Graphs of order n and diameter 2(n−1)/3 minimizing the spectral radius

Abstract The spectral radius of a graph is the largest eigenvalue of its adjacency matrix. A minimizer graph is such that minimizes the spectral radius among all connected graphs on n vertices with diameter d . The minimizer graphs are known for d ∈ { 1 , 2 } ∪ [ n / 2 , 2 n / 3 − 1 ] ∪ { n − k | k = 1 , 2 , … , 8 } . In this paper, we determine all minimizer graphs for d = 2 ( n − 1 ) / 3 .