-89. 

Randic, M. (1977). On Canonical Numbering of Atoms in a Molecule and Graph Isomorphism. J.Chem.Inf.Comput.ScL, 17,171-180. 

Fig. 59. The structure of (NH4)4[(U02)5(Mo04)7](H20) projected along the c axis (a), nodal representation of its [(1102)5(1 004)7] framework (b), nodal representation of its fundamental chain (c), and graphs isomorphous to nodal representations of fundamental chains of chiral uranyl molybdate frameworks with the U Mo ratio of 5 7, 4 5 and 6 7 (d, e and f, respectively).

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. 

Graph isomorphism algorithms can be nsed for mapping a (query) full structure onto a strncture to determine identity between two structures. 

Graph Isomorphism is a method from mathematical graph theory that can be used for mapping a structure onto another to determine the identity between two structures. 

Randic, M. (1977c) On canonical numbering of atoms in a molecule and graph isomorphism. /. Chem. Inf. Comput. Sci., 17, 171-180. 

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