• Kali Kinkar Datta

      Articles written in Journal of Chemical Sciences

    • A graph-theoretical method for stepwise factorisation of symmetric graphs for simultaneous determination of eigenvectors and eigenvalues

      Kali Kinkar Datta Asok K Mukherjee

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      A simple pictorial algorithm for factorisation of symmetric chemical graphs (weighted and unweighted) leading to simultaneous determination of their eigenvalues and eigenvectors has been devised. The method does not require group-theoretical techniques (viz. identification of the point group of the species under study, formation of symmetryadopted linear combinations using character tables etc.). It requires consideration of only one symmetry element, e.g., a reflection plane and is based on elementary row and column operations which keep the secular determinant of the adjacency matrix unchanged (except possibly for a multiplicative constant).

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

      Asok K Mukherjee Kali Kinkar Datta

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      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 (Kn) and annulenes (Cn), viz. (i)Kn has an eigenvalue −1 with (n−1)-fold degeneracy and (ii) Cn 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.

    • Topological solution of algebraic eigenvalue problem

      Kali Kinkar Datta

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      A simple graph theoretical algorithm for simultaneous determination of eigenfunctions, eigenvalues and characteristic polynomials of real symmetric matrices has been developed. The method starts with representing the matrixA−λI, whereI is an unit matrix of the size ofA, by an undirected weighted graph (G) and an assumed set of eigenfunctions. Conditions necessary to disconnect one vertex completely fromG are then developed. The method does not require any property related to the geometrical symmetry group of the graph and is applicable even to matrices containing a number of multiple eigenvalues.

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