Vertices are -spaces of a finite vector space, with edges when intersection dimension is . Basis exchanges give a path of length . Each edge changes intersection dimension with a fixed space by at most one, proving the lower bound. Equal-intersection ordered pairs are linearly conjugate, making the graph distance-transitive.
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A Grassmann graph, also known as a Grassmannian graph, is a concept from the field of combinatorial geometry and algebraic geometry that is closely related to Grassmannians. Grassmannians are spaces that parameterize all k-dimensional linear subspaces of an n-dimensional vector space. The vertices of a Grassmann graph correspond to the k-dimensional subspaces of a vector space, and the edges represent the relationships between these subspaces.