Graphs associated with orthogonal collections of k-planes over finite fields

Abstract

We study graphs coming from quadratic spaces over finite fields via orthogonality which generalize a recent result given by Bishnoi, Ihringer, and Pepe (2019). More precisely, we study the graph ${\Gamma }^{\square }(n,k,q)$ as follows: the vertex set is the set of $k$-dimensional quadratic subspaces of a fixed Lorentzian quadratic space $({\mathbb{F}}_q^n,x_1^2+\cdots +x_{n-1}^2+\lambda x_n^2)$ which are isometrically isomorphic to $x_1^2+\cdots +x_k^2$. Here $\lambda$ is a nonsquare in ${\mathbb{F}}_q$, and two vertices $x,y$ are adjacent if $x\subseteq y^{\perp }$.

5 pages, comments are welcome!