On some distance-regular graphs with many vertices

TL;DR

This paper constructs large, highly symmetric distance-regular graphs, including some new ones, using transitive group actions from Chevalley, orthogonal, and Tits groups, with graphs having up to 28431 vertices.

Contribution

It introduces new large distance-regular graphs with transitive group actions from specific algebraic groups, expanding known examples in the field.

Findings

Constructed distance-regular graphs with over 1000 vertices.

Included new graphs not previously documented.

Demonstrated transitive actions of specific algebraic groups on these graphs.

Abstract

We construct distance-regular graphs, including strongly regular graphs, admitting a transitive action of the Chevalley groups $G_2(4)$ and $G_2(5)$, the orthogonal group $O(7,3)$ and the Tits group $T=$$^2F_4(2)'$. Most of the constructed graphs have more than 1000 vertices, and the number of vertices goes up to 28431. Some of the obtained graphs are new.