On the weight distribution bound for the negative eigenvalue of polar collinearity graphs

Abstract

The weight distribution bound gives a lower bound on the weight of eigenvectors for eigenvalues of distance-regular graphs. In this note, we study when the weight distribution bound is tight for the negative eigenvalue of collinearity graphs of finite embedded polar spaces, and elliptic and hyperbolic affine polar graphs. We give a complete classification of eigenvectors meeting the weight distribution bound.