On the (Laplacian) spectral radius of weighted trees with fixed matching number q and a positive weight set

Abstract

Denote by T ( n , q , w 1 , w 2 , … , w n - 1 ) the set of n-vertex weighted trees with matching number q and fixed positive weight set W n - 1 = { w 1 , w 2 , … , w n - 1 } , where w 1 ⩾ w 2 ⩾ ⋯ ⩾ w n - 1 > 0 . Tan [S.W. Tan, On the sharp upper bound of spectral radius of weighted trees, J. Math. Res. Exposition 29 (2009) 293–301] determined the weighted tree in T ( n , q , w 1 , w 2 , … , w n - 1 ) with the largest adjacent spectral radius , whereas in [S.W. Tan, On the Laplacian spectral radius of weighted trees with a positive weight set, Discrete Math. 310 (2010) 1026–1036] Tan determined the weighted tree in T ( n , q , w 1 , w 2 , … , w n - 1 ) with the largest Laplacian spectral radius. In this paper, we use a unified approach to identify the unique weighted tree in T ( n , q , w 1 , w 2 , … , w n - 1 ) with the largest adjacent spectral radius and largest Laplacian spectral radius, respectively.