\documentclass[12pt]{article}
\usepackage{amsmath,amssymb,amsfonts}
\begin{document}
Let $k\geq1$ and $n_1,\ldots,n_k\geq1$ be some integers. Let $S(n_1,\ldots,n_k)$ be the 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 $T\setminus v\cong P_{n_1}\cup\cdots\cup P_{n_k}$ such 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 $k\geq3$. In this paper we obtain the eigenvalues of starlike trees. We obtain some bounds for the largest eigenvalue (\,for the spectral radius) of starlike trees. In particular we prove that if
$k\geq4$ and $n_1,\ldots,n_k\geq2$, then $\frac{k-1}{\sqrt{k-2}}<\lambda_1(S(n_1,\ldots,n_k))<\frac{k}{\sqrt{k-1}}$, 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 $(-2,2)$.
\end{document}