Molecular topology

General formulas for calculating the Wiener index (W)2 and the hyper-Wiener index (R)3 in spiro-graphs containing three- to six-membered rings are proposed. They are derived on the basis of Hosoya's formula4 and the Klein-Lukovitz-Gutman5 formula for evaluatingW andR, respectively, in cycle-containing graphs, by using the layer matrix of cardinality (LC).6 An extension of the Wiener number, theW* number of Gutman7 is also evaluated for these spiro-graphs.

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

[2]  H. Hosoya.  Topological Index. A Newly Proposed Quantity Characterizing the Topological Nature of Structural Isomers of Saturated Hydrocarbons , 1971 .

[3]  I. Gutman,et al.  Mathematical Concepts in Organic Chemistry , 1986 .

[4]  K. Balasubramanian.  Recent developments in tree-pruning methods and polynomials for cactus graphs and trees , 1990 .

[5]  Serge S. Tratch,et al.  Triangulanes: stereoisomerism and general method of synthesis , 1990 .

[6]  Bojan Mohar,et al.  A novel definition of the Wiener index for trees , 1993, J. Chem. Inf. Comput. Sci..

[7]  M. Randic.  Novel molecular descriptor for structure—property studies , 1993 .

[8]  Ovidiu Ivanciuc,et al.  Design of topological indices. Part 4. Reciprocal distance matrix, related local vertex invariants and topological indices , 1993 .

[9]  Ovidiu Ivanciuc,et al.  Chemical graphs with degenerate topological indices based on information on distances , 1993 .

[10]  Mircea V. Diudea.  Molecular Topology. 16. Layer Matrixes in Molecular Graphs , 1994, J. Chem. Inf. Comput. Sci..

[11]  Milan Randic,et al.  Wiener Matrix Invariants , 1994, Journal of chemical information and computer sciences.

[12]  Dejan Plavsic,et al.  The Laplacian matrix in chemistry , 1994, J. Chem. Inf. Comput. Sci..

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