  The spectral radius $\rho(G)$ of a graph $G$ is the largest eigenvalue of its
adjacency matrix. Woo and Neumaier discovered that a connected graph $G$ with
$\rho(G)\leq 3/2{\sqrt{2}}$ is either a dagger, an open quipu, or a closed
quipu. The reverse statement is not true. Many open quipus and closed quipus
have spectral radius greater than $3/2{\sqrt{2}}$. In this paper we proved the
following results. For any open quipu $G$ on $n$ vertices ($n\geq 6$) with
spectral radius less than $3/2{\sqrt{2}}$, its diameter $D(G)$ satisfies
$D(G)\geq (2n-4)/3$. This bound is tight. For any closed quipu $G$ on $n$
vertices ($n\geq 13$) with spectral radius less than $3/2{\sqrt{2}}$, its
diameter $D(G)$ satisfies $\frac{n}{3}< D(G)\leq \frac{2n-2}{3}$. The upper
bound is tight while the lower bound is asymptotically tight.
  Let $G^{min}_{n,D}$ be a graph with minimal spectral radius among all
connected graphs on $n$ vertices with diameter $D$. We applied the results and
found $G^{min}_{n,D}$ for some range of $D$. For $n\geq 13$ and $D\in
[\frac{n}{2}, \frac{2n-7}{3}]$, we proved that $G^{min}_{n,D}$ is the graph
obtained by attaching two paths of length $D-\lfloor\frac{n}{2}\rfloor$ and
$D-\lceil\frac{n}{2}\rceil$ to a pair of antipodal vertices of the even cycle
$C_{2(n-D)}$. Thus we settled a conjecture of Cioab-van Dam-Koolen-Lee, who
previously proved a special case $D=\frac{n+e}{2}$ for $e=1,2,3,4$.
