Hexagonal flakes as fused parallelograms: A determinantal formula for Zhang-Zhang polynomials of the O(2, m, n) benzenoids

We report a determinantal formula for the Zhang-Zhang polynomial of the hexagonal flake O(2, m, n) applicable to arbitrary values of the structural parameters m and n. The reported equation has been discovered by extensive numerical experimentation and is given here without a proof. Our combinatorial analysis performed on a large collection of isostructural O(2, m, n) benzenoids yielded a ZZ polynomial formula corresponding to the determinant of a certain 2 × 2 matrix referred to by us as the generalized John–Sachs path matrix, because of the striking structural similarity with the original path matrices introduced by the John–Sachs theory of Kekulé structures. The presented conjecture hints at the existence of a generalization of the John–Sachs theory applicable to characterization and enumeration of Clar covers.