On the eigenvalues of Grassmann graphs, Bilinear forms graphs and Hermitian forms graphs

TL;DR

This paper proves several conjectures about the eigenvalues of Grassmann, bilinear forms, and Hermitian forms graphs, advancing understanding in algebraic graph theory and association schemes.

Contribution

It confirms some of Brouwer et al.'s conjectures on eigenvalues of these specialized graphs, providing new theoretical insights.

Findings

Proved conjectures on eigenvalues of Grassmann graphs

Established eigenvalue properties of bilinear forms graphs

Enhanced understanding of Hermitian forms graphs eigenstructure

Abstract

Recently, Brouwer, Cioab\u{a}, Ihringer and McGinnis obtained some new results involving the eigenvalues of various graphs coming from association schemes and posed some conjectures related to the eigenvalues of Grassmann graphs, bilinear forms graphs and Hermitian forms graphs. In this paper, we prove some of their conjectures.