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

AbstractDenote by T(n,q,w1,w2,…,wn-1) the set of n-vertex weighted trees with matching number q and fixed positive weight set Wn-1={w1,w2,…,wn-1}, where w1⩾w2⩾⋯⩾wn-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,w1,w2,…,wn-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,w1,w2,…,wn-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,w1,w2,…,wn-1) with the largest adjacent spectral radius and largest Laplacian spectral radius, respectively