Laplacian Estrada index of trees

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Let G be a simple graph with n vertices and let μ₁ μ₂... μₙ ₋ ₁ μₙ = 0 be the eigenvalues of its Laplacian matrix. The Laplacian Estrada index of a graph G is defined as LEE (G) = ∑ᵢ ₌ ₁ⁿ eμⁱ. Using the recent connection between Estrada index of a line graph and Laplacian Estrada index, we prove that the path Pₙ has minimal, while the star Sₙ has maximal LEE among trees on n vertices. In addition, we find the unique tree with the second maximal Laplacian Estrada index.