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 \(\partial \) denote the path-length distance function of \(J_q(n,k)\). Pick two vertices x, y in \(J_q(n,k)\) such that \(1<\partial (x,y)<k\). 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.