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.
Similar content being viewed by others
References
Randić, M.: Theoret. Chim. Acta (Berl.) (Submitted)
Randić, M.: J. Comput. Chem.3, 421 (1982)
Hori, J., Asahi, T.: Prog. Theoret. Phys.17, 523 (1957)
Jido, Y., Inagaki, T., Fukutome, H.: Prog. Theoret. Phys.48, 808 (1972)
Tang, A.-C., Kiang, Y.-S.: Sci. Sin.19, 208 (1976); ibid.20, 595 (1977); Kiang, Y.-S.: Int. J. Quantum Chem. Symp.15, p 293 (1981)
Kaulgud, M. V., Chitgopkar, V. H.: J. Chem. Soc. Faraday Trans. 273, 1385 (1977);74, 951 (1978)
Gutman, I.: J. Chem. Soc. Faraday Trans. 276, 1161 (1980)
Graovac, A., Polansky, O. E., Tyutyulkov, N. N.: Croat. Chem. Acta (in press)
Balasubramanian, K.: Int. J. Quant. Chem.21, 581 (1982); Balasubramanian, K., Randić, M.: Theoret. Chim. Acta (Berl.)61, 307 (1982)
McClelland, B. J.: J. Chem. Soc. Faraday Trans. 278, 911 (1982)
Hosoya, H., Ohkami, N.: J. Comput. Chem. (in press)
Hosoya, H.: Bull. Chem. Soc. Jpn.44, 2332 (1971)
Hosoya, H.: Theoret. Chim. Acta (Berl.)25, 215 (1972)
Aihara, J.: J. Am. Chem. Soc.98, 2750 (1976)
Gutman, I., Milun, M., Trinajstić, N.: J. Am. Chem. Soc.99, 1692 (1977)
Farrell, E. J.: J. Combinant. TheoryB27, 75 (1979)
Gutman, I., Hosoya, H.: Theoret. Chim. Acta (Berl.)48, 279 (1978)
Schaad, L. J., Hess, B. A. Jr., Nation, J. B., Trinajstić, N., Gutman, I.: Croat. Chem. Acta52, 233 (1979)
Godsil, C. D., Gutman, I.: Croat. Chem. Acta54, 53 (1981)
Kanazawa, T., Hosoya, H., Iwata, S.: Int. J. Quant. Chem.15, 243 (1979)
Another notationL n is used in Refs. 1 and 2.
Abramowitz, M., Stegun, I. A. (ed.): Handbook of mathematical functions. Washington, D.C.: Nat. Bur. Stand. 1964
Tan, A.-C., Kiang, Y.-S., Yan, G.-S., Dai, S.-S.: Graph theoretical molecular orbitals. (in Chinese) Beijin: Publishing House of Science 1980
Hosoya, H.: Natur. Sci. Kept. Ochanomizu Univ.32, 127 (1981); see also Gutman, I.: MATCH No. 6, 75 (1979)
Collatz, L., Sinogowitz, U.: Abh. Math. Sem. Univ. Hamburg21, 63 (1957)
The expressions up ton = 12 are tabulated in Table 22.8 of Ref. 22.
Heilbronner, E.: Helv. Chim. Acta36, 170 (1952)
Lovász, L., Pelikán, J.: Perid. Math. Hung.3, 49 (1973)
Schwenk, A. J.: Lecture Notes Math.406, 153 (1973)
Lyusternik, L. A., Chervonenkis, O. A., Yanpol'skii, A. R.: Handbook for computing elementary functions, p. 163. Oxford: Pergamon 1965
Riordan, J.: Combinatorial identities, p. 101. New York: Wiley 1968.
Discussion on the Catalan numbers relevant to chemistry is given in Ohkami, N., Hosoya, H.: Bull. Chem. Soc. Jpn.52, 1624 (1979)
Although many researchers [12, 23, 34] tried to express the characteristic polynomial in terms of the Chebyshev polynomial, no systematic study seems to have been done before Refs. 1 and 2
Trinajstić, N.: Croat. Chem. Acta49, 593 (1977)
The characteristic polynomials for the lower members of trees and non-trees are tabulated in Refs. 36 and 37.
Mizutani, K., Kawasaki, K., Hosoya, H.: Natur. sci. Rept. Ochanomizu Univ. (Tokyo)22, 39 (1972)
Kawasaki, K., Mizutani, K., Hosoya, H.: Natur. Sci. Rept. Ochanomizu Univ. (Tokyo)22, 181 (1972)
This condition can further be reduced ton ≥ 2k + 2m−1.
Although the term “topological index” was originally proposed by one of the present authors [12] in 1971, currently it is used as a common noun rather than a proper noun. In this paper the “topological index” is meant by the original “Z index”
Mizutani, K., Kawasaki, K., Hosoya, H.: Bull. Chem. Soc. Jpn.45, 3415 (1972)
Narumi, H., Hosoya, H.: Bull. Chem. Soc. Jpn.53, 1228 (1980)
Hosoya, H., Hosoi, K., Gutman, I.: Theoret. Chim. Acta (Berl.)38, 37 (1975)
Hosoya, H., Murakami, M.: Bull. Chem. Soc. Jpn.48, 3512 (1975)
Hosoya, H., Hosoi, K.: J. Chem. Phys.64, 1065 (1976)
Hosoya, H.: J. Chem. Doc.12, 181 (1972)
Christofides, N.: Graph theory. An algorithmic approach. London: Academic Press 1975;
Bollobás, B.: Graph theory. An introductory course. New York: Springer-Verlag 1979
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.
Instead, the matching polynomial of graphC n is identical to the Chebyshev polynomial of the first kind [23, 24].
Extensive discussion is given in Cvetković, D. M., Doob, M., Sachs, H.: Spectra of graphs, pp. 156. New York: Academic Press 1980
Author information
Authors and Affiliations
Additional information
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
Rights and permissions
About this article
Cite this article
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/BF02394808
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02394808