Zhang–Zhang Polynomials of Ribbons
Round 1
Reviewer 1 Report
The current manuscript entitled "Zhang-Zhang Polynomials of Ribbons" descibes a closed-form formula for the ZZ polynomial of a class of elementary pericondensed benzenoids (i.e. ribbon benzenoids). The work is well excuted and the manuscript is well written and organised. The work descibed is interesting to the readers of "Symmetry". I would recommand publication in the current format.Author Response
No response is required as the referee did not ask for any modifications.
Reviewer 2 Report
The manuscript is well constructed and delivers important theoretical considerations concerning some pericondensed benzenoids reffered as ribbons.
I have two minor questions to be commented by the authors:
- Are the cited topological indices the only possible for performing the task or there are other options, not yet considered?
- Could the authors suggest some initial recommendation for future calculation of ZZ polynomials for the remaining classes of benzenoids indicated as hexagonal flakes O(k,m,n) or oblate rectangles Or(m,n)?
Author Response
We thank the referee for the two nice suggestions. To address them, we have included in the manuscript two new fragments describing (i) additional topological invariants that should/could be considered for benzenoids and (ii) definition of a good starting point for finding closed form formulas for hexagons and oblate rectangles.
(i) Stimulated by an anonymous referee's request, we would like to discuss briefly what kinds of other topological invariants could be useful for characterizing physical and chemical properties of benzenoids. The departure point for our divagations is the structure of the valence bond wave function corresponding to the ground state of these molecules, which can be considered as a linear combination of many Slater determinants (or more generally: configuration state functions (CSFs) obtained from Slater determinants by appropriate linear combinations aiming at modeling proper spin quantum number). In such a model, a single CSF would correspond to a single Kekule structure or a Clar cover of a given benzenoid. Two natural extensions to such a valence-bond CSF bases can be perceived. The first extension corresponds to the extended conjugated circuit model introduced in chemical graph theory community by Trinajstic and collaborators and including not only double bonds (i.e., K2 subgraphs) and aromatic rings (i.e., C6 subgraphs) in the description of valence resonance structures of benzenoids, but also larger cycles (C10, C14, etc). Following this idea, Zigert Pletersek conceived a generalized Zhang-Zhang polynomial, which is capable to enumerate also larger cycles. We believe that these generalized Clar covers can constitute an important class of topological invariants for benzenoids. At the moment, generalized ZZ polynomials have not been reported for any class of benzenoids and no correlation with energetics and properties of benzenoids has been studied. The second extension can be associated with radical resonance structures. It is well known that many benzenoids display pronounced radical character. Including the biradical and tetraradical Kekule structures and Clar covers in the graph-theoretical analysis of Kekulean benzenoids and single radical and triradical resonance structures in the analysis of non-Kekulean benzenoids can yield a completely new information about those structures and permit their better description in the valence bond language. Initial applications of the new version
of ZZDecomposer to study spin populations in benzenoids shows that using such radical Clar covers allows one to achieve very close correlation with spin populations determined from quantum chemical calculations.
(ii) at the end of Section 5 we inserted the following fragment: "A possible departure point for such investigations can be the interface theory of benzenoids or the recently discovered equivalence between the ZZ polynomials of regular benzenoid strips and the extended strict order polynomials of certain posets associated with those benzenoids. Yet another convenient departure point for discovering a general closed form of ZZ polynomials of O(k,m,n) (and by extension, also for Or(m,n)), can be our recently accepted study reporting structural parameters-based closed-form formula for hexagonal graphene flakes O(k,m,n) with k, m=1-7 and arbitrary n. Ultimate general formulas are still missing. "
