|
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
|
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.
|