On $Q$-polynomial association schemes of small class

TL;DR

This paper establishes inequalities related to dual eigenvalues in $Q$-polynomial association schemes of small class and characterizes dual-tight schemes of class three, with methods applicable to distance-regular graphs.

Contribution

It introduces new inequalities for dual eigenvalues and characterizes dual-tight schemes of class three, expanding understanding of $Q$-polynomial association schemes.

Findings

Derived inequalities for dual eigenvalues of $Q$-polynomial schemes

Characterized dual-tight schemes of class three

Applicable methods to distance-regular graphs

Abstract

We show an inequality involving the third largest or second smallest dual eigenvalues of $Q$-polynomial association schemes of class at least three. Also we characterize dual-tight $Q$-polynomial association schemes of class three. Our method is based on tridiagonal matrices and can be applied to distance-regular graphs as well.