Sextet Polynomial Research Articles

Construction of general forms of ordinary generating functions for more families of numbers and multiple variables polynomials

The aim of this paper is to construct general forms of ordinary generating functions for special numbers and polynomials involving Fibonacci type numbers and polynomials, Lucas numbers and polynomials, Chebyshev polynomials, Sextet polynomials, Humbert-type numbers and polynomials, chain and anti-chain polynomials, rank polynomials of the lattices, length of any alphabet of words, partitions, and other graph polynomials. By applying the Euler transform and the Lambert series to these generating functions, many new identities and relations are derived. By using differential equations of these generating functions, some new recurrence relations for these polynomials are found. Moreover, general Binet's type formulas for these polynomials are given. Finally, some new classes of polynomials and their corresponding certain family of special numbers are investigated with the help of these generating functions.

Zeros distribution of the reverse strong Turán expressions of polynomials sequences

In this paper, we first show the zeros distribution of the reverse strong Turán expressions of three recurrence polynomials sequences. Then as applications, we obtain the weak (Hurwitz) stability of the reverse strong Turán expressions of Bell polynomials, Dowling polynomials, ordered Bell polynomials, ordered Dowling polynomials and associated Lah polynomials respectively. On the other hand, the reverse strong Turán expressions of Delannoy polynomials, Morgan-Voyce polynomials, many independence polynomials and sextet polynomials have only real zeros respectively. Finally, we present some conjectures related to the Eulerian polynomials and the Narayana polynomials.

Analytic properties of sextet polynomials of hexagonal systems

In this paper we investigate analytic properties of sextet polynomials of hexagonal systems. For the pyrene chains, we show that zeros of the sextet polynomials \(P_n(x)\) are real, located in the open interval \((-3-2\sqrt{2},-3+2\sqrt{2})\) and dense in the corresponding closed interval. We also show that coefficients of \(P_n(x)\) are symmetric, unimodal, log-concave, and asymptotically normal. For general hexagonal systems, we show that real zeros of all sextet polynomials are dense in the interval \((-\infty ,0]\), and conjecture that every sextet polynomial has log-concave coefficients.

Combinatorics of Edge Symmetry: Chiral and Achiral Edge Colorings of Icosahedral Giant Fullerenes: C80, C180, and C240

We develop the combinatorics of edge symmetry and edge colorings under the action of the edge group for icosahedral giant fullerenes from C80 to C240. We use computational symmetry techniques that employ Sheehan’s modification of Pόlya’s theorem and the Möbius inversion method together with generalized character cycle indices. These techniques are applied to generate edge group symmetry comprised of induced edge permutations and thus colorings of giant fullerenes under the edge symmetry action for all irreducible representations. We primarily consider high-symmetry icosahedral fullerenes such as C80 with a chamfered dodecahedron structure, icosahedral C180, and C240 with a chamfered truncated icosahedron geometry. These symmetry-based combinatorial techniques enumerate both achiral and chiral edge colorings of such giant fullerenes with or without constraints. Our computed results show that there are several equivalence classes of edge colorings for giant fullerenes, most of which are chiral. The techniques can be applied to superaromaticity, sextet polynomials, the rapid computation of conjugated circuits and resonance energies, chirality measures, etc., through the enumeration of equivalence classes of edge colorings.

Counting Clar structures of (4, 6)-fullerenes

A (4, 6)-fullerene graph G is a plane cubic graph whose faces are squares and hexagons. A resonant set of G is a set of pairwise disjoint faces of G such that the boundaries of such faces are M-alternating cycles for a perfect matching M of G. A resonant set of G is referred to as sextet pattern whenever it only includes hexagonal faces. It was shown that the cardinality of a maximum resonant set of G is equal to the maximum forcing number. In this article, we obtain an expression for the cardinality of a maximum sextet pattern (or Clar formula) of G, which is less than the cardinality of a maximum resonant set of G by one or two. Moreover we mainly characterize all the maximum sextet patterns of G. Via such characterizations we get a formula to count maximum sextet patterns of G only depending on the order of G and the number of fixed subgraphs J1 of G, where the count equals the coefficient of the term with largest degree in its sextet polynomial. Also the number of Clar structures of G is obtained.

Chemical graph theory. II. On the graph theoretical polynomials of conjugated structures

The graph theoretical polynomials of conjugated structures are examined, namely, the characteristic polynomial, the acyclic polynomial, the Hosoya polynomial, the sextet polynomial, the cyclic polynomial, and the Wheland polynomial. Several novel relationships among them have been obtained.

Onk-Resonant Fullerene Graphs

A fullerene graph F is a 3-connected plane cubic graph with exactly 12 pentagons and the remaining faces as hexagons. Let M be a perfect matching of F. A cycle C of F is M-alternating if the edges of C appear alternately in and off M. A set $\mathcal{H}$ of disjoint hexagons of F is called a resonant pattern (or sextet pattern) if F has a perfect matching M such that all hexagons in $\mathcal{H}$ are M-alternating. A fullerene graph F is k-resonant if any i ($0\leq i \leq k$) disjoint hexagons of F form a resonant pattern. In this paper, we prove that every hexagon of a fullerene graph is resonant and all leapfrog fullerene graphs are 2-resonant. Further, we show that a 3-resonant fullerene graph has at most 60 vertices and we construct all nine 3-resonant fullerene graphs, which are also k-resonant for every integer $k>3$. Finally, sextet polynomials of the 3-resonant fullerene graphs are computed.

Zhang – Zhang Polynomial of Multiple Linear Hexagonal Chains

An explicit combinatorial expression is obtained for the Zhang-Zhang polynomial (also known as “Clar cover polynomial”) of a large class of pericondensed benzenoid systems, the multiple linear hexagonal chains Mn,m. By means of this result, various problems encountered in the Clar theory of Mn,m are also resolved: counting of Clar and Kekulé structures, determining the Clar number, and calculating the sextet polynomial.

ALGORITHM FOR SIMULTANEOUS CALCULATION OF KEKULÉ AND CLAR STRUCTURE COUNTS, AND CLAR NUMBER OF BENZENOID MOLECULES

The Chinese mathematicians Heping Zhang and Fuji Zhang conceived a combinatorial polynomial associated with benzenoid molecules. This “Zhang–Zhang polynomial” contains information on the Kekulé structure count (K), Clar structure count (C), Clar number (Cl), Hosoya-Yamaguchi sextet polynomial σ(B, x), and several other important characteristics of the underlying benzenoid molecule. We show how one can easily calculate the Zhang-Zhang polynomial, and thus arrive at an algorithm for simultaneous computation of K, C, Cl, and σ(B, x). Some applications of this algorithm are pointed out.

On the ordering of benzenoid chains and cyclo-polyphenacenes with respect to their numbers of Clar aromatic sextets

Based on Clar aromatic sextet theory [Clar, The Aromatic Serxtet (Wiley, New York, 1972)] and the concept of sextet polynomial introduced by Hosoya and Yamaguchi [Mathematical Concepts in Organic Chemistry (Springer, Berlin, 1986)], we define a new ordering of benzenoid systems. For two isomeric benzenoid systems B1 and B2, we say B1>B2 if each coefficient of sextet polynomial of B1 is no less than the corresponding coefficient of sextet polynomial of B2. In this paper, we consider the ordering of the benzenoid chains. The maximal and second maximal benzenoid chains as well as the minimal, the second minimal up to the fourth minimal benzenoid chains are determined. Furthermore, under this ordering, we determine the maximal and second maximal cyclo-polyphenacenes as well as the minimal, the second minimal, and up to the seventh minimal cyclo-polyphenacenes.

Immanants and immanantal polynomials of chemical graphs.

The much-studied determinant and characteristic polynomial and the less well-known permanent and permanental polynomial are special cases of a large class of objects, the immanants and immanantal polynomials. These have received some attention in the mathematical literature, but very little has appeared on their applications to chemical graphs. The present study focuses on these and also generalizes the acyclic or matching polynomial to an equally large class of acyclic immanantal polynomials, generalizes the Sachs theorem to immanantal polynomials, and sets forth relationships between the immanants and other graph properties, namely, Kekulé structure count, number of Hamiltonian cycles, Clar covering polynomial, and Hosoya sextet polynomial.

Clar and sextet polynomials of buckminsterfullerene

A graph-theoretical definition of Herndon and Hosoya's Clar structure is given. The Fries Kekulé structure of buckminsterfullerene ( C 60) with I h symmetry implies that every independent set of hexagon-dual graph (dodecahedron) of C 60 corresponds to a sextet pattern; and every maximal independent set to a Clar structure. By proposing the maximal independent set polynomial of a graph and developing its various calculational methods the Clar polynomial of C 60, ξ( C 60, χ)=5 χ 8+280 χ 7+10 χ 6, is given, which says that C 60 possesses a total of 295 Clar structures, and thus corrects a corresponding result recently published. Furthermore, the sextet polynomial of C 60 is also produced.

Normal Components, Kekulé Patterns, and Clar Patterns in Plane Bipartite Graphs

As a general case of molecular graphs of polycyclic alternant hydrocarbons, we consider a plane bipartite graph G with a Kekule pattern (perfect matching). An edge of G is called nonfixed if it belongs to some, but not all, perfect matchings of G. Several criteria in terms of resonant cells for determining whether G is elementary (i.e., without fixed edges) are reviewed. By applying perfect matching theory developed in plane bipartite graphs, in a unified and simpler way we study the decomposition of plane bipartite graphs with fixed edges into normal components, which is shown useful for resonance theory, in particular, cell and sextet polynomials. Further correspondence between the Kekule patterns and Clar (resonant) patterns are revealed.

The Clar covering polynomial of hexagonal systems III

The Clar covering polynomial is a recently proposed concept of hexagonal systems by which some important topological indices such as perfect matching count, Clar number, first Herndon number, etc., can be easily obtained. In this paper we establish a relationship between the Clar covering polynomial and sextet polynomial. A lower bound of the Clar number and some properties of coefficients of the Clar covering polynomial are thus deduced. It is mentioned that the summation of coefficients of Clar covering polynomial can be used to calculate the number of perfect matchings of a certain kind of polyominoes relating to crystal physics.

A semantic tree algorithm for the generation of sextet polynomials of hexagonal systems

The sextet polynomial that counts different ways of selecting varying number of resonating sextets on the hexagonal system is computed using a search based symbol manipulation algorithm. This is a #P Complete combinatorial enumeration problem, and artificial intelligence (AI) is employed for efficient enumeration. This is done by selective exploration of the semantic tree defined for that purpose. Hexagons of the graph are defined as symbols and each node of the tree is defined as a set of mutually disjoint hexagon patterns of the graph. The sextet polynomial is generated by enumerating a suitable subset of the nodes of the tree. A pruning heuristic that avoids redundant branches by a priori learning at selected intelligent branches of the semantic tree is designed.

“Crowns” und aromatische Sextette in einfachen coronoiden Kohlenwasserstoffen

A type of graphs derived from a cycle and associated with primitive coronoids are referred to as “crowns”. The characteristic polynomials and matching polynomials of crowns are studied. These notions are used to calculate the sextet polynomial for primitive coronoids. Patterns of aromatic sextets are treated in some detail.

Graph-theoretical analysis of the sextet polynomial. Proof of the correspondence between the sextet patterns and Kekul� patterns

Wenjie and Wenchen defined the sextet pattern and the super sextet and claimed to prove the one-to-one correspondence between the Kekule and sextet patterns [1]. However, the set of sextet patterns of a polyhexG by their definition cannot be obtained unless we know all the Kekule patterns ofG. In this sense, their definition does not match the theory of the sextet polynomial. Here, the whole set of sextet patterns, including the super sextets ofG, is defined from the properties ofG, not from the Kekule patterns. The one-to-one correspondence between the Kekule and sextet patterns is thus proved.

A New Method for the Calculation of the Sextet Polynomial of Unbranched Catacondensed Benzenoid Hydrocarbons

Abstract A method for the calculation of the sextet polynomial of unbranched catacondensed benzenoid molecules is proposed. It requires the multiplication of a small number of 2×2 matrices and is therefore very simple.

Clar's valence structures of benzenoid hydrocarbons

The properties of Clar's valence structures for unbranched catacondensed benzenoid systems have been considered. These molecules can be represented as caterpillar trees, i.e. ayclic graphs with side branches of length one only. The sextet polynomials of the benzenoids are simply related to the characteristic polynomials of corresponding caterpillar trees. Using the ‘ultimate pruning’ method we constructed the characteristic polynomials for several families of caterpillar trees. In some cases the characteristic polynomials are given in a compact form.

One-to-one correspondence between Kekul� and sextet patterns

The concept of super sextet is clarified and the generalized sextet polynomial in two elements is proposed. Two theorems related to Ohkami-Hosoya conjecture [1] are proved and one novel conjecture is proposed.