  A tree is said to be starlike if exactly one vertex has degree greater than
two. In this paper, we will study the spectral properties of $S(n,k \cdot 1)$,
that is, the starlike tree with $k$ branches of length 1 and one branch of
length $n$. The largest eigenvalue $\lambda_1$ of $S(n,k \cdot 1)$ satisfies
$\sqrt{k+1} \leq \lambda_1 < k/\sqrt{k-1}$. Moreover, the largest eigenvalue of
$S(n,k \cdot 1)$ is equal to the largest eigenvalue of $S(k \cdot (n+1) )$,
which is the starlike tree that has $k$ branches of length $n-1$. Using the
spectral radii of $S(n,k \cdot 1)$ we can show
