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 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 (n 4), if and Δ(T1) > Δ(T2), then λ1(T1) > λ1(T2). By applying this result, we extend the order of trees in 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].