Ordering trees by their largest eigenvalues

Let Delta(T) and lambda(1)(T) denote the maximum degree and the largest eigenvalue of a tree T, respectively. Let T-n be the set of trees on n vertices, and T-n((Delta)) = {T is an element of T-n vertical bar Delta(T) = Delta}. In the present paper, among the trees in T-n((Delta)) (n >= 4), we characterize the tree which alone minimizes the largest eigenvalue, as well as the tree which alone maximizes the largest eigenvalue when [(n-2)/(2)] <= Delta <= n-1. Furthermore, it is proved that, for two trees T-1 and T-2 in T-n (n >= 4), if Delta(T-1) >= [(2n)/(3)]-1 and Delta(T-1) > Delta(T-2), then lambda(1)(T-1) > lambda(1)(T-2). By applying this result, we extend the order of trees in T-n by their largest eigenvalues to the 13th tree when n >= 12. This extends the results of Hofmeister [Linear Algebra Appl. 260 (1997) 43] and Chang et al. [Linear Algebra Appl. 370 (2003) 175]. (c) 2006 Elsevier Inc. All rights reserved.