On the Laplacian spectral radii of trees with given domination number

Abstract

Research supported by the National Natural Science Foundation of China 10601038 and Innovation Program of Shanghai Municipal Education Commission 10YZ99. We use T_n,γ, (where n ≥ 2γ) to denote the set of trees with n vertices and domination number γ. L. Feng [Minimizing the Laplacian eigenvalues for trees with given domination number, Linear Algebra Appl., 419 (2006) 648655.] gave the tree in T_n,γ with the greatest value of the Laplacian spectral radius. In this paper, we give another proof of this result. Using our method, we can go further beyond L. Feng by giving the tree in Tn,γ (where n ≥ 2γ+1) with the second largest value of the Laplacian spectral radius.