From Wiener Index to Molecules

In this paper we present an algorithm for the generation of molecular graphs with a given value of the Wiener index. The high number of graphs for a given value of the Wiener index is reduced thanks to the application of a set of heuristics taking into account the structural characteristics of the molecules. The selection of parameters as the interval of values for the Wiener index, the diversity and occurrence of atoms and bonds, the size and number of cycles, and the presence of structural patterns guide the processing of the heuristics generating molecular graphs with a considerable saving in computational cost. The modularity in the design of the algorithm allows it to be used as a pattern for the development of other algorithms based on different topological invariants, which allow for its use in areas of interest, say as involving combinatorial databases and screening in chemical databases.

[1]  H. Wiener.  Structural determination of paraffin boiling points. , 1947, Journal of the American Chemical Society.

[2]  Julius T. Tou,et al.  Information Systems , 1973, GI Jahrestagung.

[3]  A. Balaban.  Chemical applications of graph theory , 1976 .

[4]  Danail Bonchev,et al.  Generalization of the Graph Center Concept, and Derived Topological Centric Indexes , 1980, J. Chem. Inf. Comput. Sci..

[5]  R. W. Robinson,et al.  Determination of the Wiener molecular branching index for the general tree , 1985 .

[6]  Gerta Rücker,et al.  Computer perception of constitutional (topological) symmetry: TOPSYM, a fast algorithm for partitioning atoms and pairwise relations among atoms into equivalence classes , 1990, J. Chem. Inf. Comput. Sci..

[7]  I. Gutman.  A new method for the calculation of the Wiener number of acyclic molecules , 1993 .

[8]  I. Gutman,et al.  Some recent results in the theory of the Wiener number , 1993 .

[9]  István Lukovits,et al.  On the Definition of the Hyper-Wiener Index for Cycle-Containing Structures , 1995, J. Chem. Inf. Comput. Sci..

[10]  N. Trinajstic,et al.  The Wiener Index: Development and Applications , 1995 .

[11]  John Figueras,et al.  Ring Perception Using Breadth-First Search , 1996, J. Chem. Inf. Comput. Sci..

[12]  I. Lukovits.  The Detour Index , 1996 .

[13]  Ernesto Estrada,et al.  A Topological Index Based on Distances of Edges of Molecular Graphs , 1996, J. Chem. Inf. Comput. Sci..

[14]  Reinhard Diestel,et al.  Graph Theory , 1997 .

[15]  J. Gross,et al.  Graph Theory and Its Applications , 1998 .

[16]  Ernesto Estrada,et al.  Extension of Edge Connectivity Index. Relationships to Line Graph Indices and QSPR Applications , 1998, J. Chem. Inf. Comput. Sci..

[17]  P. Kollman,et al.  Encyclopedia of computational chemistry , 1998 .

[18]  Bruno R. Preiss,et al.  Data Structures and Algorithms with Object-Oriented Design Patterns in Java , 1999 .

[19]  Ivar Jacobson,et al.  The unified modeling language reference manual , 2010 .

[20]  A. Balaban,et al.  Topological Indices and Related Descriptors in QSAR and QSPR , 2003 .

[21]  N. Trinajstic,et al.  The Detour Matrix and the Detour Index , 2000 .

[22]  Padmakar V. Khadikar,et al.  A Comparative QSAR Study Using Wiener, Szeged, and Molecular Connectivity Indices , 2000, J. Chem. Inf. Comput. Sci..

[23]  Harry P. Schultz.  Topological Organic Chemistry. 13. Transformation of Graph Adjacency Matrixes to Distance Matrixes , 2000, J. Chem. Inf. Comput. Sci..

[24]  Douglas J. Klein,et al.  Mean Wiener Numbers and Other Mean Extensions for Alkane Trees , 2000, J. Chem. Inf. Comput. Sci..

[25]  O. Ivanciuc,et al.  Matrices and Structural Descriptors Computed from Molecular Graph Distances , 2000 .

[26]  Ovidiu Ivanciuc,et al.  QSAR Comparative Study of Wiener Descriptors for Weighted Molecular Graphs , 2000, J. Chem. Inf. Comput. Sci..

[27]  A. Balaban,et al.  Reverse Wiener Indices , 2000 .

[28]  Wolfgang Linert,et al.  The Multiplicative Version of the Wiener Index , 2000, J. Chem. Inf. Comput. Sci..

[29]  A. Balaban,et al.  Vertex- and Edge-Weighted Molecular Graphs and Derived Structural Descriptors , 2000 .

[30]  O. Balaban.  The Graph Description of Chemical Structures , 2000 .

[31]  Danail Bonchev,et al.  The Overall Wiener Index-A New Tool for Characterization of Molecular Topology , 2001, J. Chem. Inf. Comput. Sci..

[32]  I. Gutman,et al.  Wiener Index of Trees: Theory and Applications , 2001 .

[33]  István Lukovits,et al.  Distance-Related Indexes in the Quantitative Structure-Property Relationship Modeling , 2001, J. Chem. Inf. Comput. Sci..

[34]  Milan Randic,et al.  On Interpretation of Well-Known Topological Indices , 2001, J. Chem. Inf. Comput. Sci..

[35]  Douglas J. Klein,et al.  Wiener Index Extension by Counting Even/Odd Graph Distances , 2001, J. Chem. Inf. Comput. Sci..

[36]  Douglas J. Klein,et al.  Computing Wiener-Type Indices for Virtual Combinatorial Libraries Generated from Heteroatom-Containing Building Blocks , 2002, J. Chem. Inf. Comput. Sci..

[37]  Douglas J. Klein,et al.  Building-block Computation of Wiener-type Indices for the Virtual Screening of Combinatorial Libraries* , 2002 .

[38]  Joel R. Bock,et al.  A New Method to Estimate Ligand-Receptor Energetics* , 2002, Molecular & Cellular Proteomics.

[39]  D. Klein,et al.  On the wiener number of thorn trees, stars, rings, and rods , 2002 .

[40]  I. Gutman,et al.  Wiener Index of Hexagonal Systems , 2002 .

[41]  Irene Luque Ruiz,et al.  Representation of the Molecular Topology of Cyclical Structures by Means of Cycle Graphs. 3. Hierarchical Model of Screening of Chemical Databases , 2004, J. Chem. Inf. Model..

[42]  I. Gutman,et al.  A class of modified Wiener indices , 2004 .

[43]  I. Gutman.  A new hyper-Wiener index , 2004 .

[44]  Evgenii A. Smolenskii,et al.  The Wiener Distance Matrix for Acyclic Compounds and Polymers , 2004, J. Chem. Inf. Model..

[45]  I. L. Ruiz,et al.  Representation of the Molecular Topology of Cyclical Structures by Means of Cycle Graphs. Part 3. Hierarchical Model of Screening of Chemical Databases. , 2005 .