Recovering of the Grassmann graph from the subgraph of non-degenerate subspaces

TL;DR

This work addresses recovering the Grassmann graph $\Gamma_k(V)$ from the subgraph $\Delta_k(V)$ induced by non-degenerate $k$-subspaces, under the conditions $1<k<n-1$ and $|\mathbb F|>n-k$. The authors identify and exploit maximal cliques (stars and tops) in $\Delta_k(V)$ and extend them by degenerate subspaces to reconstruct the full Grassmann graph, providing two reconstruction pathways via stars or via tops when $|\mathbb F|>n-k$ (and $n\ge5$). A complete treatment is given for the special case $n=4$, using a dual projective-geometry approach to classify maximal cliques and recover $\Gamma_k(V)$. They also demonstrate limitations when $|\mathbb F|\le n-k$ with concrete examples and discuss related automorphism results and connections to prior work KP2, PZ. Overall, the paper connects affine-projective geometry ideas to Grassmann graphs and offers a concrete, geometry-driven reconstruction method from a structured subgraph.

Abstract

Let ${\mathbb F}$ be a (not necessarily finite) field. A subspace of the vector space ${\mathbb F}^n$ is called {\it non-degenerate} if it is not contained in a coordinate hyperplane. We show that the Grassmann graph of $k$-dimensional subspaces of ${\mathbb F}^n$, $1<k<n-1$, can be recovered from the subgraph of non-degenerate subspaces if $|{\mathbb F}|>n-k$. In the case when ${\mathbb F}={\mathbb F}_q$ is the field of $q$ elements, this subgraph is known as the graph of non-degenerate linear $[n,k]_q$ codes.

Theorems & Definitions (37)

• Theorem 1
• Proposition 2
• proof
• Remark 3
• Proposition 4
• proof
• Example 5
• Proposition 6
• proof
• Proposition 7