Diameters of graphs with spectral radius at most [formula omitted

Abstract

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 ρ(G)⩽322 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 322. In this paper we proved the following results. For any open quipu G on n vertices (n⩾6) with spectral radius less than 322, 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 322, its diameter D(G) satisfies n3<D(G)⩽2n-23. The upper bound is tight while the lower bound is asymptotically tight.Let Gn,Dmin be a graph with minimal spectral radius among all connected graphs on n vertices with diameter D. For n⩾14 and D∈[n2,2n-53], we proved that Gn,Dmin is the unique graph obtained by attaching two paths of lengths D-⌊n2⌋ and D-⌈n2⌉ 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 D=n+e2 for e∈{1,2,3,4} and n large enough.