Isometric embeddings of dual polar graphs in Grassmann graphs over finite fields

TL;DR

This paper proves that each dual polar graph can be uniquely embedded in a corresponding Grassmann graph over the same finite field, up to automorphism, highlighting a fundamental geometric relationship.

Contribution

It establishes the uniqueness of isometric embeddings of dual polar graphs into Grassmann graphs over finite fields, up to automorphism.

Findings

Each dual polar graph has a unique isometric embedding in the corresponding Grassmann graph.

The embedding is unique up to graph automorphism.

The result applies to graphs over the same finite field.

Abstract

We consider the Grassmann graphs and dual polar graphs over the same finite field and show that, up to graph automorphism, for every dual polar graph there is the unique isometric embedding in the corresponding Grassmann graph.