Details of the reaction graphs for intramolecular isomerizations of the carboranes

From proposed mechanisms for framework reorganizations of the carboranes C₂B _n-2H _n ,n = 5–12, we present reaction graphs in which points or vertices represent individual carborane isomers, while edges or arcs correspond to the various intramolecular rearrangement processes that carry the pair of carbon heteroatoms to different positions within the same polyhedral form. Because they contain both loops and multiple edges, these graphs are actually pseudographs. Loops and multiple edges have chemical significance in several cases. Enantiomeric pairs occur among carborane isomers and among the transition state structures on pathways linking the isomers. For a carborane polyhedral structure withn vertices, each graph hasn(n -1)/2 graph edges. The degree of each graph vertex and the sum of degrees of all graph vertices are independent of the details of the isomerization mechanism. The degree of each vertex is equal to twice the number of rotationally equivalent forms of the corresponding isomer. The total of all vertex degrees is just twice the number of edges orn(n - 1). The degree of each graph vertex is related to the symmetry point group of the structure of the corresponding isomer. Enantiomeric isomer pairs are usually connected in the graph by a single edge and never by more than two edges.     

Random walks and their diagnostic value for characterization of atomic environment

A characterization of atomic environments based on counting random walks in a molecular skeleton is outlined. To each atom in a molecule a sequence of integers w₁, w₂, w₃,…, w_n is assigned, where w_i represents the number of self-returning walks of length k, the length being defined by the number of bonds traversed. Properties of the derived atom codes are discussed. The codes display an impressive diversity and are superior to atomic codes based on enumeration of self-avoiding walks (or paths) in discriminating atomic environments. In certain cases the codes of individual atoms are not unique and the same codes appear in different molecules or even within the same molecule. The occurrence of the nonunique codes can be related to special structural situations, associated with the occurrence of isospectral graphs. These isospectral graphs which have atoms with identical codes can generate additional isospectral structures by attaching any arbitrary group to such points. If nonequivalent atoms of a single molecule have identical random walk codes, substitution at the singular points alternatively will produce isospectral graphs. Examples of such situations are given.     

Method for construction of characteristic polynomials via graph linearization

A new method for construction of characteristic polynomials (CP) of complicated graphs having arbitrary edge and vertex weights has been developed. The method first converts the graph into isospectral linear chains with weighted vertices and edges and then builds up the CP coefficients recursively. Two types of graphs have been used for illustration, viz., (i) graphs that can be linearized by symmetry factorization and (ii) graphs without symmetry which are to be linearized by an algorithm involving walks of unit length. Both types have been illustrated, of which type (i) includes the Schlegel of fullerene fragment C₂₀ and another large graph with many fused rings. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65 : 199–204, 1997     

On isospectral benzenoid graphs

The recently Proposed procedure [5] for the construction of isospectral benzenoid graphs has been examined in detail. Necessary and sufficient conditions for the construction of isospectral benzenoid graphs with isomorphicH-graphs are formulated. The inapplicability of the Procedure for the construction of isospectral benzenoid graphs with an even number of vertices has been proven.     

Isorpectral multitrees

We examined trees with one multiple edge (of multiplicityk) and report all isospectral graphs found when the number of vertices was n < 9. Tile search for isospectrul multitrees was carried out systematically by constructing the characteristic polynomials of all trees having one weighted edge. For all multitrees havingn < 7 vertices, we tabulated the coefficients of the characteristic polynomial. We restricted the analysis to trees with Me maximal valencyd = 4. The number of graphs considered exceeds 300. The smallest pair of isospectral multitrees (i.e. trees with a multiple edge) hasn = 6 vertices, There is a pair of trees whenn = 7, three pairs whenn = 8, and five pads whenn = 9. In all cases, whenk = I is assumed, isospectral multitrees reduce to the same tree. Whenk = 0 is assume(], isospectral trees produce either the same disconnected graph, or an isospectral forest.Dedicated to Basil E. Gillam, Professor emeritus of the Department of Mathematics and Computer Science at Drake University.Reported in part at the 1986 International Congress of Mathematicians, Berkeley, California, USA.Operated for the U.S. Department of Energy by the Iowa State University under Contract No. W-7405-Eng-82. This work was supported in part by the Office of R.S. Hansen, Director.     

Computer generation of acyclic graphs based on local vertex invariants and topological indices. Derived canonical labelling and coding of trees and alkanes

A new procedure (GENLOIS) is presented for generating trees or certain classes of trees such as 4-trees (graphs representing alkanes), identity trees, homeomorphical irreducible trees, rooted trees, trees labelled on a certain vertex (primary, secondary, tertiary, etc.). The present method differs from previous procedures by differentiating among the vertices of a given parent graph by means of local vertex invariants (LOVIs). New graphs are efficiently generated by adding points and/or edges only to non-equivalent vertices of the parent graph. Redundant generation of graphs is minimized and checked by means of highly discriminating, recently devised topological indices based either on LOVIs or on the information content of LOVIs. All trees onN + 1 (N + 1 < 17) points could thus be generated from the complete set of trees onN points. A unique cooperative labelling for trees results as a consequence of the generation scheme. This labelling can be translated into a code for which canonical rules were recently stated by A.T. Balaban. This coding appears to be one of the best procedures for encoding, retrieving or ordering the molecular structure of trees (or alkanes).Dedicated to Professor Alexandru T. Balaban on the occasion of his 60th anniversary.     

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 Kekulé 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 Kekulé patterns and Clar (resonant) patterns are revealed.     

Cyclical Edge-Connectivity of Fullerene Graphs and (k, 6)-Cages

It is shown that every fullerene graph G is cyclically 5-edge-connected, i.e., that G cannot be separated into two components, each containing a cycle, by deletion of fewer than five edges. The result is then generalized to the case of (k,6)-cages, i.e., polyhedral cubic graphs whose faces are only k-gons and hexagons. Certain linear and exponential lower bounds on the number of perfect matchings in such graphs are also established.     

A lower bound on the number of elementary components of essentially disconnected generalized polyomino graphs

An essentially disconnected generalized polyomino graph is defined as a generalized polyomino graph with some perfect matchings and forbidden edges. The number of perfect matchings of a generalized polyomino graph G is the product of the number of perfect matchings of each elementary component in G. In this paper, we obtain a lower bound on the number of elementary components of essentially disconnected generalized polyomino graphs.     

Some results on energy of unicyclic graphs with n vertices

A unicyclic graph is a connected graph whose number of edges is equal to the number of vertices. Hou (J Math Chem 29:163–168, 2001) first considered the minimal energy for general unicyclic graphs. In this paper, we determine the unicyclic graphs with the minimal energy in U_n^l{\mathcal {U}_n^l} and the unicyclic graphs with the first forth smallest energy in U_n (n 3 13){\mathcal {U}_n\,(n\geq 13)} vertices.     

Hückel and Möbius cyclic conjugated molecules

Generalized graphs represent Hückel-type and Möbius-type polycyclic conjugated systems. We show that the number of generalized graphs with different spectra for a given parent graph is not larger than 2 ^N(R) and is equal to 2 ^N(R) if no two rings are equivalent,N(R) being the number of rings (fundamental circuits) in the parent graph. We demonstrate that the rule for the stability of generalized graphs, proved in a previuos paper, and the information on the relative magnitudes of the effects of individual circuits enable one to predict the stabilities of generalized graphs without performing numerical calculations.     

From symmetry-labeled quotient graphs of crystal nets to coordination sequences

The combinatorial topology of crystal structures may be described by finite graphs, called symmetry-labeled quotient graphs or voltage graphs, with edges labeled by symmetry operations from their space group. These symmetry operations themselves generate a space group which is generally a non-trivial subgroup of the crystal space group. The method is an extension of the so-called vector method, where translation symmetries are used as vector labels (voltages) for the edges of the graph. Non-translational symmetry operations may be used as voltages if they act freely on the net underlying the crystal structure. This extension provides a significant reduction of the size of the quotient graph. A few uninodal and binodal nets are examined as illustrations. In particular, various uninodal nets appear to be isomorphic to Cayley color graphs of space group. As an application, the full coordination sequence of the diamond net is determined.     

A graphical evaluation of characteristic polynomials of huckel trees

Characteristic polynomials of acyclic carbon chains (Huckel trees) are treated in a systematic way. Formulas of coefficients (a_k) of the polynomial are obtained in terms of connectivities that were introduced for dealing with moments in a previous paper. Based on the meaning of a_k, a graph-theoretical analysis is given such that a_k can be expressed as a linear combination of binomial factors specified by a set of graphs containing ½k edges. The numerical relationship is disclosed between each binomial factor and its specified graph. This stimulates the proposal of a novel approach for evaluating ak by simply collecting the graph set of defnite edges. The approach is equally applicable for the evaluation of matching polynomials of cyclic systems and extendable to the investigation of general trees.     

部分同谱分子

本文在HMO近似内讨论部分同谱分子。认为在一个没有对称面的分子中, 可能隐含着一个对称面。对称面将分子分成两部分。这两部分所表示的共轭体系是部分同谱分子。它们与整个分子也是部分同谱分子。文中讨论了三顶点链图, 这种链图表示了无数个部分同谱分子进行收缩的总结果, 同时用少数例子加以说明。     

On reciprocal molecular topological index

We report some properties of the reciprocal molecular topological index RMTI of a connected graph, and, in particular, its relationship with the first Zagreb index M₁. We also derive the upper bounds for RMTI in terms of the number of vertices and the number of edges for various classes of graphs, including K _r+1 -free graphs with r ≥ 2, quadrangle-free graphs, and cacti. Additionally, we consider a Nordhaus-Gaddum-type result for RMTI.     

Graph theory and the PPP method

A graph theoretical formulation of the PPP method is presented. A weighted adjacency matrix of the PPP graph is given, wherein the off-diagonal elements are the bond orders. The automorphism group of the PPP graph is defined and shown to be isomorphic with the permutational subgroup of the permutation-inversion group of the molecule. It is demonstrated that the characteristic polynomial of the adjacency matrix of the PPP bond graph is invariant in every SCF iteration. It is shown that the PPP spectra discriminate isospectral graphs.Camille and Henry Dreyfus Teacher-Scholar.     

Two metrics for a graph-theoretical model of organic chemistry

A graph-theoretical model of organic chemistry is proposed. The main idea behind this model is a molecular graph in the form of a multigraph with loops; its vertices are evaluated by vertex labels (atomic symbols). The chemical distance between two graphs from the same family of isomeric graphs is based on the maximal common subgraph. The produced reaction graph is composed of the minimal number of edges and/or loops. The reaction distance assigned to the chemical transformationG ₁ G ₂ is equal to the minimal number of the so-called elementary transformations that are necessary for the transformation ofG ₁ intoG ₂ Because these metrics are not isometric, the resulting reaction graphs may depend on the metric used.Dedicated to the memory of Professor Milan Sekanina