Mathematical Chemistry: (3,g)-Cages with Girth g, Topological Indices, and Other Graph-Theoretical Problems

Balaban, A.T.

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Tytuł artykułu

Mathematical Chemistry: (3,g)-Cages with Girth g, Topological Indices, and Other Graph-Theoretical Problems

Autorzy

Balaban A.T.

Wybrane pełne teksty z tego czasopisma

https://fi.episciences.org/

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Warianty tytułu

Języki publikacji

Abstrakty

This essay contains personal views about mathematical chemistry - mainly with non-numerical and non-quantum flavor. Similarities between mathematics and chemistry are pointed out, mainly based on the invention aspect of these two sciences, in contrast to discoveries characterizing most natural sciences. The author's involvement with chemical applications of graph theory needed a biographical background. Then five topics are reviewed: (1) enumerations of certain classes of chemical compounds initiated in collaboration with Professors Silviu Teleman and Frank Harary; (2) the (3,g)-cages with g = 10 and 11; (3) isoprenoid structures modeled in collaboration with Professor Solomon Marcus by cellular automata and picture grammars; (4) benzenoid polycyclic hydrocarbons described in collaboration with Professor Ioan Tomescu; and (5) topological indices.

Słowa kluczowe

graph theory g-cages isoprenoid structures topological index

Wydawca

IOS Press

Czasopismo

Fundamenta Informaticae

Rocznik

2005

Tom

Vol. 64, nr 1-4

Strony

1--16

Opis fizyczny

bibliogr. 78 poz., tab.

Twórcy

autor

Balaban A.T.

Texas A&M University at Galveston, marine Department, Galveston, TX, USA, balabana@tamug.edu

Bibliografia

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[2] A. Hassner and C. Stumer, Organic Syntheses Based on Name Reactions, 2nd ed. Pergamon, Amsterdam, 2002, p. 17.
[3] A.T. Balaban and C.D. Nenitzescu, Dehydrogenating condensations of aromatics (Scholl and related reactions), in Friedel-Crafts and Related Reactions, editor G.A. Olah, Wiley-Interscience, New-York, 1964, vol. 2, pp. 979–1047; C.D. Nenitzescu and A.T. Balaban, Aliphatic acylation, in the same book, vol. 3, pp. 1033–1152.
[4] A T. Balaban, Confessions and reflections of a graph-theoretical chemist. Math. Chem. (MATCH), 1993, 29, 3–17.
[5] A.T. Balaban, An attempt for the systematization of monocyclic aromatic compounds (Aromaticity I) (in Roumanian). Studii şi cercetări chim., Acad. R. P. Romania, 1959, 7, 257–295.
[6] A.T. Balaban and F. Harary, Chemical graphs. IV (Aromaticity. VI). Dihedral groups and monocyclic aromatic compounds. Rev. Roum. Chim., 1967, 12, 1511–1515.
[7] A.T. Balaban and F. Harary, Chemical graphs. V. Enumeration and proposed nomenclature of benzenoid cata-condensed polycyclic aromatic hydrocarbons. Tetrahedron, 1968, 24, 2505–2516.
[8] A.T. Balaban, R.O. Davies, F. Harary, A. Hill, and R. Westwick, Cubic identity graphs and planar graphs derived from trees. Austral. J. Math., 1970, 11, 207–215.
[9] A.T. Balaban, D. Farcasiu, and F. Harary, Chemical graphs. IX. Isotope isomerism of multiply-labelled compounds. J. Labelled Compounds, 1970, 6, 211–223.
[10] A.T. Balaban and F. Harary, The characteristic polynomial does not uniquely determine the topology of a molecule. J. Chem. Documentation, 1971, 11, 258–259.
[11] R.W. Robinson, F. Harary, and A.T. Balaban, The number of chiral and achiral alkanes and mono-substituted alkanes. Tetrahedron, 1976, 32, 355–361.
[12] A.T. Balaban, E.M. Palmer, and F. Harary, Chemical graphs. XXIX. Numbers of isomers with naphthalene and adamantane skeletons. Rev. Roum. Chim., 1977, 22, 517–523.
[13] A.T. Balaban, D. Farcasiu, and R. Bănică, Chemical graphs. II. Graphs of multiple 1,2-shifts in carbonium ions and related systems. Rev. Roum. Chim., 1966, 11, 1205–1227.
[14] A.T. Balaban, Chemical graphs. VIII. Valence isomerism and general cubic graphs. Rev. Roum. Chim., 1970, 15, 463–485.
[15] A.T. Balaban, M. Banciu, and V. Ciorba, Annulenes, Benzo-, Hetero-, Homo-Derivatives and Their Valence Isomers. CRC Press, Boca Raton, Florida, 1986, 3 volumes.
[16] A.T. Balaban, Ed., Chemical Applications of Graph Theory. Academic Press, London, 1976.
[17] A.T. Balaban, A trivalent graph of girth ten. J. Combinatorial Theory, Ser. B, 1972, 12, 1–5.
[18] M. O’Keefe and P.K. Wong, A smallest graph of girth 10 and valency 3. J. Comb. Theory, 1980, B 29, 91–105.
[19] N. Hartsfield and G. Ringel, Pearls in Graph Theory. A Comprehensive Introduction, 2nd Edition, Revised and Augmented. Academic Press, San Diego, 1990, 1994. Paperback edition, Dover Publ., Inc.,Mineola, NY 2003, pp. 80–81.
[20] A.T. Balaban, Trivalent graphs of girth nine and eleven, and relationships among cages. Rev. Roum. Math. Pures Appl., 1973, 18, 1033–1043.
[21] A.T. Balaban and D. Babic, Details about a (3,11)-cage on 112 vertices. Math. Reports (Bucharest), 2000, 2, 269–274.
[22] N.L. Biggs andM.J. Hoare, A trivalent graph with 58 vertices and girth 9. Discrete Math., 1980, 30, 299–301.
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[25] N. Biggs, Constructions for cubic graphs with large girth. The Electronic Journal of Combinatorics, 1998, 5 (1) 1–25.
[26] R.C. Read and R.J. Wilson, An Atlas of Graphs. Oxford Univ. Press, 1988, p. 272.
[27] A.T. Balaban, Chemical graphs. XIII. Combinatorial patterns. Rev. Roum. Math. Pure Appl., 1972, 17, 3–16.
[28] A.T. Balaban,M. Barasch, and S.Marcus, Computer programfor the recognition of acyclic regular isoprenoid structures. Math. Chem. (MATCH), 1979, 5, 239–261.
[29] A.T. Balaban, M. Barasch, and S. Marcus, Computer program for the recognition of standard isoprenoid structures. Math. Chem. (MATCH), 1980, 8, 215–268.
[30] A.T. Balaban, M. Barasch, and S. Marcus, Picture grammars in chemistry. Generation of acyclic isoprenoid structures. Math. Chem. (MATCH), 1980, 8, 193–213.
[31] M. Barasch, S. Marcus, and A.T. Balaban, Codification of acyclic isoprenoid structures using context-free grammars and push-down automata. Math. Chem. (MATCH), 1981, 12, 25–64.
[32] A.T. Balaban and I. Tomescu, Algebraic expressions for the number of Kekulé structures of isoarithmic cata-condensed polycyclic hydrocarbons.Math. Chem. (MATCH), 1983, 14, 155–182.
[33] A.T. Balaban and T. Tomescu, Chemical graphs. XLI. Numbers of conjugated circuits and Kekulé structures for zigzag catafusenes and _􀀀__-hexes; generalized Fibonacci numbers. Math. Chem. (MATCH), 1985, 17, 91–120.
[34] A.T. Balaban and I. Tomescu, Alternating 6-cycles in perfect matchings of graphs representing condensed benzenoid hydrocarbons.Discrete Appl.Math., 1988, 19, 5–16. Reprinted in Application of Graphs in Chemistry and Physics (J.W. Kennedy and L.W. Quintas, Eds.), North-Holland, Amsterdam, 1988, pp. 5–16.
[35] I. Tomescu and A.T. Balaban, Decomposition theorems for calculating the number of Kekulé structures in coronoids fused via perinaphthyl units. Math. Chem. (MATCH), 1989, 24, 289–389.
[36] A.T. Balaban, C. Artemi, and I. Tomescu, Algebraic expressions for Kekulé structure counts in non-branched regularly cata-condensed benzenoid hydrocarbons.Math. Chem. (MATCH), 1987, 22, 77–100.
[37] A.T. Balaban and C. Artemi, Mathematical modeling of polymers. I. Enumeration of non-redundant (irreducible) repeating sequences in stereoregular polymers, elastomers or in binary copolymers. Math. Chem. (MATCH), 1987, 22, 3–32.
[38] C. Artemi and A.T. Balaban, Mathematical modeling of polymers. II. Irreducible sequences in n-ary copolymers. Math. Chem. (MATCH), 1987, 22, 33–66.
[39] A.T. Balaban and C. Artemi, Mathematical modelling of polymers. 3. Enumeration and generation of repeating irreducible sequences in linear bi-, ter-, quater-and quinquenary copolymers and in stereoregular homopolymers.Macromol. Chem., 1988, 189, 863–870.
[40] A.T. Balaban, J.W. Kennedy, and L.V. Quintas, The number of alkanes having n carbons and a longest chain of length d. An application of a theorem of Polya. J. Chem. Educ., 1988, 65, 304–313.
[41] A.T. Balaban, Numerical and non-numerical methods in chemistry: present and future. Sigsam Bull., 1984, 18, (2), 29–30.
[42] A.T. Balaban, Symbolic computation and chemistry. In EUROCAL-85, Lecture Notes in Computer Science, No. 203 (B. Buchberger, Ed.), Springer, Berlin, 1985, pp. 68–79.
[43] A.T. Balaban, Solved and unsolved problems in chemical graph theory. Annals Discrete Math., 1993, 55, 109–126: Reprinted in Quo Vadis, Graph Theory (eds. J. Gimbel, J. W. Kennedy and L. V. Quintas), North Holland, Amsterdam, 1993.
[44] D.H. Rouvray and A.T. Balaban, Chemical applications of graph theory. In Applications of Graph Theory, (R.J. Wilson and L.W. Beineke, Eds.), Academic Press, London, 1979, pp. 177–221.
[45] D. Bonchev, O. Mekenyan, and A.T. Balaban, Algorithms for coding chemical compounds. In Mathematics and Computational Concepts in Chemistry (Internat. Symp. Dubrovnik) (N. Trinajstic, Ed.), Ellis Horwood-Wiley, Chichester, 1986, pp. 34–47.
[46] M. Randic, D.J. Klein, V. Katovic, D.O. Oakland, W.A. Seitz, and A.T. Balaban, Symmetry properties of chemical graphs X. Rearrangement of axially distorted octahedra. In Graph Theory and Topology in Chemistry (R. B. King and D. H. Rouvray, Eds.), Elsevier, Amsterdam, 1987, pp. 266–284 (Studies in Physical and Theoretical Chemistry, No. 51, Proc. Internat. Conf., Univ. of Georgia, 1987).
[47] A.T. Balaban, Numerical modelling of chemical structures: local graph invariants and topological indices. In Graph Theory and Topology in Chemistry (R.B. King and D.H. Rouvray, Eds.), Elsevier, Amsterdam, 1987, pp. 159–176 (Studies in Physical and Theoretical Chemistry, No. 51, Proc. Internat. Conf., Univ. of Georgia, 1987).
[48] A.T. Balaban, Prediction of physical properties from chemical structures. In Recent Advances in Chemical Information (H. Collier, Ed.), Royal Society of Chemistry, Special Publication No. 120, London, 1993, pp. 301–317.
[49] A.T. Balaban, Enumeration of isomers. In Chemical Graph Theory. Introduction and Fundamentals (D. Bonchev and D.H. Rouvray, Eds.), Abacus Press - Gordon and Breach, New York, 1991, pp. 177–234.
[50] A.T. Balaban, Reaction graphs. In Graph Theoretical Approaches to Chemical Reactivity (D. Bonchev and O. Mekenyan, Eds.), Kluwer Academic Publishers, Dordrecht, Netherlands, 1994, pp. 137–180.
[51] A.T. Balaban, Symmetry of graphs. In Chemical Group Theory. Introduction and Fundamentals (D. Bonchev and D.H. Rouvray, Eds.), Gordon and Breach Publishers, New York, 1994, pp. 155–208.
[52] A.T. Balaban, Symmetry in chemical structures and reactions. Comput. and Maths. with Applic., 1986, 12B, 999–1020. Reprinted in Symmetry Unifying Human Understanding (I. Hargittai, Ed.), Pergamon Press, New York, 1986, 999–1020.
[53] A.T. Balaban, Carbon and its nets. Computers and Maths. with Applic., 1989, 17, 397–416. Reprinted in Symmetry II (I. Hargittai, Ed.) Pergamon, Oxford, 1989, 397–416.
[54] A.T. Balaban, Theoretical investigation of carbon nets and molecules. In Theoretical Organic Chemistry (C. Parkanyi, Ed.), Elsevier, Amsterdam, 1998, pp. 381–404.
[55] A.T. Balaban, QSAR and computationalmethods in drug discovery. In Encyclopedia of Analytical Chemistry (R.A. Meyers, Ed.),Wiley, Chichester, 2000, vol. 8, pp. 7288–7311.
[56] O. Ivanciuc and A.T. Balaban, Graph theory in chemistry. In Encyclopedia of Computational Chemistry (P.v.R. Schleyer, N.L. Allinger, T. Clark, J. Gasteiger, P-A. Kollman, H.F. Schafer III, and P.R. Schreiner, Eds.), Wiley, Chichester, 1998, pp. 1169–1190.
[57] H. Wiener, Structural determination of paraffin boiling points. J. Am. Chem. Soc., 1947, 69, 17–20.
[58] H. Hosoya, Topological index. A newly proposed quantity characterizing the topological nature of structural isomers of saturated hydrocarbons. Bull. Chem. Soc. Japan, 1971, 44, 2332–2339.
[59] M. Randić, On characterization of molecular branching. J. Am. Chem. Soc., 1975, 97, 6609–6615.
[60] L.B. Kier and L.H. Hall, Connectivity in Chemistry and Drug Research. Academic Press, New York, 1976; Molecular Connectivity in Structure Activity Analysis. Wiley, London, 1986.
[61] A.T. Balaban, Highly discriminating distance-based topological index. Chem. Phys. Lett., 1982, 80, 3 99–404.
[62] A.T. Balaban, Topological indices based on topological distances in molecular graphs. Pure Appl. Chem., 1983, 55, 199–206.
[63] A.T. Balaban, Chemical graphs 48. Topological index 􀀀 for heteroatom-containing molecules taking into account periodicities of element properties. Math. Chem. (MATCH), 1986, 21, 115–122.
[64] A.T. Balaban and L.V. Quintas, The smallest graphs, trees, and 4-trees, with degenerate topological index 􀀀. Math. Chem. (MATCH), 1983, 14, 213–233.
[65] A.T. Balaban, N. Ionescu-Pallas, and T.S. Balaban, Asymptotic values of topological indics 􀀀 and 􀀀_ (average distance sum connectivities) for infinite acyclic and cyclic graphs. Math. Chem. (MATCH), 1985, 17, 121–146.
[66] G. Grassy, B. Calas, A. Yasri, R. Lahana, J.Woo, S. Yier,M. Kaczorek, R. Floc’h, and R. Buelow, Computerassisted rational design of immunosuppressive compounds. Nature Biotechnol., 1998, 16, 748–752.
[67] A. Thakur, M. Thakur, P.V. Khadikar, C.T. Supuran, and P. Sudele, QSAR study on benzensulphonamide carbonic anhydrase inhibitors: topological approach using Balaban index. Bioorg. Med. Chem., 2004, 12, 789–793.
[68] C.I. Bermudez, E.E. Daza, and E. Andrade, Characterization and comparison of Escherichia coli transfer RNAs by graph theory based on secondary structure. J. Theor. Biol., 1999, 197, 193–205.
[69] A.T. Balaban, I. Motoc, B. Donchev, and O. Mekenyan, Topologival indices for structure-activity correlations. In Steric Effects in Drug Design (M. Charton and I. Motoc, Eds.), Topics Curr. Chem., 1983, vol. 114, pp. 21–55, Springer, Berlin.
[70] A.T. Balaban, A personal view about topological indices for QSAR/QSPR. In QSAR/QSPR Studies byMolecular Descriptors (M. Diudea, Ed.), Huntington, New York, 2000, pp. 1–25.
[71] A.T. Balaban, A comparison between various topological indices, particularly index 􀀀 and Wiener’s. In Topology in Chemistry: Discrete Mathematics of Molecules (D.H. Rouvray and R.B. King, Eds.), Horwood Publishing Ltd., Chichester, 2002, pp. 89–112.
[72] C.S. Basak, D. Mills, B.D. Gute, G.D. Grunwald, and A.T. Balaban, Applications of topological indices in predicting property/bioactivity/toxicity of chemicals. In Topology in Chemistry: Discrete Mathematics of Molecules (D.H. Rouvray and R.B. King, Eds.), Horwood Publishing Ltd., Chichester, 2002, pp. 113–184.
[73] P.A. Filip, T.S. Balaban, and A.T. Balaban, A new approach for devising local graph invariants: derived topological indices with low degeneracy and good correlation ability. J. Math. Chem., 1987, 1, 61–83.
[74] A.T. Balaban and T.S. Balaban, New vertex invariants and topological indices of chemical graphs based on information on distances. J. Math. Chem., 1991, 8, 383–397.
[75] J. Devillers and A.T. Balaban, Eds., Topological Indices and Related Descriptors in QSAR and QSPR, with three chapters by the latter editor: A.T. Balaban and O. Ivanciuc, Historical development of topological indices (pp. 21–57); O. Ivanciuc and A.T. Balaban, The graph description of chemical structures (pp. 59–167); O. Ivanciuc, T. Ivanciuc, and A.T. Balaban, Vertex- and edge-weighted molecular graphs and derived structural descriptors (pp. 169–220). Gordon and Breach, The Netherlands, 2000.
[76] A.T. Balaban, Ed., From Chemical Topology to Three-Dimensional Geometry. Plenum Publishing Corporation, New York, 1997, with a chapter by the editor: A.T. Balaban, From chemical graphs to 3D modeling, pp. 1–24.
[77] A.T. Balaban, Applications of graph theory in chemistry. J. Chem. Inf. Comput. Sci., 1985, 25, 334–343.
[78] A.T. Balaban, Chemical graphs: looking back and a glimpse ahead. J. Chem. Inf. Comput. Sci., 1995, 35, 339–350.

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Bibliografia

Identyfikator YADDA

bwmeta1.element.baztech-article-BUS2-0005-0107