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

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 $x \subseteq y^{\perp}$.