The spectral radius of a graph is the largest eigenvalue of the adjacency matrix of the graph. Let T(n,D l) be the tree which minimizes the spectral radius of all trees of order n with exactly l vertices of maximum degree D. In this paper, T(n,D l) is determined for D=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,D 2) is a dumbbell, and T(n,D,3) is a tree consisting of three distinct stars of order D 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 Lov´asz and Pelik´an, Simi´c and To˘si´c, Wu, Yuan and Xiao, and Xu, Lin and Shu.
F. Labourie [arXiv:1212.5015] characterized the Hitchin components for PSL(n,R) for any n>1 by using the swapping algebra, where the swapping algebra should be understood as a ring equipped with a Poisson bracket. We introduce the rank n swapping algebra, which is the quotient of the swapping algebra by the (n+1)×(n+1) determinant relations. The main results are the well-definedness of the rank n swapping algebra and the "cross-ratio" in its fraction algebra. As a consequence, we use the sub fraction algebra of the rank n swapping algebra generated by these "cross-ratios" to characterize the PSL(n,R) Hitchin component for a fixed n>1. We also show the relation between the rank 2 swapping algebra and the cluster X space.