The distance-regular graphs such that all of its second largest local eigenvalues are at most one

TL;DR

This paper classifies distance-regular graphs based on their second largest local eigenvalues being at most one and explores implications for their smallest eigenvalues, extending previous classifications.

Contribution

It provides a new classification of distance-regular graphs with specific eigenvalue constraints, expanding on earlier work by the first author.

Findings

Classified distance-regular graphs with second largest local eigenvalues ≤ 1

Discussed implications for the smallest eigenvalue of such graphs

Extended previous eigenvalue classification results

Abstract

In this paper, we classify distance regular graphs such that all of its second largest local eigenvalues are at most one. Also we discuss the consequences for the smallest eigenvalue of a distance-regular graph. These extend a result by the first author, who classified the distance-regular graph with smallest eigenvalue $-1-\frac{b_1}{2}$.