The branching graphs of polyhexes: Some elementary theorems,conjectures and open questions

This paper presents a few novel results, and collects together what is known and conjectured about the branching graph of a polyhex.     

Benzenoid Molecules with Uniform Distribution of π-Electrons within Rings

In a recent work it was demonstrated that in linear hexagonal chains the distribution of π-electrons into rings (as computed by means of the Randić–Balaban method) is uniform, irrespective of the nature of the terminal fragments. We now establish that an analogous, yet somewhat more complex, uniformity in the π-electron distribution exists also in double linear hexagonal chains, as well as in some other benzenoid systems.     

Extremal hexagonal chains

Some extremal properties of the linear chainL _h ofh hexagons are pointed out. In the class of all hexagonal chains withh hexagons,L _h has minimumK,Z andx ₁ values, as well as maximum W and values;K = number of perfect matchings,Z = number of independent edge sets (Hosoya index),x ₁ = largest graph eigenvalue,W = Wiener index, = number of independent vertex sets (Merrifield-Simmons index). The extremality ofL _h with respect toZ, andx ₁ is demonstrated here for the first time.Dedicated to Frank Harary, teacher, inspirer and friend, pioneer, champion and proprietor of graph theory, on the occasion of his 70th birthday.     

Stereo- and regioselectivity in asymmetric synthesis of α-amino substituted benzocyclic compounds

The enantiomerically pure chiral benzocyclic amines 6–8 were obtained by asymmetric transamination of the corresponding prochiral ketones 9a–c. The method involves: (a) formation of chiral imines 10a–c from the prochiral ketones 9a–c and the inexpensive chiral auxiliary (R)- or (S)-phenylethylamine (PEA); (b) asymmetrically induced reduction of these imines to the diastereomeric amines 11a–c and 12a–c; (c) catalytic hydrogenation to remove the benzylic fragment of the chiral PEA auxiliary. The stereoselectivity of the imine reduction, as well as the regioselectivity of the catalytic hydrogenation, are strongly dependent on the size of the saturated ring condensed with the benzene ring. This approach was used to develop a convenient, high yielding, and stereoselective route to several practically important optically active α-amino substituted benzocyclic compounds.     

Graph theory and molecular orbitals. The loop rule

Graph theory is applied to the study of the dependence of total π-electron energy, π-electron charge distribution and free valency indices of conjugated hydrocarbons on molecular topology. It is shown that the number of loops in the molecular graph determines these quantities.     

Benzenoid Molecules with Uniform Distribution of π-Electrons within Rings

Summary. In a recent work it was demonstrated that in linear hexagonal chains the distribution of π-electrons into rings (as computed by means of the Randić–Balaban method) is uniform, irrespective of the nature of the terminal fragments. We now establish that an analogous, yet somewhat more complex, uniformity in the π-electron distribution exists also in double linear hexagonal chains, as well as in some other benzenoid systems.     

The multiplicative version of the Wiener index

The classical Wiener index, W(G), is equal to the sum of the distances between all pairs of vertexes of a (molecular) graph, G. We now consider a related topological index, pi(G), equal to the product of distances between all pairs of vertexes of G. The basic properties of the pi index are established and its possible physicochemical applications examined. In the case of alkanes, pi and W are highly correlated; a slightly curvilinear correlation exists between In pi and W.     

Phase and orientational ordering of low molecular weight rod molecules in a quenched liquid crystalline polymer matrix with mobile side chains

We study the phase diagram and orientational ordering of guest liquid crystalline (LC) rods immersed in a quenched host made of a liquid crystalline polymer (LCP) matrix with mobile side chains. The LCP matrix lies below the glass transition of the polymer backbone. The side chains are mobile and can align to the guest rod molecules in a plane normal to the local LCP chain contour. A field theoretic formulation for this system is proposed and the effects of the LCP matrix on LC ordering are determined numerically. We obtain simple analytical equations for the nematic/isotropic phase diagram boundaries. Our calculation show a nematic-nematic (N/N) first order transition from a guest stabilized to a guest-host stabilized region and the possibility of a reentrant transition from a guest stabilized nematic region to a host only stabilized regime separated by an isotropic phase. A detailed study of thermodynamic variables and interactions on orientational ordering and phases is carried out and the relevance of our predictions to experiments and computer simulations is presented.     

Theoretical study of cyclic conjugation in phenes: Some hitherto unnoticed features

Summary Cyclic conjugation in phenes is examined by means of theBosanac-Gutman-Aihara method. In contrast to the predictions based on the analysis ofKekulé orClar structures, we find that when going along the hexagons of a phene molecule, cyclic conjugation varies in a rather non-uniform manner. In contrast to a number of other homologous series of benzenoid hydrocarbons, the intensity of cyclic conjugation in phenes increases when going towards the center of the molecule.

---

Theoretische Untersuchung der cyclischen Konjugation in Phenen: Einige bisher unbekannte Eigenschaften

Zusammenfassung Die cyclische Konjugation in Phenen wird mittels derBosanac-Gutman-Aihara-Methode untersucht. Im Gegensatz zu auf der Analyse vonKekulé- oderClar-Strukturen beruhenden Voraussagen variiert die cyclische Konjugation entlang der Sechsecke eines Phenmoleküls ziemlich unregelmäßig. Anders als bei anderen homologen benzenoiden Kohlenwasserstoffen nimmt bei Phenen die Intensität der cyclischen Konjugation zum Zentrum des Moleküls hin zu.

     

Relating Total π-Electron Energy and Resonance Energy of Benzenoid Molecules with Kekulé- and Clar-Structure-Based Parameters

Within classes of isomeric benzenoid hydrocarbons various Kekulé- and Clar-structure-based parameters (Kekulé structure count, Clar cover count, Herndon number, Zhang–Zhang polynomial) are all mutually correlated. This explains why both the total π-electron energy (E), the Dewar resonance energy (DRE), and the topological resonance energy (TRE) are well correlated with all these parameters. Nevertheless, there exists an optimal value of the variable of the Zhang–Zhang polynomial for which it yields the best results. This optimal value is negative-valued for E, around zero for TRE, and positive-valued for DRE. A somewhat surprising result is that TRE and DRE considerably differ in their dependence on Kekulé- and Clar-structure-based parameters.