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The Estrada index of chemical trees

  • Aleksandar Ilić
  • Dragan Stevanović
Original Paper

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, 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.

Keywords

Estrada index Branching Broom graph Chemical trees Volkmann trees Spectral moments 

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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Faculty of Sciences and MathematicsUniversity of NišNišSerbia
  2. 2.University of Primorska—FAMNITKoperSlovenia
  3. 3.Mathematical InstituteSerbian Academy of Science and ArtsBelgradeSerbia

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