Closed-form formulas for the Zhang-Zhang polynomials of benzenoid structures: Prolate rectangles and their generalizations

Abstract

We show that the Zhang-Zhang (ZZ) polynomial of a benzenoid obtained by fusing a parallelogram M(m, n) with an arbitrary benzenoid structure ABC can be simply computed as a product of the ZZ polynomials of both fragments. It seems possible to extend this important result also to cases where both fused structures are arbitrary Kekulean benzenoids. Formal proofs of explicit forms of the ZZ polynomials for prolate rectangles Pr(m, n) and generalized prolate rectangles Pr ([m(1), m(2), ..., m(n)], n) follow as a straightforward application of the general theory, giving ZZ (Pr (m, n), x) = (1 + (1 x) . m)(n) and ZZ(Pr([m(1), m(2), ..., m(n)), n), x) = Pi(n)(k=1)(1 + (1 + x) . m(k)). (C) 2015 Elsevier B.V. All rights reserved.