A note on the Laplacian Estrada index of trees

The Laplacian Estrada index of a graph G is defined as LEE(G) = Sigma(n)(i=1) e(mu i), where mu(1) >= mu(2) >= ... >= mu(n-1) >= mu(n) = 0 are the eigenvalues of its Laplacian matrix. An unsolved problem in [19] is whether S(n)(3, n - 3) or C(n)(n - 5) has the third maximal Laplacian Estrada index among all trees on n vertices, where S(n)(3, n - 3) is the double tree formed by adding an edge between the centers of the stars S(3) and S(n-3) and C(n)(n - 5) is the tree formed by attaching n - 5 pendent vertices to the center of a path P(5). In this paper, we partially answer this problem, and prove that LEE(S(n)(3, n - 3)) > LEE(C(n)(n - 5)) and C(n)(n - 5) cannot have the third maximal Laplacian Estrada index among all trees on n vertices.