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

TL;DR

This paper identifies the graphs with order n and diameter 2(n-1)/3 that minimize the spectral radius, expanding the understanding of spectral graph optimization for specific diameter constraints.

Contribution

It determines all minimizer graphs for the case where the diameter is exactly 2(n-1)/3, a previously unresolved specific diameter value.

Findings

Identified all minimizer graphs for diameter 2(n-1)/3

Extended known classes of graphs with minimal spectral radius

Provided explicit characterization for this diameter case

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 $d=2(n-1)/3$.