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

摘要

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.