On the Relation between the Path Numbers 1Z, 2Z and the Hosoya Z Index

The path numbers of the mth order, mZ, of a connected undirected graph are introduced and studied analitically. The definition of the 1Z and 2Z indices for acyclic graphs proposed earlier is generalized in order to cover cyclic graphs. The relationships between the path numbers 1Z and 2Z and the Hosoya Z index are discussed. It is shown that the path number 1Z and the Hosoya Z index are closely related graph-theoretical invariants. The information contents of the 1Z and 2Z indices are only slightly overlapped, and hence the indices could be used as “independent” predictor variables in structure−property−activity studies.

[1]  Xiaofeng Guo,et al.  Wiener matrix: Source of novel graph invariants , 1993, J. Chem. Inf. Comput. Sci..

[2]  Milan Randić,et al.  In search of structural invariants , 1992 .

[3]  Dejan Plavšić,et al.  On the z-counting polynomial for edge-weighted graphs , 1992 .

[4]  Haruo Hosoya,et al.  Topological Index and Thermodynamic Properties. III. Classification of Various Topological Aspects of Properties of Acyclic Saturated Hydrocarbons , 1985 .

[5]  Lowell H. Hall.  Computational aspects of molecular connectivity and its role in structure-property modeling , 1990 .

[6]  Paul G. Seybold,et al.  Molecular structure: Property relationships , 1987 .

[7]  Milan Randic.  Chemical structure—What is "she"? , 1992 .

[8]  Ivan Gutman,et al.  A Regularity for the Boiling Points of Alkanes and Its Mathematical Modeling , 1986 .

[9]  M. Randic,et al.  Resistance distance , 1993 .

[10]  R. G. Woolley.  On the description of high-resolution experiments in molecular physics , 1976 .

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

[12]  Ivan Gutman.  SOME TOPOLOGICAL PROPERTIES OF BENZENOID SYSTEMS , 1975 .

[13]  Haruo Hosoya,et al.  Topological Index and Thermodynamic Properties. II. Analysis of the Topological Factors on the Absolute Entropy of Acyclic Saturated Hydrocarbons , 1980 .

[14]  N. Zefirov,et al.  Combinatorial models and algorithms in chemistry. The expanded Wiener number—a novel topological index , 1990 .

[15]  I. Gutman,et al.  Graph theory and molecular orbitals. Total φ-electron energy of alternant hydrocarbons , 1972 .

[16]  Milan Randic.  Similarity based on extended basis descriptors , 1992, J. Chem. Inf. Comput. Sci..

[17]  J. Aihara,et al.  A generalized total .pi.-energy index for a conjugated hydrocarbon , 1976 .

[18]  Haruo Hosoya,et al.  Topological Index and Thermodynamic Properties. I. Empirical Rules on the Boiling Point of Saturated Hydrocarbons , 1972 .

[19]  Dejan Plavšić,et al.  The Distance Matrix in Chemistry , 1993 .

[20]  Peter Zinn,et al.  List Operations on Chemical Graphs, 6. Comparative Study of Combinatorial Topological Indexes of the Hosoya Type , 1995, J. Chem. Inf. Comput. Sci..

[21]  Haruo Hosoya,et al.  Operator technique for obtaining the recursion formulas of characteristic and matching polynomials as applied to polyhex graphs , 1983 .

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

[23]  R. G. Woolley,et al.  Must a molecule have a shape , 1978 .

[24]  Edward C. Kirby.  Sensitivity of Topological Indexes to Methyl Group Branching in Octanes and Azulenes, or What Does a Topological Index Index? , 1994, J. Chem. Inf. Comput. Sci..

[25]  Dennis H. Rouvray,et al.  The modeling of chemical phenomena using topological indices , 1987 .

[26]  Milan Randić,et al.  Generalized molecular descriptors , 1991 .

[27]  Haruo Hosoya,et al.  Graphical enumeration of the coefficients of the secular polynomials of the Hückel molecular orbitals , 1972 .

[28]  Ivan Gutman,et al.  A topological index for the totalπ-electron energy , 1975 .

[29]  Jerry Ray Dias.  Properties and derivation of the fourth and sixth coefficients of the characteristic polynomial of molecular graphs—New graphical invariants , 1985 .

[30]  Nicholas J. Turro,et al.  Geometric and Topological Thinking in Organic Chemistry , 1986 .

[31]  Haruo Hosoya,et al.  Topological Index and Thermodynamic Properties. IV. Size Dependency of the Structure-Activity Correlation of Alkanes , 1988 .

[32]  Milan Randić.  Hosoya Matrix - A Source of New Molecular Descriptors , 1994 .