Computer generation of the characteristic polynomials of chemical graphs

A computer program based on the Frame method for the characteristic polynomials of graphs is developed. This program makes use of an efficient polynomial algorithm of Frame for generating the coefficients in the characteristic polynomials of graphs. This program requires as input only the set of vertices that are neighbors of a given vertex and with labels smaller than the label of that vertex. The program generates and stores only the lower triangle of the adjacency matrix in canonical ordering in a one-dimensional array. The program is written in integer arithmetic, and it can be easily modified to real arithmetic. The coefficients in the characteristic polynomials of several graphs were generated in less than a few seconds, thus solving the difficult problem of generating characteristic polynomials of graphs. The characteristic polynomials of a number of very complicated graphs are obtained including for the first time the characteristic polynomial of an honeycomb lattice graph containing 54 vertices.     

Computer generation of king and color polynomials of graphs and lattices and their applications to statistical mechanics

A computer program is developed in Pascal for the generation of king and color polynomials of graphs. The king polynomial was defined by Motoyama and Hosoya and was shown to be useful in dimer statistics, enumeration of Kekulé structures, etc. We show that the king polynomial of a lattice is the same as the color polynomial of the associated dualist graph, where the color polynomial is defined here as the number of ways of coloring the vertices of a graph with one type of color (say, green) such that two adjacent vertices are not colored with the same color. Applications of these polynomials to exact finite method of lattice statistics are outlined.     

Computer generation of distance polynomials of graphs

A computer program is developed to compute distance polynomials of graphs containing up to 200 vertices. The code also computes the eigenvalues and the eigenvectors of the distance matrix. It requires as input only the neighborhood information from which the program constructs the distance matrix. The eigenvalues and eigenvectors are computed using the Givens-Householder method while the characteristic polynomials of the distance matrix are constructed using the codes developed by the author before. The newly developed codes are tested out on many graphs containing large numbers of vertices. It is shown that some cyclic isospectral graphs are differentiated by their distance polynomials although distance polynomials themselves are in general not unique structural invariants.     

Computer generation of edge groups and edge colorings of graphs

A computer code and nonnumerical algorithm are developed to construct the edge group of a graph and to enumerate the edge colorings of graphs of chemical interest. The edge colorings of graphs have many applications in nuclear magnetic resonance (NMR), multiple quantum NMR, enumeration of structural isomers of unsaturated organic compounds, and in the construction of configurational integral expansion series in statistical mechanics. The code developed is applied to many NMR graphs, complete graphs containing up to 10 vertices, and the Petersen graph.     

Computer generation of characteristic polynomials of edge-weighted graphs,heterographs, and directed graphs

The computer code developed previously (K. Balasubramanian, J. Computational Chem., 5 , 387 (1984)) for the characteristic polynomials of ordinary (nonweighted) graphs is extended in this investigation to edge-weighted graphs, heterographs (vertex-weighted), graphs with loops, directed graphs, and signed graphs. This extension leads to a number of important applications of this code to several areas such as chemical kinetics, statistical mechanics, quantum chemistry of polymers, and unsaturated systems containing heteroatoms which include bond alternation. The characteristic polynomials of several edgeweighted graphs which may represent conjugated systems with bond alternations, heterographs (molecules with heteroatoms), directed graphs (chemical reaction network), and signed graphs and lattices are obtained for the first time.     

Computer generation of chemical structures from known fragments

The interactive generation of chemical structures from given fragments is described and discussed. It is implemented as a part of our expert system CARBON, based on C-13 NMR spectra. As it is designed, this program can also be a useful tool in the structure elucidation process when information on parts of the structure is obtained by other means (IR, mass and other spectrometries, chemical analysis, other relevant information). The topological characteristics of candidate fragments are first chosen interactively and then the elements are connected in all topologically possible ways. In the following step, the topological building blocks are substituted by chemical structural fragments resulting in a set of all chemical structures consistent with the input information.     

Two new graph-theoretical methods for generation of eigenvectors of chemical graphs

Two new graph-theoretical methods, (A) and (B), have been devised for generation of eigenvectors of weighted and unweighted chemical graphs. Both the methods show that not only eigenvalues but also eigenvectors have full combinatorial (graph-theoretical) content. Method (A) expresses eigenvector components in terms of Ulam’s subgraphs of the graph. For degenerate eigenvalues this method fails, but still the expressions developed yield a method for predicting the multiplicities of degenerate eigenvalues in the graph-spectrum. Some well-known results about complete graphs (K _n) and annulenes (C _n ), viz. (i)K _n has an eigenvalue −1 with (n−1)-fold degeneracy and (ii) C _n cannot show more than two-fold degeneracy, can be proved very easily by employing the eigenvector expression developed in method (A). Method (B) expresses the eigenvectors as analytic functions of the eigenvalues using the cofactor approach. This method also fails in the case of degenerate eigenvalues but can be utilised successfully in case of accidental degeneracies by using symmetry-adapted linear combinations. Method (B) has been applied to analyse the trend in charge-transfer absorption maxima of the some molecular complexes and the hyperconjugative HMO parameters of the methyl group have been obtained from this trend.     

Distance-restricted matching extendability of fullerene graphs

A fullerene graph is a 3-connected cubic plane graph whose all faces are bounded by 5- or 6-cycles. In this paper, we show that a matching M of a fullerene graph can be extended to a perfect matching if the following hold: (i) three faces around each vertex in \(\{x,y:xy\in M\}\) are bounded by 6-cycles and (ii) the edges in M lie at distance at least 13 pairwise.     

Computer generation of spectra of graphs: Applications to C60 clusters and other systems

A computer code based on the Givens–Householder matrix diagonalization method is used to calculate the spectra of graphs containing a large number of vertices. The code is most general in that it can handle graphs containing 200 or more vertices. Further, the code can be used to generate the spectra of weighted graphs. The program requires as input only the neighborhood table of the graph. The spectra of many graphs are generated for the first time in less than a few minutes of computer time. Applications to a number of chemical systems including two forms (foot and hand) of the recently synthesized C₆₀ cluster and the effect of bond alternation on these systems are discussed. In addition, the spectra of square and honeycomb lattices and the characteristic polynomials of the foot and hand forms of the C₆₀ cluster are obtained.     

Homomorphism polynomials of chemical graphs

We say that a graphG ishomomorphic to a graphH if there is a mappingp from the vertices of G onto the vertices ofH such thatp(u) andp() are adjacent inH wheneveru and are adjacent in G. Thehomomorphism polynomial of a graphG is a polynomial in two variables that counts the number of homomorphisms ofG onto the complete graph of each order. This polynomial can be computed recursively in an analog to the chromatic polynomial. In this paper, we present some results regarding the homomorphism polynomials of the graphs of chemical compounds — in particular, alkane isomers. The coefficients of the homomorphism polynomial can be used to predict the rankings of compounds with respect to several chemical properties. Our results seem to refine those obtained by Randi et al. from path lengths.     

Symmetry groups of chemical graphs

The symmetry groups of all trees are shown to be expressible as generalized wreath products by a tree pruning algorithm. The symmetry groups of certain cyclic graphs which can be expresssed as generalized compositions are also shown to be generalized wreath products. The symmetry groups of complete multipartite graphs can be obtained in a similar manner. Character tables of symmetry groups of certain chemical graphs are also obtained.     

Adjacency matrices and chemical transformation graphs

The vast numbers of molecular objects and chemical transformations of a substrate cannot be evaluated even intuitively, which necessitates the development of simple reaction monitoring methods. Here this problem is formulated for the first time, and possible solutions are substantiated.     

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.     

Computational algorithms for matching polynomials of graphs from the characteristic polynomials of edge-weighted graphs

Computational algorithms are described which provide for constructing the set of associated edge-weighted directed graphs such that the average of the characteristic polynomials of the edge-weighted graphs gives the matching polynomial of the parent graph. The weights were chosen to be unities or purely imaginary numbers so that the adjacency matrix is hermitian. The computer code developed earlier by one of the authors (K.B.) is generalized for complex hermitian matrices. Applications to bridged and spirographs, some lattices and all polycyclic graphs containing up to four cycles are considered.     

Extended connectivity in chemical graphs

The true nature of the extended connectivity, used in Morgan algorithm for the canonical numerotation of points in chemical graphs, is discussed. An alternative method for its calculation based on the number of walks is described and shown to yield results identical to Morgan's method.     

Configurations,regular graphs and chemical compounds

The mathematical structures of a configuration and a regular graph and certain chemical compounds are compared. Some recursive construction methods for these structures are described. A short survey on results about configurations concludes the paper.     

Computer enumeration of walks on directed graphs

A vectorized computer code is developed for the enumeration of walks through the matrix power method for directed graphs. Application of this code to several graphs is considered. It is shown that the coefficients in the generating functions for signed graphs are much smaller in magnitude. It is shown that self-avoiding walks on some graphs can be enumerated as a linear combination of walk GFs of directed paths and rooted-directed paths.