An excess theorem for spherical 2-designs

TL;DR

This paper establishes an excess theorem for spherical 2-designs, providing a characterization of Q-polynomial association schemes using eigenvalues and excess, analogous to the spectral excess theorem for graphs.

Contribution

It introduces a dual excess theorem for spherical 2-designs, characterizing Q-polynomial association schemes in terms of spectral properties.

Findings

Characterization of Q-polynomial association schemes among spherical 2-designs

Dual version of the spectral excess theorem for graphs

Provides spectral criteria for association schemes

Abstract

We give an excess theorem for spherical 2-designs. This theorem is a dual version of the spectral excess theorem for graphs, which gives a characterization of distance-regular graphs, among regular graphs in terms of the eigenvalues and the excess. Here we give a characterization of Q-polynomial association schemes among spherical 2-designs.