Canonical indexing and constructive enumeration of molecular graphs

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“…Kvasnicka and Pospichal [24] extended this idea and published an implementation of the resulting theorem for application to the MCES problem. This line-graph induced isomorphism concept has served as the basis for the development of the MCES program TopSim [25,26] as well as the work of Koch [27] and Raymond et.…”

“…In their respective papers, Levi suggested using a clique detection procedure due to Grasselli [37], and Barrow and Burstall proposed using an algorithm due to Bron and Kerbosch [38]. Until recently [28], the BronKerbosch algorithm was the clique detection procedure of choice for clique-based MCIS applications [1,24,[39][40][41][42][43][44].…”

Combinatorial enumeration in chemistry

“…67,71,72 Balasubramanian 73,74 also carried out combinatorial and group-theoretical analyses of a C 48 N 12 dodecaazafullerene. The IUPAC name of this compound is rather unwieldy (8,13,18,23,26,29,32,35,40,45,50)-dodeca[60-S 6 ]fullerene. Balasubramanian 75 found earlier that there are 233,227,974,475 possible isomers of this compound.…”

“…Gordon and Davison 136 initiated the path counting method. In the path counting method the number of Kekule´structures K of a benzenoid B is equal to the number of the mutually self-avoiding directed peak-to-valley paths: 138,139 K ¼ det |P| (8) where P is a matrix whose elements (P) ij represent counts of self-avoiding paths in B starting at peak(s) and ending at valley(s). A peak is a vertex on the upper perimeter of B that lies above its adjacent vertices, whilst a valley is a vertex lying below its nearest neighbors on the lower perimeter of B.…”

“…It should be noted that the constructive enumeration means counting objects by constructing each object separately using a suitable code. 7,8 In his work, Lukovits represented alkanes by chemical trees. Chemical trees are graph-theoretical trees 9 that represent the carbon skeleton of alkanes and in which no vertex has a degree greater than 4.…”