Wolfgang R. Müller

Molecular topological index

The molecular topological index (MTI) has been systematically tested for counterexamples (two or more nonisomorphic structures with the same MTI number). The analysis was carried out for alkane trees with up to 16 atoms. The search for counterexamples was positive : the first pair of alkane trees with identical MTI numbers was found in the octane family. The more disturbing finding was that two nonisomorphic alkane trees of different sizes may also possess the same MTI value. An attempt to redefine the MTI in terms of only the distance matrix and the valency matrix was abortive

On the concept of the weighted spanning tree of dualist

The concepts of dual, inner dual and dualist are reviewed. The application of these concepts to polyhexes is briefly described. The concepts of a spanning tree and a weighted spanning tree of dualist are introduced. The uses of the weighted spanning tree of dualist in coding and enumerating polyhex hydrocarbons are outlined.

Use of small computers for large computations: Enumerations of polyhex hydrocarbons

The enumeration of polyhex hydrocarbons with up to 16 hexagons is reported. These results are now for the first time available

On irreducible endospectral graphs

The paper considers endospectral trees, a special class of graphs associated with the production of numerous isospectral graphs. Endospectral graphs have been considered in the literature sporadically (the name was suggested very recently [M. Randic, SIAM J. Algebraic Discrete Meth. 6, 145 (1985)]). They are characterized by the presence of a pair of special vertices that, if replaced by any fragment, produce an isospectral pair of graphs. Recently Jiang [Y. Jiang, Sci. Sin. 27, 236 (1984)] and Randic and Kleiner (M. Randic and A. F. Kleiner, ‘‘On the construction of endospectral trees,’’ submitted to Ann. NY Acad. Sci.) considered alternative constructive approaches to endospectral trees and listed numerous such graphs. The listing of all such trees having n=16 or fewer vertices has been undertaken here. It has been found that relatively few endospectral trees have novel structural features and cannot be reduced to some already known endospectral tree. These few have been named ‘‘irreducible endospectral...

On the determinant of the adjacency-plus-distance matrix as the topolgical index for characterizing alkanes

The determinant of the adjacency-plus-distance matrix has been systematically tested for examples that would disprove a supposition that it is a unique topological index (TI) for alkanes. The analysis was carried out for alkane trees with up to 20 sites. The search was fruitful: The first pair of alkane trees with identical TI numbers was already found in the nonane family

Enumeration and classification of coronoid hydrocarbons: note to a note

Abstract A double coronoid is a polyhex with two holes. Altogether 161 such systems with up to 15 hexagons have been generated and enumerated with the use of a computer. The systems are classified into Kekulean (normal and essentially disconnected) and non-Kekulean, and catacondensed (including primitive ) and Pericondensed. The distribution into symmetry groups is specified. The forms of all the generated systems are displayed, and the Kekule structure counts are given.

Accurate values for the nonrelativistic energies of the lowest singlet and triplet S-states of the4He-Isotop

Accurate lower and upper bounds for the nonrelativistic lowest energies1E0 and3E0 of the singlet and triplet-system of the4He-Isotop are calculated with the linearized method of variance minimization. The same was done for1E1 the energy of the first excitedS-state 21S.The results especially for1E0 and3E0 in a.u. are −2.90330769975 ≤1E0 ≤ −2.90330769218 −2.17493242637 ≤3E0 ≤ −2.17493242459 i.e. the values are determined with an absolute error smaller than 0.00167 cm−1 for1E0 and 0.00039 cm−1 for3E0.

THE CHARACTERISTIC POLYNOMIAL AS A STRUCTURE DISCRIMINATOR

We investigate use of the characteristic polynomial for discrimination of graphs. Here we consider acyclic graphs (trees) only, and in particular we consider trees without bridging vertices because no isospectral trees in which both graphs are without bridging vertices have been hitherto reported. However, we found the smallest such isospectral trees when n = 15. When the maximal valence is limited to four, so that graphs correspond to molecules of organic chemistry, the smallest isospectral (nonbridging) trees have n = 19 vertices. The smallest pair of isospectral binary trees were found among trees having n = 26 vertices. By combining contraction of trees to proper trees (i.e., trees without bridging vertices) and subsequent pruning of proper trees one can in many cases determine if two trees are isomorphic or not from comparison of their characteristic polynomials.

On the number of square-cell configurations

SummaryThe numbers of simply and multiply connected square-cell configurations are computed. The computation is based on the original algorithm for constructive enumeration of animals which is founded on the DAST (dualist angle-restricted spanning tree) code.

ON THE CLASSIFICATION AND ENUMERATION OF PLANAR POLYHEX HYDROCARBONS

Abstract A novel classification of planar polyhexes based on their graph-theoretical characteristics is proposed. The members of each class of planar polyhexes are enumerated with up to 12 hexagons.