Haruo Hosoya

On some counting polynomials in chemistry

Abstract Various counting polynomials suggested by chemical and physical problems are discussed. Mathematical relations among them and physico-chemical interpretations are given.

Graphical enumeration of the coefficients of the secular polynomials of the Hckel molecular orbitals

A general and simple method for graphical enumeration of the coefficients of the secular polynomials of the Hückel molecular orbitals is presented with examples. The essential procedure is to count for the graph and its subgraphs the non-adjacent numbers, p(G, k)s, which appear as the coefficients of the Z-counting polynomial Q(Y). Two composition principles are proposed and shown to simplify these procedures to a great extent. Application to biphenylene shows the superiority of this method to others, e.g., the method of polygons.ZusammenfassungEine einfache und allgemeine Methode zur graphischen Bestimmung der Koeffizienten des Säkularpolynoms bei Hückelproblemen wird angegeben. Das Wesentliche für den Graphen und seine Untergraphen ist es, die Nichtnachbar-Zahlen abzuzählen. Letztere erscheinen im Z-zählenden Polynom. Zwei einfache Aufbau-Schemata werden vorgeschlagen. Am Beispiel von Biphenylen werden die Vorzüge diesen Verfahrens demonstriert.

Graph-theoretical analysis of the clar's aromatic sextet : Mathematical properties of the set of the kekulé patterns and the sextet polynomial for polycyclic aromatic hydrocarbons

Abstract The mathematical structure of a set of the Kekule patterns for a polycyclic aromatic hydrocarbon has been analysed graph-theoretically. By defining the proper and improper sextets, sextet pattern, Clar transformation, and sextet rotation, one can prove the important property of the sextet polynomial BG(x) as BG(1) = K(G), where K(G) is the number of the Kekule patterns for thin polyhex graph G. For fat polyhex graphs such as coronene the above relation is found to be also valid by introducing the concept of a super sextet. All the Kekule patterns for a given G are shown to form a hierarchical tree structure by the sextet rotation. The theory developed in this paper gives a mathematical basis and interpretation for the concept of the Clars aromatic sextet.

King and domino polynomials for polyomino graphs

For the purposes of treating several enumeration problems of lattice dynamics, king and domino polynomials are defined for a chessboard, polyomino, or square lattice of arbitrary size and shape. These polynomials are shown to be closely related to the partition function of the dimer statistics, the number of Kekule structures, or maximum matching number. Several recursion formulas are found. Interpretation of these newly proposed quantities is given, and the possibility of extending them to the important physical models is discussed.

Topological index as applied to π‐electronic systems. III. Mathematical relations among various bond orders

Mathematical relations among the Coulson (pCl), Ham and Ruedenberg (pHRl), Pauling (pPl), and topological (pTl) bond orders have been derived by combining the enumeration technique of the graph theory and the complex integral developed by Coulson and Longuet‐Higgins. If one defines a function FG,l(y) for bond l of graph G as FG,l(y) =Δr,s(iy)/Δ (iy)(l=rs¯) the following relations are proved for alternant hydrocarbons: pCl=2/πF∞0FG,l(y) dy, pHRl=pPl=FG,l(0), pTl=FG,l(1) for tree graphs, pTl=FG,l(1) for nontree graphs. Linear relationship pCl∝pTl+ApPl (A≪1) is also proved. Abnormal bond orders for 4n‐membered ring systems are discussed. Graphical methods for decomposing determinants and adjuncts are proposed.

Matching and symmetry of graphs

Abstract Matching is a mathematical concept that deals with the way of spanning a given graph network with a set of pairs of adjacent points. It is pointed out that in many different areas of science and culture (e.g., physics, chemistry, games, etc.), computing the perfect and imperfect matching numbers is commonly performed but under different names, such as partition function for dimer statistics, Kekule structures of molecules, paving domino problem. This paper demonstrate the mathematically beautiful but somewhat mystic relation between the symmetry of a graph and the factorable nature of its perfect matching number. There is introduced another interesting relation between the certain series of graphs and a family of orthogonal polynomials through the matching polynomial and topological index that are defined for counting the matching numbers.

Topological dependency of the aromatic sextets in polycyclic benzenoid hydrocarbons. Recursive relations of the sextet polynomial

Mathematical meaning of the Clars “aromatic sextet” is clarified by analysing the topological dependency of the sextet polynomial. Generalised recursive method for obtaining the sextet polynomial of a polyhex graph is presented. It is shown that the concept of the “super sextet” is necessarily introduced, if one-to-one correspondence between the Kekulé and sextet patterns is assumed. Topological dependency of the maximum number of resonant sextets is clarified and discussed in relation to the aromaticity and stability of polycyclic benzenoid hydrocarbons.

Mo- and Vb-benzene characters : Analysis of the “clar's aromatic sextet” in polycyclic aromatic hydrocarbons

Abstract In order to analyse the mode of distribution of π-electrons in a given hexagon in a polycyclic aromatic hydrocarbon, normalised VB- and MO- (HMO and PPP) benzene characters are defined and calculated for a number of molecules. Their relation to the topological structure of the molecule was studied in detail. By combining these quantities it was shown that the π-electrons in the three different classes (primary, secondary and tertiary) of hexagons have different distribution over the benzene ring. It was found that the value of the benzene character is determined by the local topological structure up to the third next hexagons. These findings qualitatively support and quantitatively figure out the concept of the Clars aromatic sextet, which can be interpreted as the “higher order π-electronic correlation” that the π-electronic system of polycyclic aromatic hydrocarbons can be described by the cooperative interaction of the units of closely correlated six π-electrons.

Topological Index and Thermodynamic Properties. 5.† How Can We Explain the Topological Dependency of Thermodynamic Properties of Alkanes with the Topology of Graphs?

Correlations between the thermodynamic properties of alkanes and topological indices of integer values (ITI) for the corresponding graphs are studied. A set of higher order Z-indices, Zn, for n-ste...

An effective algorithm for obtaining polynomials for dimer statistics. Application of operator technique on the topological index to two‐ and three‐dimensional rectangular and torus lattices

Recursion formulas on the numbers of ways for placing an arbitrary number of indistinguishable dumbbells on the various rectangular lattice spaces (3×n, 4×n) and torus spaces (2×n, 3×n, 2×2×n) were obtained by the use of the operator technique to the counting polynomial on the topological characteristics (e.g., nonadjacent number, Kekule number, and topological index). Perfect matching numbers or Kekule numbers were also obtained for larger lattices such as the 4×n torus and the 2×3×n lattice. By deriving these new results the utility of the proposed method for dimer statistics is demonstrated and some of their mathematical features are discussed.