Reciprocal spanning-tree density: A new index for characterising the intricacy of a (poly)cyclic molecular-graph

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Volumn , Issue 48, 2003, Pages 97-116

  Mallion, R B ^a   Trinajstic N ^b  

^a King's School   (United Kingdom)

^b RUDER BO KOVI INSTITUTE   (Croatia)

Author keywords

[No Author keywords available]

Indexed keywords

ARTICLE; CHEMICAL STRUCTURE; DENSITY; ENTROPY; MATHEMATICAL COMPUTING; MATHEMATICS; MOLECULAR DYNAMICS; QUANTITATIVE ANALYSIS;

EID: 0142231843     PISSN: 03406253     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (91)

1. Waldrop, M. M. Complexity; Touchstone/Simon & Schuster: New York, 1992.
2. Bertz, S.H. The First General Index of Molecular Complexity. J. Am. Chem Soc. 1981, 103, 3599-3601.
3. Nikolić, S.; Trinajstić, N.; Tolić, I.M. Complexity of Molecules. J. Chem. Inf. Comput. Sci. 2000, 40, 920-926.
4. Randić, M. On Complexity of Transitive Graphs Representing Degenerate Arrangements. Croat. Chem. Acta 2001, 74, 683-705. See also Randić, M.; Plavšić, D. On the Concept of Molecular Complexity. Croat. Chem. Acta 2002, 75, 107-116.
5. Randić, M. On Complexity of Transitive Graphs Representing Degenerate Arrangements. Croat. Chem. Acta 2001, 74, 683-705. See also Randić, M.; Plavšić, D. On the Concept of Molecular Complexity. Croat. Chem. Acta 2002, 75, 107-116.
6. Barone, R.; Chanon, M. A New and Simple Approach to Chemical Complexity: Application to the Synthesis of Natural Products. J. Chem. Inf. Comput. Sci. 2001, 41, 269-272.
7. Gutman, I.; Rücker, C.; Rücker, G. On Walks in Molecular Graphs. J. Chem. Inf. Comput. Sci. 2001, 41, 739-745.
8. Mowshowitz, A. Entropy and the Complexity of Graphs. I. An Index of the Relative Complexity of a Graph. Bull. Math. Biophys. 1968, 30, 175-204.
9. Biggs, N. L. Algebraic Graph Theory; Cambridge University Press: Cambridge (England, United Kingdom), 1974, p. 38.
10. Runge, F.; Sachs, H. Berechnung der Anzahl der Gerüste von Graphen und Hypergraphen mittels deren Spektren. Mathematica Balkanica 1974, 4, 529-536.
11. Sachs, H. On the Number of Spanning Trees. In Proceedings of the Fifth British Combinatorial Conference, Aberdeen, 1975; Nash-Williams, C, St. J. A.; Sheehan, J. Eds.; Winnipeg (Canada), 1976, pp. 529-535.
12. Minoli, D. Combinatorial Graph Complexity. Atti Acad. Naz. Lincei Rend. Cl. Sci. Fiss. Mat. Natur. (Ser. 8) 1975, 59, 651-661.
13. rd edition; Johann Ambrosius Barth: Heidelberg & Leipzig, 1995.
14. Kaye, B. Chaos & Complexity; VCH: Weinheim (Germany), 1993.
15. Gutman, I.; Mallion, R. B.; Essam, J. W. Counting the Spanning Trees of a Labelled Molecular-Graph. Mol. Phys. 1983, 50, 859-877.
16. Brown, T. J. N.; Mallion, R. B.; Pollak, P.; de Castro, B. R. M.; Gomes, J. A. N. F. The Number of Spanning Trees in Buckminsterfullerene. J. Comput. Chem. 1991, 12, 1118-1124.
17. Gutman, I; Mallion, R. B. On Spanning Trees in Catacondensed Molecules. Z. Naturforsch. 1993, 48a, 1026-1030.
18. Kirby, E. C.; Mallion, R. B.; Pollak, P. On the Question of Counting Spanning Trees in Labelled Nonplanar Graphs. Mol. Phys. 1994, 83, 599-602.
19. John, P. E.; Mallion, R. B. An Algorithmic Approach to the Number of Spanning Trees in Buckminsterfullerene. J. Math. Chem. 1994, 15, 261-271. (See also Corrigendu'm: idem ibid. 1994, 16, 389-390).
20. Trinajstić, N.; Nikolić, S.; Mihalić, Z. On the Complexity of the Platonic Solids. Bull. Chem. Technol. Macedonia 1994, 13, 61-68.
21. Brown, T. J. N.; Mallion, R. B.; Pollak, P.; Roth, A. Some Methods for Counting the Spanning Trees in Labelled Molecular Graphs, Examined in Relation to Certain Fullerenes. Discrete Appl. Math. 1996, 67, 51-66.
22. John, P. E.; Mallion, R. B. Calculating the Number of Spanning Trees in a Labeled Planar Graph Whose Inner Dual is a Tree. Int. J. Quantum Chem. 1996, 60, 59-66.
23. Nikolić, S.; Trinajstić, N.; Jurić, A.; Mihalić, Z.; Krilov, G. Complexity of Some Interesting (Chemical) Graphs. Croat. Chem. Acta 1996, 69, 883-897.
24. John, P. E.; Mallion, R. B.; Gutman, I. An Algorithm for Counting Spanning Trees in Labeled Molecular Graphs Homeomorphic to Cata-Conjugated Systems. J. Chem. Inf. Comput. Sci. 1998, 38, 108-112.
25. Bonchev, D.; Rouvray, D. H., Eds.; Chemical Complexity; Taylor & Francis: London, in press.
26. Bonchev, D.; Temkin, O. N.; Kamenski, D. On the Complexity of Linear Reaction Mechanisms. React. Kinet. Catal. Lett. 1980, 15, 119-124.
27. Bonchev, D.; Kamenski, D.; Temkin, O. N. Complexity Index for the Linear Mechanisms of Chemical Reactions. J. Math. Chem. 1987, 1, 345-388.
28. st edition; Oliver & Boyd: Edinburgh, 1972.
29. nd edition; CRC Press: Boca Raton (Florida, United States of America), 1992.
30. n) Représentant les Hydrocarbures Annulaires. Bull. Soc. Chim. France 1974, 2799-2800.
31. Harary, F. Graph Theory; Addison-Wesley: Reading (Massachusetts, United States of America), 1969.
32. Gutman, I.; Polansky, O. E.; Mathematical Concepts in Organic Chemistry; Springer-Verlag: Berlin (West), 1986.
33. Graph Theory and Topology; King, R.B.; Rouvray, D.H., Eds.; Elesevier: Amsterdam, 1987.
34. Sylvester, J. J. On the Change of Systems of Independent Variables. Quart. J. Math. 1857, 1, 42-56 & 126-134.
35. Borchardt, C. W. Ueber eine der Interpolation entsprechende Darstellung der Eliminations-Resultante. J. reine angewandte Math. ("Borchardt' s Journal") 1860, 57, 111-121.
36. Cayley, A. A Theorem on Trees. Quart. J. Pure Appl. Math. 1889, 23, 376-378.
37. 5)={(1/12) (5-3) (54)}= {(1/6)} = 1. The above formula is discussed, for instance, in ref. 30 (p. 118) and in ref. 27 (p. 71), and is proved in ref. 37 formula is discussed, for instance, in ref. 30 (p. 118) and in ref. 27 (p. 71), and is proved in ref. 37.
38. Harris, B., Ed.; Graph Theory and its Applications; Academic Press: New York, 1970.
39. See, for example, p. 64 of ref. 27.
40. Kuratowski, K. Sur le Problème des Courbes Gauches en Topologie. Fund. Math. 1930, 15, 271-283.
41. Klein, D. J.; Kirby, E. C.; Mallion, R. B.; Pollak, P.; Sachs. H. A Theorem for Counting Spanning Trees in General Chemical Graphs and its Particular Application to Toroidal Fullerenes. Manuscript in Preparation.
42. See p. 219 of ref. 12.
43. Dudeney, H. E. Amusements in Mathematics; Nelson: London, 1917.
44. Biggs, N. L.; Lloyd, E. K.; Wilson, R. J. Graph Theory 1736-1936; Clarendon Press: Oxford (England, United Kingdom), 1976.
45. 3,3) = {(1/4) (3-2) (3-2)} = {(1/4)} = 1.
46. See p. 72 of ref. 27.
47. Petersen, J. Sur le Théorème de Tait, L'Intermé diaire des Mathématiciens 1898, 5, 225-227.
48. Tait, P. G. Note on a Theorem in the Geometry of Position. Trans. Roy. Soc. Edinburgh 1880, 29, 657-660.
49. See ref. 43, Chapter 10 (pp. 187-207).
50. See ref. 30, pp. 89-90.
51. Dunitz, J. D.; Prelog, V. Umordnung von Liganden an trigonal bipyramidalen Atomen. Angew. Chemie 1968, 80, 700-701.
52. King, R. B. Applications of Topology and Graph Theory in Understanding Inorganic Molecules. In From Chemical Topology to Three-Dimensional Geometry; Balaban, A. T., Ed.; Plenum Press: New York, 1997, pp. 343-414.
53. Wilson, R. J.; Watkins, J. J. Graphs: An Introductory Approach; John Wiley & Sons, Inc.: New York, 1990, p. 225.
54. th, 1999.
55. Blanuša, D. Problem četiriju boja (The Four-Colour Problem). Glasnik mat.-fiz. i astr., ser 2, 1946, 1, 31-42.
56. Rücker, G.; Rücker, C. Symmetry-Aided Computation of the Detour Matrix and the Detour Index. J. Chem. Inf. Comput. Sci. 1998, 38, 710-714.
57. Gielen, M. Applications of Graph Theory to Organometallic Chemistry. In Chemical Applications of Graph Theory; Balaban, A. T., Ed.; Academic Press: London, 1976, pp. 261-298.
58. Randić, M.; Klein, D. J.; Katović, V.; Oakland, D. O.; Seitz, W. A.; Balaban, A. T. Symmetry Properties of Chemical Graphs. X. Rearrangements of Axially Distorted Octahedra. In Graph Theory and Toplogy in Chemistry; King, R. B.; Rouvray, D. H. Eds., Elsevier, Amsterdam, 1987, pp. 266-284.
59. Klein, D. J.; Graovac, A.; Mihalić, Z.; Trinajstić, N. Excitation Spectra for Degenerate Rearrangements. J. Mol. Struct. (Theochem) 1995, 341, 157-164.
60. Živković, T. P. Bullvalene Reaction Graph. Croat. Chem. Acta 1996, 69, 215-222.
61. Alkorta, L.; Elguero, J.; Rozas, I.; Balaban, A. T. Theoretical Studies of Aza Analogues of Platonic Hydrocarbons. Part I. Cubane and its Aza Derivatives. J. Mol. Struct. (Theochem) 1990, 206, 67-75.
62. Alkorta, L.; Elguero, J.; Rozas, I.; Balaban, A. T. Theoretical Studies of Aza Analogues of Platonic Hydrocarbons. Part II. Tetrahedrane and its Aza Derivatives. J. Mol. Struct. (Theochem) 1990, 208, 63-77.
63. Alkorta, L.; Elguero, J.; Rozas, I.; Balaban, A. T. Theoretical Studies of Aza Analogues of Platonic Hydrocarbons. Part III. Dodecahedrane and its Aza Derivatives. J. Mol. Struct. (Theochem) 1991, 228, 47-60.
64. Lukovits, I.; Nikolić, S.; Trinajstić, N. Resistance Distance in Regular Graphs. Int. J. Quantum Chem. 1999, 71, 217-225.
65. Ball, D. W. On the ΔHf-Values of Tetrahedrane and Cubane: Density Functional Theory Calculations. J. Mol. Struct. (Theochem) 1996, 364, 183-188.
66. Trinajstić, N.; Nikolić, S.; Babić, D.; Mihalić, Z. The Vertex- and Edge-Connectivity Indices of Platonic and Archimedian Molecules. Bull. Chem. Technol. Macedonia 1997, 16, 43-51.
67. King, R. B. Unusual Permutation Groups in Negative Curvature Carbon and Boron Nitride Structures. Croat. Chem. Acta 2000, 73, 993-1015.
68. de la Vassière, D.; Fowler, P. W.; Deza, M. Codes of Platonic, Archimedian, Catalan, and Related Polyhedra: a Model for Maximum Addition Patterns in Chemical Cages. J. Chem. Inf. Comput. Sci. 2001, 41, 376-386.
69. The cyclomatic number, μ, of a polycyclic graph, G, is the minimum number of edges that have to be removed from G in order to convert it into an acyclic graph. The cyclomatic number is given by the expression μ = e - v + 1.
70. 60.
71. Manolopoulos, D.; Fowler, P. W. Molecular Graphs, Point Groups and Fullerenes. J. Chem. Phys. 1992, 96, 7603-7614.
72. Liu, X.; Schmalz, T. G.; Klein, D. J. Favorable Structures for Higher Fullerenes. Chem. Phys. Letters 1992, 188, 550-554.
73. Manolopoulos, D. Comment on "Favorable Structures for Higher Fullerenes". Chem. Phys. Letters 1992, 192, 330-330.
74. Liu, X.; Schmalz, T. G.; Klein, D. J. Reply to 'Comment on "Favorable Structures for Higher Fullerenes"'. Chem. Phys. Letters 1992, 192, 331-331.
75. Babić, D.; Trinajstić, N. Pyracylene Rearrangement Classes of Fullerene Isomers. J. Comput. Chem. 1993, 17, 271-275.
76. Klein, D. J.; Liu, X. Elemental Carbon Isomerism. Int. J. Quantum Chem. 1994, S28, 501-523.
77. Randić, M. On Characterization of Cyclic Structures. J. Chem. Inf. Comput. Sci. 1997, 37, 1063-1071.
78. Babić, D.; Klein, D. J.; Lukovits, I.; Nikolić, N.; Trinajstić. N. Resistance-Distance Matrix: a Computational Algorithm and its Application. Int. J. Quantum Chem. In Press.
79. (a) Wiener, H. Structural Determination of Paraffin Boiling Points. J. Am. Chem. Soc. 1947, 69, 17-20
80. (b) Note that the Wiener index, W, is based on the distance matrix of a graph.
81. (a) Balaban, A. T. Highly Discriminating Distance-Based Topological Index. Chem. Phys. Letters 1982, 89, 399-404
82. (b) The Balaban index J is also based on the distance matrix of a graph, whilst the other Balaban index, Jω, is founded on the resistance-distance matrix.
83. The Wiener-sum index, WS, is derived from the 'quotient' matrix denoted "D/ω", which is obtained by dividing the off-diagonal elements of the distance matrix, D, by the corresponding elements of the resistance-distance matrix, ω. The diagonal elements in both matrices are zero, and the diagonal elements in the 'quotient' matrix are defined to be zero, too.
84. (a) Bonchev, D.; Balaban, A. T.; Liu, X.; Klein, D. J. Molecular Cyclicity and Centricity of Polycyclic Graphs. I. Cyclicity Based on Resistance Distances or Reciprocal Distances. Int. J. Quantum Chem. 1994, 50, 1-20
85. (b) The Kirchhoff index, Kf, is based on the resistance-distance matrix, whilst the Kirchhoff-sum index, KfS, depends on the 'quotient' matrix, "ω/D". The diagonal elements of the 'quotient' matrix are all zero, by definition.
86. Ivanciuc, O.; Balaban, A.T. Design of Topological Indices. Part 8. Path Matrices and Derived Molecular Graph Invariants. Comm. Math. CHEM. (MATCH) 1994, 30, 141-142.
87. Amić, D.; Trinajstić, N. On the Detour Matrix. Croat. Chem. Acta 1995, 68, 53-62.
88. Rouvray, D. H., Ed.; Fuzzy Logic in Chemistry; Academic Press: San Diego (California, United States of America), 1997.
89. Rouvray, D. H., Ed.; Concepts in Chemistry: a Contemporary Challenge; Research Studies Press: Taunton, Somerset (England, United Kingdom) and John Wiley & Sons Ltd.: Chichester (England, United Kingdom), 1997.
90. Rouvray, D. H. Are the Concepts of Chemistry all Fuzzy? Chapter 1 (pp. 1-15) of ref. 85.
91. Bonchev, D.; Seitz, W. A. The Concept of Complexity in Chemistry. Chapter 11 (pp. 353-381) of ref. 85.