The smallest eigenvalues of Hamming graphs, Johnson graphs and other distance-regular graphs with classical parameters

TL;DR

This paper proves conjectures regarding the smallest eigenvalues of Hamming and Johnson graphs, and extends the analysis to eigenvalues of graphs from classical association schemes, advancing spectral graph theory.

Contribution

It confirms conjectures on eigenvalues of specific distance-regular graphs and generalizes the results to graphs of classical association schemes.

Findings

Proved the conjecture by Van Dam and Sotirov on Hamming graphs.

Confirmed Karloff's conjecture on Johnson graphs.

Analyzed eigenvalues in classical P- and Q-polynomial association schemes.

Abstract

We prove a conjecture by Van Dam and Sotirov on the smallest eigenvalue of (distance-$j$) Hamming graphs and a conjecture by Karloff on the smallest eigenvalue of (distance-$j$) Johnson graphs. More generally, we study the smallest eigenvalue and the second largest eigenvalue in absolute value of the graphs of the relations of classical $P$- and $Q$-polynomial association schemes.