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.     

Application of molecular orbital graph theory to vibrational problems of finite chain systems

A generating function approach based on molecular orbital graph theory is presented that provides a straightforward way of obtaining the secular polynomials and energy bands for repeated unit systems from polynomial recurrence expressions. The possibility of obtaining the analytical energy-level spectrum of the system can also be predicted. These results are then used to discuss the vibrational problems of finite chain systems with single- and double-component lattices. It seems to be the first report describing the vibrational states of an (AB)_N chain.     

Spectra of chemical trees

A method is developed for obtaining the spectra of trees of NMR and chemical interests. The characteristic polynomials of branched trees can be obtained in terms of the characteristic polynomials of unbranched trees and branches by pruning the tree at the joints. The unbranched trees can also be broken down further till we obtain a tree containing just two vertices. This effectively reduces the order of the secular determinant of the tree we started with to determinants of orders atmost equal to the number of vertices in the branch containing the largest number of vertices. An illustrative example of a NMR graph is given for which the 22 × 22 secular determinant is reduced to determinants of orders atmost 4 × 4 in just the second step of the algorithm. The tree pruning algorithm can be applied even to trees with no symmetry elements and such a factoring can be achieved. Methods developed here can be elegantly used to find if two trees are cospectral and to construct cospectral trees.     

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     

Contemplation on the heats of combustion of isomeric hydrocarbons

Within the Hückel molecular orbital theory, the heats of combustion of isomeric hydrocarbons are related to some topological factors. The standard heats of combustion values of alternant hydrocarbons, expressed as kcal/g, seem to be related to a₄ coefficient of their secular polynomials.     

Localized impurity states in the Hartree-Fock,LCAO approximation. I.

One-electron energies and wave functions for deep trap impurity electrons in a crystal are calculated by the Hartree-Fock, single determinant method. The interactions arising from a many-electron single determinant crystal wave function, with automatic inclusion of exchange effects, are those which determine the one-electron functions and energies. The crystal plus impurity system has no translational symmetry and hence the Bloch theorem is not applicable for the solution of the essentially infinite Hartree-Fock eigenvalue matrix. Thus we develop a technique in which the Hamiltonian and overlap matrices are written in terms of bordered matrices, with the interaction of the impurity functions with the rest of the crystal environment contained in the bordering rows and columns. The resulting secular equation explicitly includes the effects of orthogonalization of the entire basis set, including the impurity functions. This technique could be used in an iterative calculation of the electronic structure of a small number of electrons, assuming that the rest of the electrons in the environment are fixed according to an initial estimate.     

Matrix formulation of the generalized Hartree-Fock methods

Spin-projected one-particle density and spin density matrices are presented as polynomials of suitable unprojected quantities with generalized Sasaki-Ohno coefficients. Thus an explicit form of Harriman's theorems is given. For the two-particle spatial density matrix an expansion in direct products of powers of unprojected residual electron and spin density matrices is given. For these basic matrices of the scheme the variational spin-extended equations are formulated.     

An exponential Galerkin method for solutions of HIV infection model of CD4+ T-cells

In this study, we consider a nonlinear first order model about the infection of CD4⁺ T-cells by HIV. In order to solve it numerically, we present a new method based on exponential polynomials reminiscent of the Galerkin method. Considering the approximate solutions in the form of exponential polynomials, we first substitute these approximate solutions in the original model. Some relations are thus obtained, which we express in terms of matrices. Taking inner product of a set of exponential functions with these matrix expressions then yields a nonlinear system of algebraic equations. The solution of these equations gives the approximate solutions of the model. Additionally, the technique of residual correction, which aims to reduce the error of the approximate solution by estimating this error, is discussed in some detail. The method and the residual correction technique are illustrated with an example. The results are also compared with numerous existing methods from the literature.     

Characteristic polynomials of organic polymers and periodic structures

It is shown that the Frame's method (also, Le Verrier-Faddeev's method) for characteristic polynomials of chemical graphs can be extended to periodic graphs and structures. The finite periodic structures are represented by cyclic structures in the Born-von Kárman boundary condition which leads to complex matrices. In this article we demonstrate that our earlier computer program (based on Frame's method) can be extended to these periodic networks. The characteristic polynomials of several lattices such as polydiacetylenes, one-dimensional triangular, square, and hexagonal lattices are obtained. These polynomials can be obtained with very little computer time using this method.     

Gegenbauer polynomials in a theoretical study of the vibrational structure of the Ca+–H2 system

This work presents an application of Gegenbauer polynomials in vibrational calculations. We illustrated that by example calculations of vibrational structure of the Ca⁺–H₂ exciplex, in the state correlated with 3D calcium ion state. For this case Gegenbauer polynomials are used for formation of a basis set for a bending mode. We showed that this basis set leads to a faster convergence of results than a basis set formed from Legendre polynomials. Additionally we compared vibrational structure obtained in this manner with results of discrete variable representation-distributed Gaussian basis (DVR–DGB) method.     

The characteristic polynomials of structures with pending bonds

It is well known [1] that the calculation of characteristic polynomials of graphs of interest in Chemistry which are of any size is usually extremely tedious except for graphs having a vertex of degree 1. This is primarily because of numerous combinations of contributions whether they were arrived at by non-imaginative expansion of the secular determinant or by the use of some of the available graph theoretical schemes based on the enumeration of partial coverings of a graph, etc. An efficient and quite general technique is outlined here for compounds that have pending bonds (i.e., bonds which have a terminal vertex). We have extended here the step-wise pruning of pending bonds developed for acyclic structures by one of the authors [2] for elegant evaluation of the characteristic polynomials of trees by accelerating this process, treating pending group as a unit. Further, it is demonstrated that this generalized pruning technique can be applied not only to trees but to cyclic and polycyclic graphs of any size. This technique reduces the secular determinant to a considerable extent. The present technique cannot handle only polycyclic structures that have no pending bonds. However, frequently such structures can be reduced to a combination of polycyclic graphs with pending bonds [3] so that the present scheme is applicable to these structures too. Thus we have arrived at an efficient and quite a simple technique for the construction of the characteristic polynomials of graphs of any size.     

On the representations of the rotation group

The matrices of the irreducible representations of the 3-dimensional rotation group are shown to be related to Krawtchouk's orthogonal polynomials of a discrete variable x = j – m', whose degrees are given by n = j + m. The relation follows directly from the recurrence formulas satisfied by the matrix elements and permits a concise development of the formal properties of the rotation matrices. In particular, an asymptotic relation for large j is developed that generalizes a formula first discussed for a special case by Wigner.     

Symmetry factoring of the characteristic equations of graphs corresponding to polyhedra

A systematic procedure is described which uses two-and three-fold symmetry elements in graphs to reduce their adjacency matrices to lead to corresponding factorings of their characteristic polynomials. A graph splitting algorithm based on this matrix reduction procedure is described. Applications of these methods to the factoring of the characteristic polynomials of 28 polyhedra with nine or less vertices are given. General expressions for the eigenvalues of prisms, pyramids, and bipyramids in terms of the eigenvalues of their basal or equatorial regular polygons are calculated by closely related matrix methods.