Characteristic polynomials of alternant edge weighted linear chains with subsequent application to some linear poly (p-phenylene) graphs

Abstract

The algorithms for the calculation of the characteristic polynomial coefficients for the alternant edge weighted graphs of linear chains have been developed. These algorithms have been utilized to calculate the characteristic polynomials of the linear poly (p-phenylene) and the methylene substituted linear poly (p-phenylene) compounds. These compounds are found to be important materials for electro-optical and electronic applications. The recurrence relation to obtain the sum of the CP coefficients \({\left({S_{\rm CP}^{N_r }}\right)}\) of any such poly (p-phenylene) in terms of the respective sums of its lower analogs has been derived. The number of Kekule valence structures (K) for such molecular graphs have been presented. Excellent linear correlations of logarithm of sum of CP coefficients \(({\rm log} \, {S_{\rm CP}^{N_r }})\) and logarithm of Kekule valence structures count (log K) with the ambient conditions density and bulk modulus of linear poly (p-phenylene) have also been found.