Original scientific paper

Acyclic and Characteristic Polynomial of Regular Conjugated Polymers and Their Derivatives

Ante Graovac ; Max-Planck-Institut fur Strahlenchemie, Stiftstr, 34-36, D-4330 Millheim a. d. Ruhr, F. R. Germany
Oskar E. Polansky ; Max-Planck-Institut fur Strahlenchemie, Stiftstr, 34-36, D-4330 Millheim a. d. Ruhr, F. R. Germany
Nikolay N. Tyutyulkov ; Max-Planck-Institut fur Strahlenchemie, Stiftstr, 34-36, D-4330 Millheim a. d. Ruhr, F. R. Germany

Abstract

A method to study the acyclic and characteristic polynomial
of regular conjugated polymers is described.
For a regular polymer with l bonds linking the monomer
units, one first builds a 2'X21 polynomial matrix T1. Its matrix
elements are acyclic polynomials of the monomer unit graph and
its subgraphs obtained by successive deletion of atoms serving as
the linking sites. The acyclic polynomials of the fasciagraph (representing
an open polymeric chain) and some of its subgraphs are
then obtained as the appropriate matrix elements of Ti" where
M stands for the degree of polymerization of the polymer under
consideration. For the rotagraph (representing the polymeric chain
closed on itself) the acyclic polynomial equal the trace of T1".
It is proved that the acyclic polynomials of regular polymers
and some of their derivatives satisfy recursion formulae of the
same form which contain 21 + 1 terms. The coefficients appearing
in the recursion are derived only from the knowledge of the
matrix Ti and are, therefore, independent of M.
As far as the characteristic polynomial of a regular polymer
is concerned, here we apply an analogon of the Ti-formalism only
for the special case of l = 1 and reproduce an already known
recursion formula. However, a new determinantal representation
of the characteristic polynomial of a polymer as well as its
explicit expression in terms of the characteristic polynomials of
monomer graph and its subgraphs is established for this special
case.

Keywords

Hrčak ID:

194203

URI

https://hrcak.srce.hr/194203

Publication date:

1.10.1983.

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