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 $\Gamma<sup>{\square}(n,k,q)$ as follows: the vertex set is the set of $k$-dimensional quadratic subspaces of a fixed Lorentzian quadratic space $$(\mathbb{F}<em>{q}^{n},x</em>{1}^{{2}+\cdots+x_{n-1}^{2}+\lambda} x_{n}<sup>{2})$$ which are isometrically isomorphic to $x_{1}<sup>{2}+\cdots+x_{k}<sup>{2}$. Here $\lambda$ is a nonsquare in $\mathbb{F}_{q}$, and two vertices $x,y$ are adjacent if $x \subseteq y<sup>{\perp}$.