Zhang-Zhang Polynomials of Regular 3-and 4-tier Benzenoid Strips

Abstract

We present compact, closed-form expressions for Zhang-Zhang (ZZ) polynomials of regular 3- and 4-tier benzenoid strips. It is possible to unify the ZZ polynomials of 11 classes of regular 3- and 4-tier benzenoid strips into a single, universal, three-parameter formula Sigma(Cl)(k=0) Sigma(2)(l=0) a(l) (n +1(k)) (n - l + Cl - k n-1)x(k) where Cl is an element of{2,3,4,5,6}, a(0) = 1, a(1) is an element of{0,1,2,3}, and a(2) is an element of{0,1}. The parameters a(1) and a(2) partition the 3- and 4-tiers benzenoid strips into four superfamilies; a(1) and a(2) are constant within a given superfamily and Cl enumerates subsequent benzenoid structures. Our finding provides also a compact and universal expression for the number of Kekule structures for regular 3- and 4-tier benzenoid strips given by K = Sigma(2)(l=0) a(l)(n - l + Cl Cl) These expressions are expected to be readily applicable also to wider regular benzenoid strips.