Counting edges of different types in a local graph of a Grassmann graph

TL;DR

This paper classifies and counts different types of edges in the local graph of a Grassmann graph, using projective geometry and group actions to understand their structure.

Contribution

It introduces a new classification of edges in Grassmann graphs based on subspace intersections and provides formulas for counting each type within local graphs.

Findings

Counts of each edge type in local graphs are derived.

The action of the stabilizer group partitions the local graph into five orbits.

Results connect subspace intersections with edge classifications in Grassmann graphs.

Abstract

Let $\mathbb{F}_q$ denote a finite field with $q$ elements. Let $n,k$ denote integers with $n>2k\geq 6$. Let $V$ denote a vector space over $\mathbb{F}_{q}$ that has dimension $n$. The vertex set of the Grassmann graph $J_q(n,k)$ consists of the $k$-dimensional subspaces of $V$. Two vertices of $J_q(n,k)$ are adjacent whenever their intersection has dimension $k-1$. Let $\partial$ denote the path-length distance function of $J_q(n,k)$. Pick a vertex $y$. In this paper we define three types of edges in $X$, namely type $0$, type $+$, and type $-$; for adjacent vertices $w,z$ such that $\partial(w,y)=\partial(z,y)$, the type of the edge $wz$ depends on the subspaces $w+z,w,z,w\cap z$ and their intersections with $y$. Pick a vertex $x$ such that $1<\partial(x,y)<k$. Let $\Gamma(x)$ denote the local graph of $x$ in $J_q(n,k)$. Our general goal is to count the number of edges in $\Gamma(x)$ for each type. Consider a two-vertex stabilizer $\text{Stab}(x,y)$ in $GL(V)$; it is known that the $\text{Stab}(x,y)$-action on $\Gamma(x)$ has five orbits. Pick two orbits $\mathcal{O},\mathcal{N}$ that are not necessarily distinct; for a given $w\in \mathcal{O}$, we find the number of vertices in $z\in \mathcal{N}$ such that the edge $wz$ has (i) type $0$, (ii) type $+$, (iii) type $-$. To find these numbers, we use many results that involve a projective geometry $P_q(n)$, which is the set of all subspaces of $V$.