Bholanath Mandal

A PASCAL'S TRIANGLE-LIKE APPROACH FOR THE DETERMINATION OF CHARACTERISTIC POLYNOMIAL COEFFICIENTS OF RECIPROCAL GRAPHS

Characteristic polynomial coefficients of three classes of graph, namely Ln + n(p), Cn + n(p) and K 1,n-1 + n(p), which are known to have reciprocal pairs of eigenvalues, have been shown to be generated by simple manipulation of the Pascals triangle.

Algorithms to calculate the distance numbers and the Wiener indices of linear and cylindrical poly (p-phenylene) in terms of number of hexagonal rings

The compounds of poly (p-phenylene) are important materials for electro-optical and electronic applications. Algorithms for the calculation of the distances and hence the Wiener indices (W) of linear poly (p-phenylene) and cylindrical poly (p-phenylene) systems have been developed that require only the number of hexagonal rings present in the systems. Ambient conditions density and the bulk modulus of linear poly (p-phenylenes) with two to six phenyl rings have correlated with W 1/4. Cylindrical poly (p-phenylene) systems with 5 and 6 phenyl rings are found to occur as intervening strands in between two caps of fullerene cages.

Wiener and Hosoya indices of reciprocal graphs

Two types of topological indices, Wiener and Hosoya indices, of three classes of reciprocal graphs (namely , and ) have been shown to be expressed in terms of the number of pendant vertices (n). The Wiener index is expressed in analytical form whereas the Hosoya index is expressed either by recurrence relations or directly by matrix products for which a facile computer program can be written. It has also been shown that the Hosoya index of can be obtained in analytical form.

Graph theoretical analysis on the kinetic rate equations of linear chain and cyclic reaction networks.

Graph theoretical solutions for kinetic rate equations of some reaction networks involving linear chains and cycles have been derived; condensation polymerization and long chain of radioactive decay come under the purview of the former whereas the interconversion of the species in cycles under the later. The reactions for the linear chains considered here proceed monotonically to the steady states with time whereas the cycle with all irreversible steps has been found to have either periodic or monotonic time evaluation of concentrations depending on the values of rate constants of the involved paths. In case of a cyclic reaction having all reversible paths, the condition for the microscopic reversibility has been derived on the basis of the assumption that the decay constants obtained for this case are all real.

Graph theoretical solutions for the coupled kinetic rate equations.

A graph theoretical procedure for solving multistep coupled kinetic rate equations and thereby obtaining the concentrations of the species involved in the reaction has been developed. The method so developed has been illustrated with some well-known reaction schemes.

Eigensolutions of cyclopolyacene graphs

A graph of {X, Y}-cyclopolyacene with n of hexagonal rings has been presented that contains four orbits, of which orbits 1 and 4 are occupied by the X-type of vertex and orbits 2 and 4 are occupied by the Y-type, or vice versa. Eigensolutions for such a graph have been derived in analytical form through the use of rotational symmetry followed by a plane of symmetry. Varying X ( = C, N, B, …) and Y ( = C, N, B, …) several types of cyclopolyacene graph may be obtained. Eigenvalue-expressions for such systems containing C, N and B have been shown in analytical form and their total π-electron energies with 2–6 hexagonal rings have been calculated with the help of the expressions developed.

Use of symmetry plane fragmentation and graph squaring techniques to express the eigenspectra for some vertex-weighted graphs of linear chains and cycles in analytical form

Three types of graphs of linear chains, viz. linear chains with unit increment or decrement in weight on one terminal vertex, linear chains with unit increment or decrement in weight on both the terminal vertices and linear chains with unit increment in weight on one terminal vertex and decrement in that on the other terminal vertex, have been considered. The symmetry plane fragmentation and graph squaring techniques have been exploited to express the eigenspectra of such graphs of linear chains in analytical form, and have subsequently been used to express the eigenspectra of graphs of linear chains and cycles with alternant vertex weights. The derived expressions for the eigenspectra have been used to obtain the eigenspectra of linear polyacenes, methylene-substituted linear polyacenes and cylindrical polyacene strips in analytical form.

Symmetry-adapted linear combinations for the eigenvalues and eigenvectors of reciprocal graphs

ABSTRACT Three classes of reciprocal graphs, viz. monocycle (GCn), linear chain (GLn) and star (GKn) with reciprocal pairs of eigenvalues (λ, 1/λ), are well known. Reciprocal graphs of monocycle (GCn) and linear chain (GLn) are obtained by putting a pendant vertex to each vertex of simple monocycle (Cn) and simple linear chain (Ln), respectively. A star graph of such kind is obtained by attaching a pendant vertex to the central vertex and to each of the (n − 1) peripheral vertices of the star graph (K1, (n−1)). An n-fold rotational axis of symmetry for GCn and (n − 1)-fold rotational axis of symmetry for GKn have been exploited for obtaining their respective condensed graphs. The condensed graph for GLn has been generated from that of GCn incorporating proper boundary conditions. Condensed graphs are lower dimensional graphs and are capable of keeping all eigeninformation in condensed form. Thus the eigensolutions (i.e. the eigenvalues and the eigenvectors) in analytical forms for such graphs are obtained by solving 2 × 2 or 4 × 4 determinants that in turn result in the charge densities and bond orders of the corresponding molecules in analytical forms. Some mathematical properties of the eigenvalues of such graphs have also been explored.

Cardinalities of poly(p-phenylene) graphs

Recurrence relation for the cardinalities of linear and cylindrical poly(p-phenylene) (PPP) compounds has been developed that requires the cardinalities of two of their immediate lower homologues. Such recurrence relation reduces into analytical expressions for the cardinalities under transfer matrix formalism. Ambient condition density and bulk modulus of linear PPPs are found to bear excellent linear correlation with the inverse of logarithm of their cardinalities. Topological bond orders obtained from the cardinalities of such PPPs have been found to have good linear correlations with the respective Hückel bond orders.

Construction and studies of a new class of reciprocal trees: interknitting of the Pascal's triangle

A method of construction of a new class of trees with reciprocal pairs of eigenvalues (λ, 1/λ) has been developed. They are derived from star graphs and can be symbolized as K 1, n −1 + n(p) + mK 2 (1 ≤ m ≤ n − 1 except for n = 1). The trees are minimally Kekulenoid and hence contain reciprocal pairs of eigenvalues in their eigenspectra. The characteristic polynomial coefficients of these trees with given values of n and m are shown to be obtainable by appropriate use of the Pascals triangle. A general formula for this purpose has been developed. An analytical formula for the Wiener indices of such trees in terms of m and n has been derived and some consequences of this formula are presented. The relevance of these trees to real molecular structures is discussed. The trees have been shown to be useful in observing the subspectrallity of two series of IPR fullerenes of formulae C50+10 n and C60+12 n (n is a positive integer).