The Estrada index of chemical trees

Let G be a simple graph with n vertices and let λ1, λ2, . . . , λn 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 Bn,Δ, consisting of a star SΔ+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(Sn) = EE(Bn,n−1) > EE(Bn,n−2) > . . . > EE(Bn,3) > EE(Bn,2) = EE(Pn). 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.

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