A note on the Laplacian Estrada index of trees
نویسندگان
چکیده
Abstract The Laplacian Estrada index of a graphG is defined as LEE(G) = ∑n i=1 e μi , where μ1 ≥ μ2 ≥ · · · ≥ μn−1 ≥ μn = 0 are the eigenvalues of its Laplacian matrix. An unsolved problem in [19] is whether Sn(3, n − 3) or Cn(n − 5) has the third maximal Laplacian Estrada index among all trees on n vertices, where Sn(3, n − 3) is the double tree formed by adding an edge between the centers of the stars S3 and Sn−3 and Cn(n− 5) is the tree formed by attaching n − 5 pendent vertices to the center of a path P5. In this paper, we partially answer this problem, and prove that LEE(Sn(3, n − 3)) > LEE(Cn(n − 5)) and Cn(n− 5) cannot have the third maximal Laplacian Estrada index among all trees on n vertices.
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