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).

Keywords:

• spectral radius
• starlike tree
• Perron vector
• weighted graph

AMS Subject Classifications::

• 05C50
• 15A18

Acknowledgements

The authors express their sincere gratitude to the referee for a very careful reading of this article and for all insightful comments, which led a number of improvements to this article. This work is financially supported by the National Natural Science Foundation of China (Grant No. 11071096) and Special Fund for Basic Scientific Research of Central Colleges (CCNU11A02015).