Zhang-Zhang Polynomials of Phenylenes and Benzenoid Graphs

Abstract

The aim of this paper is to study some variations of the Zhang-Zhang polynomial for phenylenes, which can be obtained as special cases of the multivariable Zhang-Zhang polynomial. Firstly, we prove the equality between the first Zhang-Zhang polynomial of a phenylene and the generalized Zhang-Zhang polynomial of some benzenoid graph, which enables us to prove also the equality between the first Zhang-Zhang polynomial and the generalized cube polynomial of the resonance graph. Next, some results on the roots of the second Zhang-Zhang polynomial of phenylenes are provided and another expression for this polynomial is established. Finally, we give structural interpretation for (partial) derivatives of different Zhang-Zhang polynomials.