Graph theoretical characterization and computer generation of certain carcinogenic benzenoid hydrocarbons and identification of bay regions

An algorithm is developed for generating and characterizing carcinogenic catacondensed benzenoid hydrocarbons. The bay regions in these structures are identified by a technique that we developed at Johns Hopkins. Using the three-digit code proposed by Balaban, and the concept of ring adjacency matrix expounded here, we generate catacondensed benzenoid hydrocarbons in the computer and identify the number of potentially carcinogenic bay regions in each of them. The results of computer generation agree with the combinatorial enumeration of Harary and Read. All structures containing up to five rings and some with six rings and the number of bay regions in these are presented. Computer results for the structures and bay regions of all seven-, eight-, and nine-membered unbranched catacondensed benzenoid hydrocarbons and the number of bays are available from the authors.     

Calculating the number of spanning trees in a labeled planar molecular graph whose inner dual is a tree

The quantum mechanical relevance of the concept of a spanning tree extant within a given molecular graph—specifically, one that may be considered to represent the carbon-atom connectivity of a particular (planar) conjugated system—was first explicitly pointed out by Professor Roy McWeeny in his now-classic 1958 memoire entitled “Ring Currents and Proton Magnetic Resonance in Aromatic Molecules.” In a recent work, Gutman and one of the present authors proposed a scheme for calculating the number of spanning trees in the graph associated with catacondensed, benzenoid molecules which, by definition, contain rings of just the one size (six-membered); here, we present an algorithmic approach that enables the determination of the number of spanning trees in the molecular graph of any catacondensed system (which, in general, has rings of more than one size, and these may be of any size). An illustrative example is given, in which the algorithm devised is applied to a (hypothetical) pentacyclic catacondensed structure comprising a five-membered ring, a six-membered ring, a seven-membered ring, and two four-membered rings. © 1996 John Wiley & Sons, Inc.     

Ring signatures for benzenoids with up to seven rings,Part 1: Catacondensed systems

We revisited the π‐electron ring partitions in catacondensed benzenoids and re‐examined structural regularities reported for the ring partitions in these compounds, seeking the origin of the observed regularities. We examined the distribution of the π‐electron ring partitions by counting the contributions arising from benzene rings having different assigned numbers of π‐electrons. This has led to a better insight into the underlying structure of the π‐electron ring partitions and also to a novel and unique π‐electron “signature” of individual rings in benzenoid hydrocarbons. The physical interpretation of the local π‐electron ring partition represents the local π‐electron density of benzenoid molecules, and is not an alternative index of local aromaticity as one may erroneously assume. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

Partitioning of pi-electrons in rings of polycyclic conjugated hydrocarbons. 5. Nonalternant compounds

All possible nonalternant hydrocarbons with a total of two, three, or four 5-, 6-, and 7-membered rings have been examined for the partition of their pi-electrons by averaging over all Kekulé structures (considered to contribute equally to the electron distribution) the pi-electrons in each ring in accordance to the rules introduced earlier: for each double bond shared with another ring one pi-electron is taken into account, and for double bonds that are not shared two pi-electrons are added. The trends observed for the partitions are discussed, and a comprehensive bibliography is provided as Supporting Information for all such systems, including both experimental and theoretical published data.     

Algebraic Kekulé formulas for benzenoid hydrocarbons

By assigning two pi-electrons of CC double bonds in a Kekulé valence structure to a benzene ring if not shared by adjacent rings and one pi-electron if CC double bond is shared by two rings we arrived at numerical valence formulas for benzenoid hydrocarbons. We refer to numerical Kekulé formulas as algebraic Kekulé valence formulas to contrast them to the traditional geometrical Kekulé valences formulas. The average over all numerical Kekulé valence structures results in a single numerical structure when a benzenoid hydrocarbon molecule is considered. By ignoring numerical values the novel quantitative formula transforms into a qualitative one which can replace incorrectly used notation of pi-electron sextets to indicate aromatic benzenoids by placing inscribed circles in adjacent rings-which contradicts Clar's characterization of benzenoid hydrocarbons.     

Wiener numbers of benzenoid hydrocarbons: two theorems

Two exact results on the Wiener numbers of catacondensed benzenoid hydrocarbons are obtained. All catacondensed isomers have Wiener numbers congruent modulo 8. Among unbranched catacondensed systems, helicenes have the minimum and linear polyacenes the maximum Wiener number.     

Global Forcing Number of Benzenoid Graphs

A global forcing set in a simple connected graph G with a perfect matching is any subset S of E(G) such that the restriction of the characteristic function of perfect matchings of G on S is an injection. The number of edges in a global forcing set of the smallest cardinality is called the global forcing number of G. In this paper we prove several results concerning global forcing sets and numbers of benzenoid graphs. In particular, we prove that all catacondensed benzenoids and catafused coronoids with n hexagons have the global forcing number equal to n, and that for pericondensed benzenoids the global forcing number is always strictly smaller than the number of hexagons.     

Benzenoid rings resonance energies and local aromaticity of benzenoid hydrocarbons

In this article, we consider partitioning of the analytical expression for resonance energy (RE) in smaller benzenoid hydrocarbons, to individual benzenoid rings of polycyclic molecules. The analytical expression for molecular RE, available since 1976, is given by the count of all linearly independent conjugated circuit in all Kekulé structures in a molecule. Analytical expression for local ring RE (RRE) is given by counting all linearly independent conjugated circuits involving single benzenoid ring in all Kekulé structures, which when added, gives the molecular RE. If for benzene ring the RRE is taken to be 1.000, rings in polycyclic benzenoid hydrocarbons have their ring RRE, which give the degree of their local aromaticity, smaller than 1.000. The difference to 1.000 is a measure of the similarity of a ring to benzene in this one-dimensional (1-D) representation of local aromaticities of benzenoid hydrocarbons. The plot of RRE against the distance of the same ring from benzene in the Local Aromaticity Map, in which benzenoid rings are characterized ring bond orders and average variations of adjacent CC bonds, shows linear correlation (with r = 0.91), reducing the local aromaticity in benzenoid hydrocarbons to 1-D molecular property. © 2018 Wiley Periodicals, Inc.     

Coding and ordering Kekulé structures

The concept of numerical Kekulé structures is used for coding and ordering geometrical (standard) Kekulé structures of several classes of polycyclic conjugated molecules: catacondensed, pericondensed, and fully arenoid benzenoid hydrocarbons, thioarenoids, and [N]phenylenes. It is pointed out that the numerical Kekulé structures can be obtained for any class of polycyclic conjugated systems that possesses standard Kekulé structures. The reconstruction of standard Kekulé structures from the numerical ones is straightforward for catacondensed systems, but this is not so for pericondensed benzenoid hydrocarbons. In this latter case, one needs to use two codes to recover the geometrical Kekulé structures: the Wiswesser code for the benzenoid and the numerical code for its Kekulé structure. There is an additional problem with pericondensed benzenoid hydrocarbons; there appear numerical Kekulé structures that correspond to two (or more) geometrical Kekulé structures. However, this problem can also be resolved.     

Flag graphs and their applications in mathematical chemistry for benzenoids

Flag graphs have been used in the past for describing maps on closed surfaces. In this paper we use them for the first time in mathematical chemistry for describing benzenoids and some other similar structures. Examples include catacondensed and pericondensed benzenoids. Several theorems are included. Symmetries of benzenoid systems, flag graphs, and symmetry type graphs are briefly discussed.     

Asymmetric annellation effects—II

The comparison of the U.V. absorption spectra of acenes, 1:2–3:4-dibenzacenes and tetrabenzacenes shows a strong asymmetric annellation effect. This is explained on the basic assumption that an aromatic sextet or benzenoid ring can transfer only two electrons to another ring. Three benzenoid rings can thus produce an induced aromatic sextet in an included ring of the type of the central ring in triphenylene.

The synthesis of tetrabenzotetracene is described.     

Recursive Formulae for Enumeration of LM-Conjugated Circuits in Structurally Related Benzenoid Hydrocarbons

The linearly independent and minimal conjugated (LM-conjugated) circuits of benzenoid hydrocarbons play the central role in the conjugated circuit model. For a general case, the enumeration of LM-conjugated circuits may be tedious as it requires construction of all Kekule structures. In our previous work, a recursive method for enumeration of LM-conjugated circuits of benzenoid hydrocarbons was established. In this paper, we further extend the recursive formulae for enumerations of LM-conjugated circuits for both catacondensed benzenoid hydrocarbons and some families of structurally related pericondensed benzenoid hydrocarbons.     

π-Electron currents in fixed π-sextet aromatic benzenoids

In view of different patterns of π-electron density currents in benzenoid aromatic compounds it is of interest to investigate the pattern of ring currents in various classes of compounds. Recently such a study using a graph theoretical approach to calculating CC bond currents was reported for fully benzenoid hydrocarbons, that is, benzenoid hydrocarbons which have either π-sextets rings or “empty” rings in the terminology of Clar. In this contribution we consider π-electron currents in benzenoid hydrocarbons which have π-electron sextets and C=C bonds fully fixed. Our approach assumes that currents arise from contributions of individual conjugated circuits within the set of Kekulé valence structures of these molecules.     

On the number of kekulé structures of unbranched cata-condensed benzenoid chains

It is known that an alternative algorithm to the Gordon-Davidson algorithm for counting the Kekulé valence structures of catacondensed non-branched benzenoid hydrocarbons is a reformulation of the original algorithm.     

A new yardstick for benzenoid polycyclic aromatic hydrocarbons

The newly introduced signature of benzenoids (a sequence of six real numbers Si with i = 6-1) shows the composition of the pi-electron partition by indicating the number of times all rings of the benzenoid are assigned 6, 5, 4, 3, 2, or 1 pi-electrons. It allows the introduction of a new ordering criterion for such polycyclic aromatic hydrocarbons by summing some of the terms in the signature. There is an almost perfect linear correlation between sums S6 + S5 and S4 + S3 for isomeric cata- or peri-fused benzenoids, so that one can sort such isomers according to ascending s 6 + S5 or to descending S4 + S3 sums (the resulting ordering does not differ much and agrees with that based on increasing numbers of Clar sextets and of Kekule structures). Branched cata-condensed benzenoids have higher S6 + S5 sums than isomeric nonbranched systems. For nonisomeric peri-condensed benzenoids, both sums increase with increasing numbers of benzenoid rings and decrease with the number of internal carbon atoms. Other partial sums that have been explored are S6 + S5 + S3 And S6 + S2 + S1, and the last one appears to be more generally applicable as a parameter for the complexity of benzenoids and for ordering isomeric benzenoids.     

π‐Electron currents in larger fully aromatic benzenoids

Recently, we have reported on calculation of π‐electron ring currents in several smaller fully benzenoid hydrocarbons having up to eight fused benzene rings and five Clar π‐aromatic sextets. In contrast to early HMO ring current calculations and more recent ab initio calculations of π‐electron density, our current calculations are based on a graph theoretical model in which contributions to ring currents comes from currents associated with individual conjugated circuits. In this contribution, we consider several larger fully benzenoid hydrocarbons having from 9 to 13 fused rings and from six or seven π‐aromatic sextets. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Growth in catacondensed benzenoid graphs

The generating function of the sequence counting the number of graph vertices at a given distance from the root is called the spherical growth function of the rooted graph. The vertices farthest from the root form an induced subgraph called the distance-residual graph. These mathematical notions are applied to benzenoid graphs which are used in graph theory to represent benzenoid hydrocarbons. An algorithm for calculating the growth in catacondensed benzenoids is presented, followed by some examples.     

On the construction of the matching polynomial for unbranched catacondensed benzenoids

An algorithm for obtaining the matching polynomial of an arbitrary catacondensed unbranched benzenoid molecule is presented. It is based on multiplication of only three 5 x 5 transfer matrices I, J, K, and an appropriate terminal vector. The choice of the matrices is dictated by the history of the growth of the hexagonal “animals” (i.e., by the pattern of the successive fusions of the benzene rings). The approach also gives the number of Kekule valance structures, the count of conjugated circuits, the values of the topological index Z, and the characteristic polynomials.