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 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.     

Parallel algorithm for the computation of characteristic polynomials of chemical graphs

A parallel algorithm is developed for the first time based on Frame's method to compute the characteristic polynomials of chemical graphs. This algorithm can handle all types of graphs: ordinary, weighted, directed, and signed. Our algorithm takes only linear time in the CRCW PRAM model with O(n³) processors whereas the sequential algorithm takes O(n⁴) time. Especially when the number of vertices of the graph is large this method will be more efficient than the recently developed vectorized Frame and Givens–Householder methods.     

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.     

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.     

The use of Frame’s method for the characteristic polynomials of chemical graphs

The evaluation of characteristic polynomials of graphs of any size is an extremely tedious problem because of the combinatorial complexity involved in this problem. While particular elegant methods have been outlined for this problem, a general technique for any graph is usually tedious. We show in this paper that the Frame method for the characteristic polynomial of a matrix is extremely useful and can be applied to graphs containing large numbers of vertices. This method reduces the difficult problem of evaluating the characteristic polynomials to a rather simple problem of matrix products. The coefficients in the characteristic polynomial are generated as traces of matrices generated in a recursive product of two matrices. This method provides for an excellent and a very efficient algorithm for computer evaluation of characteristic polynomials of graphs containing a large number of vertices without having to expand the secular determinant of the matrix associated with the graph. The characteristic polynomials of a number of graphs including that of a square lattice containing 36 vertices are obtained for the first time.     

Computer generation of matching polynimials of chemical graphs and lattices

A computer program in Pascal is developed for computing the matching polynomials of graphs and lattices. This program is based on the recursive relation for matching polynomials outlined by Hosoya [Bull. Chem. Soc. Jpn., 44 , 2332 (1971)], Gutman and Hosoya [Theor. Chim. Acta, 48 , 279 (1978)], and others. The graph whose matching polynomial is of interest is reduced recursively until the graph reduces to several trees. The characteristic polynomial of a tree is the same as the matching polynomial. The characteristic polynomials of resulting trees are computed using the computer program based on Frame's method developed by Balasubramanian [Theor. Chim. Acta, 65 , 49 (1984)]; J. Comput. Chem., 5 , 387 (1984). The resulting polynomials are then assembled to compute the matching polynomial of the initial graph. The program is especially useful in generating the matching polynomials of graphs containing a large number of vertices. The matching polynomials thus generated are potentially useful in several applications such as lattice statistics (dimer covering problem), aromaticity, valence bond methods (enumeration of perfect matchings) in the calculation of grand canonical partition functions, in the computation of thermodynamic properties of saturated hydrocarbons, and in chemical documentation.     

On the characteristic polynomials of spirographs and related graphs

General recursive techniques are used to determine recurrence relations for the characteristic polynomials of graphs associated with various ring compounds.     

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.     

Moments and characteristic polynomials for square lattice graphs

The method of moments is used to derive closed analytical expressions for the characteristic polynomials of square lattice graphs. We obtain exact analytical formulae for three-dimensional cubic lattices, square lattices, tubular square lattices, and cylindrical square lattices containing any number of vertices.Camille and Henry Dreyfus Teacher-Scholar.     

Moments and characteristic polynomials of bipartite Hückel graphs

Moments (u _k ) and coefficients of characteristic polynomials (a _k ) have been evaluated in terms of molecular fragments up tok = 12 for bipartite Hückel graphs. Based on combinatorial analysis, each coefficient can be derived as a combination of binomial factors mapping to the corresponding multi-component graphs. The general formula becomes lengthy as k increases, but can be considerably simplified for a homologous series. This has been illustrated by dealing with the cata-condensed benzenoid hydrocarbons as a corollary where a rather compact set ofa _k has been deduced. On combining the present result with Coulson's formula, one gains insight into the relative stability of isomers in relationship to the energy contribution of fragments classified as stabilized and destabilized species.received by the Publisher 20 September 1989     

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.     

Characteristic polynomials of chemical graphs via symmetric function theory

The characteristic polynomial corresponding to the adjacency matrix of a graph is constructed by using the traces of the powers of the adjacency matrix to calculate the coefficients of the characteristic polynomial via Newton's identities connecting the power sum symmetric functions and the elementary symmetric functions of the eigenvalues. It is shown that Frame's method, very recently employed by Balasubramanian, is nothing but symmetric functions and Newton's identities.     

Imminant polynomials of graphs

Summary The imminant polynomials of the adjacency matrices of graphs are defined. The imminant polynomials of several graphs [linear graphs (L _n ), cyclic graphs (C _n ) and complete graphs (K _n )] are obtained. It is shown that the characteristic polynomials and permanent polynomials become special cases of imminant polynomials. The connection between the Schur-functions and imminant polynomials is outlined.Cammile & Henry Dreyfus Teacher-scholar     

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 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.     

Characteristic polynomials of large graphs. On alternate form of characteristic polynomial [1]

A recursion exists between the absolute magnitudes of the coefficients of the characteristic polynomials of certain families of cyclic and acyclic graphs which makes their computation quite easy for very large graphs using a pencil-and-a-paper approach. Structural requirements are given for such families of graphs which are of interest to the problem of recognition defined in [1].