The twisted Grassmann graph is the block graph of a design

TL;DR

This paper demonstrates that the twisted Grassmann graph is the block graph of a specific design and establishes an isomorphism between their automorphism groups, linking graph theory and design theory.

Contribution

It proves the twisted Grassmann graph is the block graph of a known design and shows their automorphism groups are isomorphic, revealing a deep structural connection.

Findings

Twisted Grassmann graph is the block graph of a certain design.

Automorphism groups of the design and the graph are isomorphic.

Establishes a link between graph and design automorphisms.

Abstract

In this note, we show that the twisted Grassmann graph constructed by van Dam and Koolen is the block graph of the design constructed by Jungnickel and Tonchev. We also show that the full automorphism group of the design is isomorphic to the full automorphism group of the twisted Grassmann graph.