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 $d\in\{1,2\}\cup [n/2,2n/3-1]\cup\{n-k\mid k=1,2,...,8\}$. In this paper, we determine all minimizer graphs for $d=2(n-1)/3$.