Abstract
Let G be a simple graph with n vertices and let λ1, λ2, . . . , λ n be the eigenvalues of its adjacency matrix. The Estrada index of G is a recently introduced molecular structure descriptor, defined as \({EE (G) = \sum_{i = 1}^n e^{\lambda_i}}\), proposed as a measure of branching in alkanes. In order to support this proposal, we prove that among the trees with fixed maximum degree Δ, the broom B n,Δ, consisting of a star S Δ+1 and a path of length n−Δ−1 attached to an arbitrary pendent vertex of the star, is the unique tree which minimizes even spectral moments and the Estrada index, and then show the relation EE(S n ) = EE(B n,n−1) > EE(B n,n−2) > . . . > EE(B n,3) > EE(B n,2) = EE(P n ). We also determine the trees with minimum Estrada index among the trees with perfect matching and maximum degree Δ. On the other hand, we strengthen a conjecture of Gutman et al. [Z. Naturforsch. 62a (2007), 495] that the Volkmann trees have maximal Estrada index among the trees with fixed maximum degree Δ, by conjecturing that the Volkmann trees also have maximal even spectral moments of any order. As a first step in this direction, we characterize the starlike trees which maximize even spectral moments and the Estrada index.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Estrada E.: Chem. Phys. Lett. 319, 713 (2000)
Estrada E.: Bioinformatics 18, 697 (2002)
Estrada E.: Proteins 54, 727 (2004)
Estrada E.: Phys. Rev. E 75, 016103 (2007)
Estrada E., Rodríguez-Valázquez J.A., Randić M.: Int. J. Quantum Chem. 106, 823 (2006)
Gutman I., Furtula B., Marković V., Glišić B.: Z. Naturforsch. 62a, 495 (2007)
Estrada E.: Chem. Phys. Lett. 463, 422 (2008)
Estrada E., Rodríguez-Valázquez J.A.: Phys. Rev. E 71, 056103 (2005)
Estrada E., Rodríguez-Valázquez J.A.: Phys. Rev. E 72, 046105 (2005)
Gutman I., Estrada E., Rodríguez-Valázquez J.A.: Croat. Chem. Acta 80, 151 (2007)
Gutman I., Radenković S., Graovac A., Plavšić D.: Chem. Phys. Lett. 447, 233 (2007)
Zhang B., Zhou B.: Z. Naturforsch. 61a, 536 (2006)
Gutman I., Radenković S.: Kragujevac J. Sci. 29, 67 (2007)
Zhou B.: MATCH Commun. Math. Comput. Chem. 60, 485 (2008)
Heuberger C., Wagner S.G.: J. Graph Theory 58, 49 (2008)
Heuberger C., Wagner S.G.: J. Math. Chem. 46, 214 (2009)
Marković Z., Ivanov-Petrović V., Gutman I.: J. Mol. Struct. (Theochem) 629, 303 (2003)
Fischermann M., Gutman I., Hoffmann A., Rautenbach D., Vidović D., Volkmann L.: Z. Naturforsch. 57a, 49 (2002)
Lovász L., Pelikán J.: Period Math. Hung. 3, 175 (1973)
Deng H.: MATCH Commun. Math. Comput. Chem. 62, (2009)
Entringer R.C., Jackson D.E., Snyder D.A.: Czech. Math. J. 26, 283 (1976)
Gutman I.: Theor. Chim. Acta 45, 79 (1977)
Lin W., Guo X.: J. Math. Chem. 42, 1057 (2007)
Stevanović D.: MATCH Commun. Math. Comput. Chem. 61, 407 (2009)
Yan W., Ye L.: Appl. Math. Lett. 18, 1046 (2005)
Yu A., Lv X.: Linear Algebra Appl. 418, 625 (2006)
Lepović M., Gutman I.: J. Chem. Inf. Comput. Sci. 38, 823 (1998)
Cvetković D., Doob M., Sachs H.: Spectra of Graphs—Theory and Applications. Academic Press, New York (1980)
de la Peña J.A., Gutman I., Rada J.: Linear Algebra Appl. 427, 70 (2007)
Gutman I., Graovac A.: Chem. Phys. Lett. 436, 294 (2007)
Ginosar Y., Gutman I., Mansour T., Schork M.: Chem. Phys. Lett. 454, 145 (2008)
Fischermann M., Hoffmann A., Rautenbach D., Székely L., Volkmann L.: Discrete Appl. Math. 122, 127 (2002)
Simić S., Tošić D.: MATCH Commun. Math. Comput. Chem. 54, 351 (2005)
Gutman I., Das K.C.: MATCH Commun. Math. Comput. Chem. 50, 83 (2004)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ilić, A., Stevanović, D. The Estrada index of chemical trees. J Math Chem 47, 305–314 (2010). https://doi.org/10.1007/s10910-009-9570-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10910-009-9570-0