Majorization and the spectral radius of starlike trees

A starlike tree is a tree with exactly one vertex of degree greater than two. The spectral radius of a graph G, that is denoted by (lambda (G)), is the largest eigenvalue of G. Let k and (n_1,ldots ,n_k) be some positive integers. Let (T(n_1,ldots ,n_k)) be the tree T (T is a path or a starlike tree) such that T has a vertex v so that (T{setminus } v) is the disjoint union of the paths (P_{n_1-1},ldots ,P_{n_k-1}) where every neighbor of v in T has degree one or two. Let (P=(p_1,ldots ,p_k)) and (Q=(q_1,ldots ,q_k)), where (p_1ge cdots ge p_kge 1) and (q_1ge cdots ge q_kge 1) are integer. We say P majorizes Q and let (Psucceq _M Q), if for every j, (1le jle k), (sum _{i=1}^{j}p_ige sum _{i=1}^{j}q_i), with equality if (j=k). In this paper we show that if P majorizes Q, that is ((p_1,ldots ,p_k)succeq _M(q_1,ldots ,q_k)), then (lambda (T(q_1,ldots ,q_k))ge lambda (T(p_1,ldots ,p_k))).