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

Abstract

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.

Zhang–Zhang Polynomials of Multiple Zigzag Chains Revisited: A Connection with the John–Sachs Theorem

“…We hope that the presented here results will stimulate the graph-theoretical community to discover further examples of generalized John–Sachs matrices also for other benzenoids, which will pave the road to conception and formulation of the generalization of John–Sachs theorem [ 75 , 76 , 77 , 100 , 101 , 102 , 103 , 104 , 105 , 106 ] to the world of Clar covers. We also hope that the presented here techniques will suggest an appropriate line of attack on the most difficult unsolved problem in the theory of ZZ polynomials: discovering the closed-form formula for the ZZ polynomial of hexagonal flake with arbitrary set of parameters [ 37 , 41 , 107 ].…”

“…The most convenient way of representing the subsequence is given in the form of its generating function which is most often referred to as the Clar covering polynomial or, from the names of its inventors, as the Zhang–Zhang polynomial or the ZZ polynomial of [ 9 , 10 , 11 , 12 , 13 , 14 , 15 ]. Substantial research effort has been invested in the determination of for elementary families of benzenoids [ 8 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 ]. The rapid development of Clar theory stimulated by these discoveries in recent years has led to many new interesting applications and connections to other branches of chemistry, graph theory, and combinatorics [ 8 , 17 , 18 , 19 , 21 , 28 ,…”

Energy Decomposition Scheme for Rectangular Graphene Flakes

“…The Kekulé count and Clar count of a multiple zigzag chain graphene flake can be computed directly from the Zhang–Zhang polynomial of . (See Equation (21) of [ 70 ] and Equations (7) and (8) of [ 71 ].) We have …”