Using a Grassmann graph to recover the underlying projective geometry

Let $n,k$ denote integers with $n>2k\ge 6$. Let ${\mathbb{F}}_q$ denote a finite field with $q$ elements, and let $V$ denote a vector space over ${\mathbb{F}}_q$ that has dimension $n$. The projective geometry $P_q(n)$ is the partially ordered set consisting of the subspaces of $V$; the partial order is given by inclusion. For the Grassmann graph $J_q(n,k)$ the vertex set consists of the $k$-dimensional subspaces of $V$. Two vertices of $J_q(n,k)$ are adjacent whenever their intersection has dimension $k-1$. The graph $J_q(n,k)$ is known to be distance-regular. Let $\mathrm{\partial }$ denote the path-length distance function of $J_q(n,k)$. Pick two vertices $x,y$ in $J_q(n,k)$ such that $1<\mathrm{\partial }(x,y)<k$. The set $P_q(n)$ contains the elements $x,y,x\cap y,x+y$. In our main result, we describe $x\cap y$ and $x+y$ using only the graph structure of $J_q(n,k)$. To achieve this result, we make heavy use of the Euclidean representation of $J_q(n,k)$ that corresponds to the second largest eigenvalue of the adjacency matrix.