Spectral radius of a star with one long arm

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