Stars of graphs of projective codes

TL;DR

This paper characterizes the structure of maximal cliques in a subgraph of the Grassmann graph formed by projective codes, identifying two types: stars and tops, and providing a detailed description of stars.

Contribution

It provides a complete classification of maximal cliques in the projective code subgraph of the Grassmann graph, focusing on stars and tops.

Findings

Maximal cliques in the subgraph are of two types: stars and tops.

Stars are characterized by all projective codes containing a fixed (k-1)-dimensional subspace.

The paper offers a full description of the structure of stars.

Abstract

Let $\Gamma_k(V)$ be the Grassmann graph whose vertex set is formed by all $k$-dimensional subspaces of an $n$-dimensional vector space $V$ over the finite field $F_q$ consisting of $q$ elements. We discuss its subgraph $\Pi(n,k)_q$ formed by projective codes. We show that there are precisely two types of maximal cliques in $\Pi(n,k)_q$: stars and tops. We give a complete description of stars, i.e., maximal cliques consisting of all $k$-dimensional projective codes containing a certain $(k-1)$-dimensional subspace of $V$.