Ordering trees by their largest eigenvalues

Let Δ(T) and λ1(T) denote the maximum degree and the largest eigenvalue of a tree T, respectively. Let Tn be the set of trees on n vertices, and . In the present paper, among the trees in (n ⩾ 4), we characterize the tree which alone minimizes the largest eigenvalue, as well as the tree which alone maximizes the largest eigenvalue when . Furthermore, it is proved that, for two trees T1 and T2 in Tn (n ⩾ 4), if and Δ(T1) > Δ(T2), then λ1(T1) > λ1(T2). By applying this result, we extend the order of trees in Tn 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].