Laplacian Estrada index of trees
  Let $G$ be a simple graph with $n$ vertices and let $\mu_1 \geqslant \mu_2 \geqslant...\geqslant \mu_{n - 1} \geqslant \mu_n = 0$ be the eigenvalues of its Laplacian matrix. The Laplacian Estrada index of a graph $G$ is defined as $LEE (G) = \sum\limits_{i = 1}^n e^{\mu_i}$. Using the recent connection between Estrada index of a line graph and Laplacian Estrada index, we prove that the path $P_n$ has minimal, while the star $S_n$ has maximal $LEE$ among trees on $n$ vertices. In addition, we find the unique tree with the second maximal Laplacian Estrada index. 
Aleksandar Ilic and Bo Zhou
math.CO