Let k≥1\documentclass[12pt]{minimal}
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\begin{document}$$k\ge 1$$\end{document} and n1,…,nk≥1\documentclass[12pt]{minimal}
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\begin{document}$$n_1,\ldots ,n_k\ge 1$$\end{document} be some integers. Let S(n1,…,nk)\documentclass[12pt]{minimal}
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\begin{document}$$S(n_1,\ldots ,n_k)$$\end{document} be a tree T such that T has a vertex v of degree k and T\v\documentclass[12pt]{minimal}
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\begin{document}$$T{\setminus } v$$\end{document} is the disjoint union of the paths Pn1,…,Pnk\documentclass[12pt]{minimal}
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\begin{document}$$P_{n_1},\ldots ,P_{n_k}$$\end{document}, that is T\v≅Pn1∪⋯∪Pnk\documentclass[12pt]{minimal}
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\begin{document}$$T{\setminus } v\cong P_{n_1}\cup \cdots \cup P_{n_k}$$\end{document} so that every neighbor of v in T has degree one or two. The tree S(n1,…,nk)\documentclass[12pt]{minimal}
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\begin{document}$$S(n_1,\ldots ,n_k)$$\end{document} is called starlike tree, a tree with exactly one vertex of degree greater than two, if k≥3\documentclass[12pt]{minimal}
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\begin{document}$$k\ge 3$$\end{document}. 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 k≥4\documentclass[12pt]{minimal}
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\begin{document}$$k\ge 4$$\end{document} and n1,…,nk≥2\documentclass[12pt]{minimal}
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\begin{document}$$n_1,\ldots ,n_k\ge 2$$\end{document}, then k-1k-2<λ1(S(n1,…,nk))<kk-1\documentclass[12pt]{minimal}
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\begin{document}$$\frac{k-1}{\sqrt{k-2}}<\lambda _1(S(n_1,\ldots ,n_k))<\frac{k}{\sqrt{k-1}}$$\end{document}, where λ1(T)\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _1(T)$$\end{document} is the largest eigenvalue of T. Finally we characterize all starlike trees that all of whose eigenvalues are in the interval (-2,2)\documentclass[12pt]{minimal}
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\begin{document}$$(-2,2)$$\end{document}.
