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

Abstract

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).