On the eigenvalues and spectral radius of starlike trees

Let (kge 1) and (n_1,ldots ,n_kge 1) be some integers. Let (S(n_1,ldots ,n_k)) be a tree T such that T has a vertex v of degree k and (T{setminus } v) is the disjoint union of the paths (P_{n_1},ldots ,P_{n_k}), that is (T{setminus } vcong P_{n_1}cup cdots cup P_{n_k}) so that every neighbor of v in T has degree one or two. The tree (S(n_1,ldots ,n_k)) is called starlike tree, a tree with exactly one vertex of degree greater than two, if (kge 3). In this paper we obtain the eigenvalues of starlike trees. We find some bounds for the largest eigenvalue (for the spectral radius) of starlike trees. In particular we prove that if (kge 4) and (n_1,ldots ,n_kge 2), then (frac{k-1}{sqrt{k-2}}, where (lambda _1(T)) is the largest eigenvalue of T. Finally we characterize all starlike trees that all of whose eigenvalues are in the interval ((-2,2)).