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Internet Electronic Journal of Molecular Design - IEJMD, ISSN 1538-6414, CODEN IEJMAT
ABSTRACT - Internet Electron. J. Mol. Des. March 2003, Volume 2, Number 3, 160-178

On the Hosoya Z Index of General Graphs
Ante Milicevic, Sonja Nikolic, Dejan Plavsic, and Nenad Trinajstic
Internet Electron. J. Mol. Des. 2003, 2, 160-178

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Abstract:
The Z index, proposed by Haruo Hosoya in 1971, has initially been applied only to simple graphs representing saturated hydrocarbons. Our aim in this report was to extend the Z index to general graphs that may be used to represent unsaturated systems and heterosystems and to investigate the behavior of the Z index in regard to their structural characteristics such as size, branching, cyclicity, multiple edges and loops. Chemical graph-theoretical concepts were used. The Z index was calculated by means of the Z counting polynomial and the matching polynomial. These polynomials were constructed using the appropriate recurrence relations. The structural behavior of the Z index was tested against the total walk count (twc) index. The Z index was obtained for a number of simple graphs and general graphs. It is shown that the Z index of these graphs follows their structural changes, that is, the Z index increases with the size, loops, multiple edges, cycles and is sensitive to branching. The twc index supports in most cases the structural behavior of the Z index. The relationship between the Z counting polynomial and the matching polynomial is discussed. The edge decomposition of the Z index is also commented. The range of applicability of the Hosoya Z index is extended to general graphs. It is shown that this index accounts well for their structural characteristics.

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