Abstract
We address systematics for the enumeration of substitutional isomers when there is constrained positioning of ligands on a molecular skeleton. One constraint involves ‘restrictive ligands’ where two of the same kind are forbidden to occupy adjacent sites in a molecular skeleton. This may arise because of steric hindrance, or because of groups which in neighbor proximity react to eliminate one. For instance, no pair of –OH groups attach to the same C atom in a molecular skeleton. In another case, malonic acid residues decarboxylate leaving no more than one decarboxylation in each residue. The enumeration with such restrictive ligands may be addressed via a Polya-theoretic cycle index hybridized with the graph-theoretic independence polynomial (when there is just a single such neighbor-excluding ligand and another which is not), while more generally a hybridization with the chromatic polynomial is needed. Another substitional-isomer constraint involves bidentate ligands, with each ligand-part occupying adjacent sites, and possibly also with additional separate unidentate ligands. Here, the set of all pure & mixed such ligand placements is analytically represented by a ‘symmetry-reduced’ matching polynomial (which is a hybrid now of the matching polynomial and Polya’s cycle index). This result gives the generating function for isomer enumeration, taking into account every possible so-restricted assortment of the employed ligands. Here we make such novel hybridizations (for these and other graphtheoretic polynomials) to deal with such oft-encountered chemical problems, which nevertheless transcend typical earlier unconstrained formulizations. Further subsymmetry classification & enumerations, along with examples are considered in a further article.
Similar content being viewed by others
References
G. Pólya, Kombinatorische Anzahlbestimungen für Gruppen, Graphen und chemische Verbindungen. Acta Math. 68, 145–254 (1937). See English translation in: G. Pólya and R. C. Read, Combinatorial Enumeration of Groups, Graphs, and Chemical Compounds, Springer, Berlin, 1987. Russian translation in: Perechislitel’nye Zadachi Kombinatornogo Analiza (Sbornik Perevodov) = Enumeration Problems of Combinatorial Analysis (Collections of Translations) (G. P. Gavrilov, ed.), Mir, Moscow, 1979, pp. 36–136
Kerber A.: Algebraic Combinatorics via Finite Group Actions. Wissenschaftsverlag, Manheim, Wien (1991)
Kerber A.: Applied Finite Group Actions. Springer, Berlin (1999)
Harary F., Palmer E.M.: Graphical Enumeration. Academic Press, New York (1973)
Ruch E., Hässelbarth W., Richter B.: Doppelnebenklassen als Klassenbegriff und Nomenklaturprinzip für Isomere und ihre Abzälung. Theoret. Chim. Acta 19(3/5), 288–300 (1970)
De Bruijn N.G.: A Survey of generalizations of Pólya’s enumeration theorem. Niew Archief voor Wiskunde 19(2), 89–112 (1971)
Balaban , A.T. (eds): Chemical Applications of Graph Theory. Academic Press, London (1976)
Balasubramanian K.: Applications of combinatorics and graph theory to spectroscopy and quantum chemistry. Chem. Rev. 85, 599–618 (1985)
Fujita S.: Symmetry and Combinatorial Enumeration in Chemistry. Springer, Berlin (1991)
El-Basil S.: Combinatorial Organic Chemistry. Nova Science Publishers, Huntington (1999)
Kerber A.: Enumeration under finite group action, basic tools, results and methods. MATCH Commun. Math. Comput. Chem. 46, 151–198 (2002)
Klein D.J., Cowley A.H.: Permutational isomerism with bidentate ligands and other constraints. J. Am. Chem. Soc. 100, 2593–2599 (1978)
D.J. Klein, A. Ryzhov, and V. Rosenfeld, Permutational isomers on a molecular skeleton with neighbor-excluding ligands. J. Math. Chem. 45(4), 892–909 (2008). Erratum, ibid., p. 910
Rota G.-C., Smith D.A.: Enumeration under group action. Ann. Sci. Norm. Super. Pisa. Cl. Sci., Ser. 4, 637–646 (1977)
V.R. Rosenfeld, Yet another generalization of Pólya’s theorem: Enumerating equivalence classes of objects with a prescribed monoid of endomorphisms. MATCH Commun. Math. Comput. Chem. 43, 111–130 (2001) (see Erratum on p. 125: A lost multiplier s just after the sign ∑ in (16)!)
V.R. Rosenfeld, Resolving a combinatorial problem of crystallography. Deposited in VINITI, 16.09.82, no. 4877–82 Dep., 1982 (Russian)
V.R. Rosenfeld, The generalized Pólya’s theorem (A novel approach for enumerating equivalence classes of objects with a prescribed automorphism subgroup), in: Proceedings of The Sixth Caribbean Conference in Graph Theory and Computing, The University of the West Indies, St. Augustine, Trinidad, West Indies, 1991, The University of West Indies, St. Augustine, Trinidad, 1991, p. 240–259
Klein D.J.: Rigorous results for branched polymer models with excluded volume. J. Chem. Phys. 75, 5186–5189 (1981)
Hässelbarth W., Ruch E.: Permutational isomers with chiral ligands. Israel J. Chem. 15, 112–115 (1976)
Ruch E., Klein D.J.: Double cosets in chemistry and physics. Theor. Chim. Acta 63, 447–472 (1983)
Klein D., Misra A.D.: Topological isomer generation & enumeration: Application for polyphenacenes. MATCH Commun. Math. Comput. Chem. 46, 45–69 (2002)
M. Gionfriddo, Alcuni resultati relativi alle colorazioni L s d’un grafo. Riv. Mat. Univ. Parma. 6, 125–133 (1980) (Italian)
M. Gionfriddo, Su un problema relativo alle colorazioni L 2 d’un grafo planare e colorazioni L s . Riv. Mat. Univ. Parma. 6, 151–160 (1980) (Italian)
Vukičević D., Graovac A.: On functionalized fullerenes C60X n . J. Math. Chem. 45(2), 557–562 (2009)
Farrell E.J.: On a general class of graph polynomials. J. Combin. Theory B 26(1), 111–122 (1979)
Klein D.J.: Variational localized-site cluster expansions. I. General theory. J. Chem. Phys. 64(12), 4868–4872 (1976)
Farrell E.J.: On a class of polynomials obtained from circuits in a graph and its application to characteristic polynomials of graphs. Discrete Math. 25, 121–133 (1979)
Farrell E.J., Grell J.C.: The circuit polynomial and its relation to other polynomials. Caribean J. Math. 2(1/2), 15–24 (1982)
Farrell E.J.: An introduction to matching polynomials. J. Combin. Theory Ser. B 27, 75–86 (1979)
Gutman I.: On the theory of the matching polynomial. MATCH Commun. Math. Comput. Chem. 6, 75–91 (1979)
V.R. Rosenfeld, The circuit polynomial of the restricted rooted product G(Γ) of graphs with a bipartite core G. Discrete Appl. Math. 156, 500–510 (2008). (It supersedes an earlier version with this Title; see arXiv:math.CO/0304190, v. 1, 15 Apr 2003, 21 p.)
Farrell E.J., Rosenfeld V.R.: Block and articulation node polynomials of the generalized rooted product of graphs. J. Math. Sci. (India) 11(1), 35–47 (2000)
Rosenfeld V.R., Diudea M.V.: The block polynomials and block spectra of dendrimers. Internet Electron. J. Mol. Design 1(3), 142–156 (2002)
Klein D.J., Schmalz T.G., Hite G.E., Metropolous A., Seitz W.A.: The poly-polyphenanthrene family of multi-phase pi-network polymers in a valence-bond picture. Chem. Phys. Lett. 120, 367–371 (1985)
Klein D.J., Hite G.E., Schmalz T.G.: Transfer-matrix method for subgraph enumeration: Application to polypyrene fusenes. J. Comput. Chem. 7, 443–456 (1986)
Heilmann O.J., Lieb E.H.: Theory of monomer-dimer systems. Commun. Phys. 25(3), 190–232 (1972)
V.R. Rosenfeld, The enumeration of admissible subgraphs of the n-dimensional Ising problem (n ≥ 1), in: Calculational Methods in Physical Chemistry, ed. by Yu. G. Papulov (KGU (Kalinin’s State University), Kalinin, 1988), p. 15–20 (Russian)
Rosenfeld V.R., Gutman I.: A novel approach to graph polynomials. MATCH Commun. Math. Comput. Chem. 24, 191–199 (1989)
Rosenfeld V.R., Gutman I.: On the graph polynomials of a weighted graph. Coll. Sci. Papers Fac. Sci. Kragujevac 12, 49–57 (1991)
Gutman I., Harary F.: Generalizations of the matching polynomial. Utilitas Mathematica 24, 97–106 (1983)
Hoede C., Li X.: Clique polynomials and independent set polynomials of graphs. Discrete Math. 125, 219–228 (1991)
L.W. Beineke, Derived graphs of digraphs, in: Beiträge zur Graphentheorie, ed. by H. Sachs, H.-J. Voss, H.-J. Walter (Teubner, Leipzig, 1968), p. 17–33
L.W. Beineke, L.W., Characterizations of derived graphs. J. Combin. Theory 9, 129–135 (1970); MR0262097
Gutman I.: Some relations for the independence and matching polynomials and their chemical applications. Bull. Acad. Serbe Sci. Arts 105, 39–49 (1992)
Scott A.D., Sokal A.D.: The repulsive lattice gas, the independence-set polynomial, and the Lovász local lemma. J. Stat. Phys. 118, 1151–1261 (2005)
Scott A.D., Sokal A.D.: On dependency graphs and the lattice gas. Combin. Probab. Comput. 15, 253–279 (2006)
G.D. Birkhoff, A determinant formula for the number of ways of coloring a map. Ann. Math. 14, 42–46 (1912). See also W.T. Tutte. Graph Theory, Addison-Wesley, 1984
Thomassen C.: Chromatic roots and Hamiltonian paths. J. Combin. Theory B 80, 218–224 (2000)
Harary F.: Graphical exposition of the Ising problem. J. Austral. Math. Soc. 12, 365–377 (1971)
Kasteleyn P.W.: Graph theory and crystal physics. In: Harary , F. (eds) Graph Theory and Theoretical Physics, pp. 43–110. Academic Press, London (1967)
Montroll E.W.: Lattice statistics. In: Beckenbach, E.E. (eds) Applied Combinatorial Mathematics, pp. 96–143. Wiley, New York (1964)
Feynman R.: Statistical Mechanics, A Set of Lectures. W.A. Benjamin Inc., Reading, MA (1972)
Litvin D.B.: The icosahedral point groups. Acta Cryst. A 47, 70–73 (1991)
Boyle L.L., O zgo Z.: Icosahedral irreducible tensors and their applications. Int. J. Quantum Chem. 7, 383–404 (1973)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Rosenfeld, V.R., Klein, D.J. Enumeration of substitutional isomers with restrictive mutual positions of ligands: I. Overall counts. J Math Chem 51, 21–37 (2013). https://doi.org/10.1007/s10910-012-0056-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10910-012-0056-0