Abstract
The structural dependency (effect of branching and cyclisation) of an alternative form, the Chebyshev expansion, for the characteristic polynomial were investigated systematically. Closed forms of the Chebyshev expansion for an arbitrary star graph and a bicentric tree graph were obtained in terms of the “structure factor” expressed as the linear combination of the “step-down operator”. Several theorems were also derived for non-tree graphs. Usefulness and effectiveness of the Chebyshev expansion are illustrated with a number of examples. Relation with the topological index (Z G ) was discussed.
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This condition can further be reduced ton ≥ 2k + 2m−1.
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For a star graph the problem is to find the coefficient of\(\hat d^4 \) from Eq. (29). See also Eq. (36) for a bicentric graph, where the\(\hat d^4 \) term comes only from the branching points but not from the crossed terms.
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Operated for the U.S. Department of Energy by ISU under contract no. W-ENG-7405-82. Supported in part by the Office of Director
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Hosoya, H., Randić, M. Analysis of the topological dependency of the characteristic polynomial in its chebyshev expansion. Theoret. Chim. Acta 63, 473–495 (1983). https://doi.org/10.1007/BF00552651
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DOI: https://doi.org/10.1007/BF00552651