On Distance-Regular Graphs with Smallest Eigenvalue at Least $-m$

TL;DR

This paper proves that for any fixed integer m ≥ 2, only finitely many non-geometric distance-regular graphs with certain spectral and structural properties exist, highlighting limitations on their diversity.

Contribution

It establishes finiteness results for non-geometric distance-regular graphs with specified eigenvalue bounds, diameter, and intersection number c_2.

Findings

Finiteness of non-geometric distance-regular graphs with smallest eigenvalue ≥ -m

Characterization of geometric vs. non-geometric graphs in this class

Constraints on diameter and intersection numbers for such graphs

Abstract

A non-complete geometric distance-regular graph is the point graph of a partial geometry in which the set of lines is a set of Delsarte cliques. In this paper, we prove that for fixed integer $m\geq 2$, there are only finitely many non-geometric distance-regular graphs with smallest eigenvalue at least $-m$, diameter at least three and intersection number $c_2 \geq 2$.