On the Estrada index conjecture

Let G be a simple graph of order n with m edges. Let the adjacency spectrum be {λ₁, λ₂, ..., λ_{n - 1}, λ_n} of G, where λ₁ ≥ λ₂ ≥ ⋯ ≥ λ_{n - 1} ≥ λ_n. The Estrada index of a graph G is EE (G) = ∑_{i = 1}ⁿ e^λi. In [J.A. Peña, I. Gutman, J. Rada, Estimating the Estrada index, Linear Algebra Appl. 427 (2007) 70-76], Peña et al. posed a conjecture that the star S_n has maximum Estrada index for any tree of order n and the path P_n has minimum Estrada index for any tree of order n or any connected graph of order n. In this paper, we have proved that the star has maximum Estrada index for any tree. Also, we obtain that the path has minimum Estrada index for any connected graph with m ≥ 1.8 n + 4 or m ≥ n² / 6. Moreover, we give better lower bound on Estrada index for any connected graph.