Exploring the limits of graph invariant- and spectrum-based discrimination of (sub)structures

Journal of Chemical Information and Computer Sciences

Volumn 42, Issue 3, 2002, Pages 640-650

  Rücker, Christoph ^a   Rücker, Gerta ^a   Meringer, Markus ^a  

^a UNIVERSITY OF BAYREUTH   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

SUBSTRUCTURES;

COMPUTER SOFTWARE; EIGENVALUES AND EIGENFUNCTIONS; MOLECULAR STRUCTURE; TREES (MATHEMATICS); UNSATURATED COMPOUNDS;

HYDROCARBONS;

EID: 0036589085     PISSN: 00952338     EISSN: None     Source Type: Journal    
DOI: 10.1021/ci010121y     Document Type: Article

References (40)

1. (a) Bonchev, D. The Overall Wiener Index - A New Tool for Characterization of Molecular Topology. J. Chem. Inf. Comput. Sci. 2001, 41, 582-592.
2. (b) Bertz, S.H. Complexity of Molecules and their Synthesis. In Complexity in Chemistry; Bonchev, D., Rouvray, D.H., Eds.; Mathematical Chemistry Series, Vol. 7, in press.
3. (c) Varmuza, K.; Scsibrany, H. Substructure Isomorphism Matrix. J. Chem. Inf. Comput. Sci. 2000, 40, 308-313.
4. (d) Poroikov, V.V.; Filimonov, D.A.; Borodina, Yu.V.; Lagunin, A.A.; Kos, A. Robustness of Biological Activity Spectra Predicting by Computer Program PASS for Noncongeneric Sets of Chemical Compounds. J. Chem. Inf. Comput. Sci. 2000, 40, 1349-1355.
5. 1H 2D NMR Correlations. Fresenius J. Anal. Chem. 2001, 369, 709-714.
6. (b) Varmuza, K.; Penchev, P.N.; Scsibrany, H. Maximum Common Substructures of Organic Compounds Exhibiting Similar Infrared Spectra. J. Chem. Inf. Comput. Sci. 1998, 38, 420-427.
7. (c) Seebass, B.; Pretsch, E. Automated Compatibility Tests of the Molecular Formulas or Structures of Organic Compounds with Their Mass Spectra. J. Chem. Inf. Comput. Sci. 1999, 39, 713-717.
8. Pricker, G.; Rücker, C. Automatic Enumeration of Ali Connected Subgraphs. MATCH - Commun. Math. Comput. Chem. 2000, 41, 145-149.
9. Rücker, G.; Rücker, C. On Finding Nonisomorphic Connected Subgraphs and Distinct Molecular Substructures. J. Chem. Inf. Comput. Sci. 2001, 41, 314-320; 865.
10. Rücker, G.; Rücker, C. Substructure, Subgraph, and Walk Counts as Measures of the Complexity of Graphs and Molecules. J. Chem. Inf. Comput. Sci. 2001, 41, 1457-1462.
11. Balaban, A.T. Higly Discriminating Distance-Based Topological Index. Chem. Phys. Lett. 1982, 89, 399-404.
12. (a) Balaban, A.T.; Quintas, L.V. The Smallest Graphs, Trees, and 4-Trees with Degenerate Topological Index J. MATCH - Commun. Math. Comput. Chem. 1983, 14, 213-233.
13. (b) Razinger, M.; Chrétien, J.R.; Dubois, J.E. Structural Selectivity of Topological Indexes in Alkane Series. J. Chem. Inf. Comput. Sci. 1985, 25, 23-27.
14. (a) Bonchev, D.; Mekenyan, O.; Trinajstić, N. Isomer Discrimination by Topological Information Approach. J. Comput. Chem. 1981, 2, 127-148.
15. (b) Balasubramanian, K.; Basak, S.C. Characterization of Isospectral Graphs Using Graph Invariants and Derived Orthogonal Parameters. J. Chem. Inf. Comput. Sci. 1998, 38, 367-373.
16. Randić, M.; Vracko, M.; Nović, M. Eigenvalues as Molecular Descriptors. In Diudea, M.V., Ed.; QSPR/QSAR Studies by Molecular Descriptors; Nova Science Publ.: Huntington, NY, 2001.
17. Balaban, A.T.; Harary, F. The Characteristic Polynomial Does Not Uniquely Determine the Topology of a Molecule. J. Chem. Doc. 1971, 11, 258-259.
18. Cvetković, D.; Rowlinson, P.; Simić, S. Eigenspaces of Graphs; Cambridge University Press: Cambridge, 1997.
19. Schwenk, A.J. Almost All Trees are Cospectral. In New Directions in the Theory of Graphs; Harary, F., Ed.; Academic Press: New York, 1973; pp 275-307.
20. Mihalić, Z.; Veljan, D.; Amić, D.; Nikolić, S.; Plavsić, D.; Trinajstić, N. The Distance Matrix in Chemistry. J. Math. Chem. 1992, 11, 223-258.
21. Cvetković, D.M.; Doob, M.; Gutman, I.; Torgasev, A. Recent Results in the Theory of Graph Spectra; Elsevier: Amsterdam, 1988; p 128.
22. McKay, B.D. On the Spectral Characterisation of Trees. Ars Combinatorica 1977, 3, 219-232.
23. Cai, J.-Y.; Fürer, M.; Immerman, N. An Optimal Lower Bound on the Number of Variables for Graph Identification. Combinatorica 1992, 4, 389-410.
24. Shrikhande, S.S. On a Characterization of the Triangular Association Scheme, Ann. Math. Statist. 1959, 30, 39-47.
25. Weisfeiler, B. On Construction and Identification of Graphs; Lecture Notes in Mathematics No. 558, Springer: Berlin, 1976.
26. Mathon, R. Proc. 9th S.-E. Conf. Combinatorics, Graph Theory and Computing 1978, 499.
27. Read, R.C.; Wilson, R.J. An Atlas of Graphs; Clarendon Press: Oxford: 1998; pp 4 and 7.
28. Recall that cubane, cuneane, and octabisvalene are pentacyclic octanes (n = 8, m = 12, c = 5). A heptacyclic octane (n = 8, m = 14, c = 7) would require two additional bonds in a cubane skeleton, for example.
29. Kerber, A.; Lane, R.; Grüner, T.; Meringer, M. MOLGEN 4.0. MATCH - Commun. Math. Comput. Chem. 1998, 37, 205-208.
30. J and eigenvalues were calculated as double precision numbers, for comparisons eight decimal places and seven decimal places were used for J and eigenvalues, respectively, for reasons detailed in ref 4.
31. 1,5]heptane.
32. In fact, this was not so surprising, after these graphs could not be pairwise distinguished using J and the distance eigenvalues.
33. (a) Hosoya, H.; Nagashima, U.; Hyugaji, S. Topological Twin Graphs. Smallest Pair of Isospectral Polyhedral Graphs with Eight vertices. J. Chem. Inf. Comput. Sci. 1994, 34, 428-431.
34. (b) Hosoya, H.; Ohta, K.; Satomi, M. Topological Twin Graphs II. Isospectral Polyhedral Graphs With Nine and Ten vertices. MATCH - Commun. Math. Comput. Chem. 2001, 44, 183-200.
35. 5,8]-octane.
36. If index values are compared in a sample of several or all edge numbers m within a constant vertex number n, i.e., without prior sorting by m, then additional degeneracies will occur. For instance, cyclooctane (n = 8, m = 8) and cubane (n = 8, m = 12) share the J value 2.000. Since NIMSG always sorts by n and m, we are not interested in such degeneracies here. It is well-known that graphs of different n can share the same J value, e.g. cyclohexane (n = 6, m = 6) also has J = 2.000.
37. Initially we were concerned that J and the distance spectrum, both being derived from the distance matrix, might tend to exhibit degeneracies for the same pairs of graphs. Fortunately, as foreseen already from the results shown in Figures 1 and 2, such concerns did net materialize to a large extent. However, the moderate improvement in the resolution of the distance spectrum on addition of J compared to the large improvement in the resolution of the adjacency spectrum on addition of J (Tables 15/16) may be interpreted to be partially due to such an effect.
38. The lower homologues of 13 and 14 lacking the methyl group are neither J-equivalent nor isospectral.
39. Grüner, T. Program GRADPART; Diploma Thesis, University of Bayreuth, 1995.
40. Balaban, A.T. Topological Index J for Heteroatom-Containing Molecules Taking into Account Periodicities of Element Properties. MATCH - Commun. Math. Comput. Chem. 1986, 21, 115-122.