# IJPAM: Volume 5, No. 3 (2003)

**ON CHARACTERISTIC POLYNOMIALS OF**

MOLECULAR GRAPHS WITH HETEROATOMS

MOLECULAR GRAPHS WITH HETEROATOMS

Institute of Algebra and Computational Mathematics

Vienna University of Technology

Wiedner Hauptstraße 8-10, A-1040 Wien, AUSTRIA

e-mail: d.dorninger@tuwien.ac.at

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
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 or vertices and edges of are
substituted. Exploiting these recursions, if 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