The spectral excess theorem for distance-regular graphs having distance-$d$ graph with fewer distinct eigenvalues

TL;DR

This paper generalizes the spectral excess theorem to characterize distance-regular graphs with a Kneser graph having fewer eigenvalues, including antipodal and half-antipodal cases, based on spectral properties and vertex distances.

Contribution

It provides a new spectral characterization of partially antipodal distance-regular graphs, extending the spectral excess theorem to broader classes of graphs.

Findings

Characterization of partially antipodal distance-regular graphs via spectrum.

Extension of the spectral excess theorem to graphs with fewer eigenvalues.

Identification of conditions for bipartite, antipodal, and strongly regular Kneser graphs.

Abstract

Let $\Gamma$ be a distance-regular graph with diameter $d$ and Kneser graph $K=\Gamma_d$, the distance-$d$ graph of $\Gamma$. We say that $\Gamma$ is partially antipodal when $K$ has fewer distinct eigenvalues than $\Gamma$. In particular, this is the case of antipodal distance-regular graphs ($K$ with only two distinct eigenvalues), and the so-called half-antipodal distance-regular graphs ($K$ with only one negative eigenvalue). We provide a characterization of partially antipodal distance-regular graphs (among regular graphs with $d$ distinct eigenvalues) in terms of the spectrum and the mean number of vertices at maximal distance $d$ from every vertex. This can be seen as a general version of the so-called spectral excess theorem, which allows us to characterize those distance-regular graphs which are half-antipodal, antipodal, bipartite, or with Kneser graph being strongly regular.