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The spectral radius $\rho(G)$ of a graph $G$ is the largest eigenvalue of itsadjacency 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 closedquipu. The reverse statement is not true. Many open quipus and closed quipushave spectral radius greater than $3/2{\sqrt{2}}$. In this paper we proved thefollowing results. For any open quipu $G$ on $n$ vertices ($n\geq 6$) withspectral 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 $3/2{\sqrt{2}}$, itsdiameter $D(G)$ satisfies $\frac{n}{3}< D(G)\leq \frac{2n-2}{3}$. The upperbound is tight while the lower bound is asymptotically tight.  Let $G^{min}_{n,D}$ be a graph with minimal spectral radius among allconnected graphs on $n$ vertices with diameter $D$. We applied the results andfound $G^{min}_{n,D}$ 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 graphobtained 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, whopreviously proved a special case $D=\frac{n+e}{2}$ for $e=1,2,3,4$.

Diameters of Graphs with Spectral Radius at most $3/2\sqrt{2}$