Matching of graphs

Balaban, A.T. and Tomescu, T. (1988). Alternating 6-Cycles in Perfect Matchings of Graphs Representing Condensed Benzenoid Hydrocarbons. In Application of Graphs in Chemistry and Physics (Kennedy, J.W. and Quintas, L.V., eds.), North-Holland, Amsterdam (The Netherlands), pp. 5-16. 

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 ... 

The central purpose of a graph is to present, summarize, and/or highlight trends in data or sets of data. Graphs of various types (e.g., scatter plots, contour plots, two- and three-dimensional line graphs, and bar graphs) are used for different purposes thus, authors must match their purpose with the appropriate type of graph. 

A perfect matching or 1-factor of a graph G is a matching of G that covers all vertices. If, in particular, G is a BS then a 1-factor of G is usually called a Kekule structure (KS). 

Since a benzenoid system H is a bipartite graph, the existence of Kekule structures of H is equivalent to the existence of 1-factors (perfect matchings) of a bipartite graph. In 1935, P. Hall found the following necessary and sufficient conditions. 

Several applications of the BG-concept were put forward by Kirby [12, 13]. It has been shown recently [23] that the number of 2-factors of a benzenoid system equals the number of 1-factors ( = perfect matchings) of the branching graph. This latter result is of considerable relevance in Clar s aromatic sextet theory. 

Each Kekule valence-bond structure corresponds to a perfect match of the edges of the polyhex graph, see ref- 38-J- Riordan, An Introduction to Combinatorial Analysis Wiley, New York (1958) C-L- Liu, Introduction to Combinatorial Mathematics, McGraw-Hill, New York (1968)- A chemical version on rook boards may be found in C-D- Godsil and 1- Gutman, Croat- Chem-Acta- 54, 53 (1981). 

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. 

Polanski, J. and Bonchev, D. (1986). The Wiener Number of Graphs. I. General Theory and Changes Due to Graph Operations, MATCH (Comm.Math.Comp.Chem.),21,133-186. Polanski, J. and Bonchev, D. (1987). The Minimum Distance Number of Trees. MATCH (Comm. Math.Comp.Chem.), 21, 314-344. 

Skorobogatov, V.A. and Dobrynin, A.A. (1988). Metric Analysis of Graphs. MATCH (Comm. Math.Comp.Chem.), 23,105-151. 

A.T. Balaban, Ed., Chemical Application of Graph Theory. Academic, New York, 1976 Math. Chem. (MATCH) 1,33 (1975). 

Unfortunately, even these rules are not ready for use in an integrator tool as described in the previous section. In case of non-deterniinistic transformations between interdependent documents, it is crucial that the user is made aware of conflicts between applicable rules. A conflict occurs, if multiple rules match the same increment as owned increment. Thus, we have to consider all applicable rules and their mutual conflicts before selecting a rule for execution. To achieve this, we have to give up atomic rule execution, i.e., we have to decouple pattern matching from graph transformation [33, 255]. 

Characteristic polynomials belong to a more general class of graph polynomials, which are used to encode some information on molecular graphs. Among these, there are Z-counting polynomial, —> matching polynomial, and Wiener polynomial. 

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