Two new graph-theoretical methods for generation of eigenvectors of chemical graphs |
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Authors: | Asok K Mukherjee and Kali Kinkar Datta |
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Institution: | (1) Chemistry Department, Burdwan Raj College, 713 104 Burdwan, India |
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Abstract: | 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. |
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Keywords: | Graph-theory Ulam’ s subgraphs characteristic polynomials eigenvectors multiplicity of degenerate eigenvalues complete graphs cofactor polynomials |
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