An inequality involving the second largest and smallest eigenvalue of a distance-regular graph

TL;DR

This paper establishes a new inequality relating the second largest and smallest eigenvalues of distance-regular graphs, providing insights into their spectral properties and structure, especially for graphs with fixed second largest eigenvalue.

Contribution

The paper introduces a novel inequality involving eigenvalues of distance-regular graphs and characterizes cases of equality, advancing understanding of their spectral structure.

Findings

The inequality (rac{1+1)(rac{D+1)<= -b1 holds for distance-regular graphs.

Equality occurs only when the graph's diameter is two.

The study offers new bounds for graphs with fixed second largest eigenvalue.

Abstract

For a distance-regular graph with second largest eigenvalue (resp. smallest eigenvalue) \mu1 (resp. \muD) we show that (\mu1+1)(\muD+1)<= -b1 holds, where equality only holds when the diameter equals two. Using this inequality we study distance-regular graphs with fixed second largest eigenvalue.