Isomer enumeration

An earlier paper [PolG35] contained the essence of Polya s Theorem in its three-variable form, together with an informal proof of it. This paper also entered into the field of isomer enumeration by presenting a generating function for the substituted benzenes. 

A paper in the same journal [PolG36b] elaborated on isomer enumeration and the corresponding asymptotic results. Here the functional equations for the generating functions for four kinds of rooted trees were presented without proof. They were, in a slightly different notation, formulae (8), (4), and (7) in the introduction to Polya s main paper, and one form of the functional equation for the generating function for rooted trees. From these results a number of asymptotic formulae were derived. These results were all incorporated into the main paper. 

In the light of these long traditions, extensive enumerations of the isomers of benzenoid hydrocarbons is a very new area. A systematic investigation can be dated to 1982 with the first paper of Dias [7] (but see also below). He published an article series in ten parts [7-16] entitled A Periodic Table for Polycyclic Aromatic Hydrocarbons and more recent works [17, 18]. With the invention of the periodic table, Dias created orderness in the chaotic myriads of chemical formulas for benzenoid hydrocarbons, which may be written. He has also written a monograph [19] with relevance to this topic and some other reviews [20-22], Two years before Dias, Elk [23] published a paper on benzenoids, which contains explicitly the enumeration of isomers up to h = 5. It seems that the work of Elk has largely been overlooked in the context of benzenoid isomer enumeration. 

Beachhead. However, basically this discrepance does not disparage the explorations of the English mathematician. Cayley was the first to represent a molecule as a topological graph and to start developing the methods of isomer enumeration on the basis of the graph theory. Moreover, Cayley forwarded the notion of enumerative polynomial for rooted trees, i.e. the polynomial whose coefficients Ai specify the number of rooted trees with i vertices ... 

As to isomer enumeration, the various attempts made in the 19th century to develop the Cayley approach any further failed. Only in 1930s the American scientists H. R. Henze and C. M. Blair derived the formulas for determination of the number of isomers in aliphatic corn-... 

Bytautas, L. and Klein, D.J. (1998). Chemical Combinatorics for Alkane-Isomer Enumeration and More. J.Chem.Inf.Comput.Sci.,38,1063-1078. [R]... 

Hansen, P.J. and Jurs, P.C. (1988b). Chemical Applications of Graph Theory. Part II. Isomer Enumeration. J.Chem.Educ., 65,661-664. 

Isomer Enumeration. - In the past two years (2003-2005) there were somewhat less reports on the enumeration of isomeric structures unlike the periods between 1999 2001 and 2001 2003. 

P, j, Hansen and P. C. Jurs, /. Chem. Ediic., 65, 574 (1988), Chemical Applications of Graph Theory II Isomer Enumeration,... 

Dias, J.R. 1982. A Periodic Table of Polycyclic Aromatic Hydrocarbons. Isomer Enumeration of Fused Polycyclic Aromatic Hydrocarbons. Journal of Chemical Information and Computer Sciences 22 15-22. 

Generating functions were first employed in isomer enumeration work by the mathematician Cayley. In 1857, Cayley [81] had devised a method for enumerating rooted trees which centered on a generating function of the general form ... 

Isomer enumeration methods have been reviewed several times. The literature till 1973 is summarized in Rouvray s excellent and comprehensive review [3] which emphasizes the dassical methods for enumerating alkanes and alkyl derivatives a similar approach is contained in Trinaj-stid s recent book [4]. In a book edited by the present author in 1976, three chapters were devoted to this problem one dealing with Pdlya s contributions to the field [5a], a second with the enumeration of acyclic systems [5b], and a third with the enumeration of cyclic systems [5c]. Since that time considerable progress has been made in the latter area. Accordingly, the present review will stress these more recent applications, while trying to reduce repetition of previously reviewed work to the minimum necessary to make the present chapter intelligible and self-contained. 

Several other, less extensive, reviews on isomer enumeration have also appeared [6,7). 

Isomerism opened the gates for structure theory in organic chemistry, for Werner s theory of complexes in inorganic chemistry, and for stereochemistry. The above review has highlighted some aspects of isomer enumeration, with the inevitable bias of the author s interest in organic chemistry. 

Chapter 5 by Balaban deals with the methodology of isomer enumeration, a held that continues to challenge us with many fascinating problems. Balaban s discussion embraces not only the traditional techniques adopted for enumeration of chemical species, such as the now classic Pdlya method, but also delves into more recent approaches developed by workers such as Ruch and De Bru n, Haraiy and Palmer. Our final Chapter by Trinajstic elucidates the interplay between graph theory and molecular orbital theoiy from the standpoint of spectral graph theory, and highlights in particular the concept of topological resonance in molecular species. 

By general structural isomer enumeration, we mean the enmneration of all molecular graphs corresponding to a molecular formula. We do not include here solutions that construct molecular structures from additional constraints, such as the presence or the absence of substructural fragments. Enmneration with constraints is reviewed in the next subsection. 

Alkanes and alkane-like substances have captured the interest of researchers in isomer enumeration for a long time because of their commercial importance. For example, Henze and Blair published the first isomer enumeration of alkanes in 1931. Here we provide, for reference, tables that list the number of isomers of alkanes, alkenes, alkynes, and stereoalkanes (Table 4), ketones and esters (Table 5), and primary, secondary, and tertiary alcohols (Table 6) up to 25 carbon atoms. 

Isomer Enumeration of Catafusenes, C4 +2H2n+4 Benzenoid and Helicenic Hydrocarbons. 

Isomer Enumeration of Unbranched Catacondensed Polygonal Systems with Pentagons and Heptagons. 

Cayley was the first to develop a systematic mathematical approach to isomer enumeration. He listed alkanes and alkyl radicals up to those with 13 carbon atoms. Although his counts for n = 12 and n = 13 are now known to be incorrect, his pioneering work led the way. He also indicated that he could not reduce his scheme to a single formula for enumeration of alkane isomers. 

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