On the spectral radius of weighted trees with given number of pendant vertices and a positive weight set

摘要

Let T(n, r; W _(n-1)) be the set of all n-vertex weighted trees with r vertices of degree 2 and fixed positive weight set W _(n-1), P(n, γ; W _(n-1)) the set of all n-vertex weighted trees with q pendants and fixed positive weight set W _(n-1), where W _(n-1) = {w _1, w _2,..., w _(n-1)} with w _1 ≥ w _2 ≥ ··· ≥ w _(n-1) > 0. In this article, we first identify the unique weighted tree in T(n, r; W n-1) with the largest adjacency spectral radius. Then we characterize the unique weighted trees with the largest adjacency spectral radius in P(n, γ; _(W n-1)).