Distance-regular graphs where the distance-$d$ graph has fewer distinct eigenvalues

TL;DR

This paper investigates the spectral properties of distance-regular graphs, focusing on cases where the distance-d graph has fewer eigenvalues, especially in the half antipodal scenario, revealing new algebraic insights.

Contribution

It characterizes when the Bose-Mesner algebra of a distance-regular graph is not generated by the Kneser graph's adjacency matrix, with significant results for half antipodal graphs.

Findings

Identifies conditions under which the Bose-Mesner algebra is not generated by the Kneser graph.

Provides strong algebraic results in the half antipodal case.

Enhances understanding of eigenvalue structures in distance-regular graphs.

Abstract

Let the Kneser graph $K$ of a distance-regular graph $\Gamma$ be the graph on the same vertex set as $\Gamma$, where two vertices are adjacent when they have maximal distance in $\Gamma$. We study the situation where the Bose-Mesner algebra of $\Gamma$ is not generated by the adjacency matrix of $K$. In particular, we obtain strong results in the so-called `half antipodal' case.