Some results on the largest eigenvalues of starlike trees

A starlike tree is a tree with exactly one vertex of degree greater than two. Let S(n1; ..., nk) be the starlike tree T with a vertex v of degree k≥ 3 such that T nv is the disjoint union of the paths Pn1, ...., Pnk , that is T \ v = Pn1 U ... U Pnk . In this paper we study the largest eigenvalue of starlike trees. In particular, we obtain some bounds for the largest eigenvalue of starlike trees. Let ) be the largest eigenvalue of T. We find that if k ≥ 4 and n1,.... nk ≥ 2, then .k-1/√k-2 λ1 (S(n1, ... , nk)) k/√k-1 Finally we study the starlike trees T such that λ1(T) 2.