Diameters of graphs with spectral radius at most

The spectral radius ρ(G) of a graph G is the largest eigenvalue of its adjacency matrix. Woo and Neumaier discovered that a connected graph G with 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 radii greater than . In this paper we proved the following results. For any open quipu G on n vertices (n⩾6) with spectral radius less than , its diameter D(G) satisfies D(G)⩾(2n-4)/3. This bound is tight. For any closed quipu G on n vertices (n⩾13) with spectral radius less than , its diameter D(G) satisfies . The upper bound is tight while the lower bound is asymptotically tight.Let be a graph with minimal spectral radius among all connected graphs on n vertices with diameter D. For n⩾14 and , we proved that is the unique graph obtained by attaching two paths of lengths and to a pair of antipodal vertices of the even cycle C2(n-D). This result is tight. Thus we settled a conjecture of Cioabaˇ–van Dam–Koolen–Lee, who previously proved a special case for e∈{1,2,3,4} and n large enough.