Trees with given maximum degree minimizing the spectral radius

The spectral radius of a graph is the largest eigenvalue of the adjacency matrix of the graph. Let T (n,∆, l) be the tree which minimizes the spectral radius of all trees of order n with exactly l vertices of maximum degree ∆. In this paper, T (n,∆, l) is determined for ∆ = 3, and for l ≤ 3 and n large enough. It is proven that for sufficiently large n, T (n, 3, l) is a caterpillar with (almost) uniformly distributed legs, T (n,∆, 2) is a dumbbell, and T (n,∆, 3) is a tree consisting of three distinct stars of order ∆ connected by three disjoint paths of (almost) equal length from their centers to a common vertex. The unique tree with the largest spectral radius among all such trees is also determined. These extend earlier results of Lovász and Pelikán, Simić and Tos̆ić, Wu, Yuan and Xiao, and Xu, Lin and Shu.