IJPAM: Volume 5, No. 3 (2003)


Dietmar Dorninger$^1$, Helmut Länger$^2$
$^{1,2}$Institute of Algebra and Computational Mathematics
Vienna University of Technology
Wiedner Hauptstraße 8-10, A-1040 Wien, AUSTRIA
$^1$e-mail: d.dorninger@tuwien.ac.at
$^2$e-mail: h.laenger@tuwien.ac.at

Abstract.We consider molecular graphs whose vertices and edges are both weighted by real numbers corresponding to Coulomb and resonance integrals of the underlying chemical compounds. Given a molecular graph $G$ we derive recursive procedures to find the characteristic polynomials of graphs that are obtained when new vertices and edges carrying new weights are added to $G$ or vertices and edges of $G$ are substituted. Exploiting these recursions, if $G$ is a cycle or path, we obtain explicit formulas for the characteristic polynomials of the compounds that arise. Since the zeros of the characteristic polynomials approximately correspond to the energy values of electrons it is of interest to know about the influence of heteroatoms when added to a given compound or when atoms of a compound are substituted. For this end we determine factors of the characteristic polynomials that do not depend on the weights of the newly introduced atoms and their bonds which leads to the investigation of common divisors of Chebyshev polynomials of the first and second kind.

Received: February 2, 2003

AMS Subject Classification: 92E10, 05C50

Key Words and Phrases: molecular graph, characteristic polynomial, Chebyshev polynomial

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2003
Volume: 5
Issue: 3