Distance-transitive graph

An automorphism group is transitive on ordered pairs of vertices at every fixed distance. Grassmann graphs provide examples with the finite general linear group as a distance-transitive automorphism group.

A subgroup of the graph automorphism group acts distance-transitively when it is transitive on the ordered pairs of vertices at each fixed graph distance. Thus two vertex pairs at the same distance can be matched by one automorphism from that subgroup. Existence of such a subgroup implies that the full automorphism group has the same property, making the graph a distance-transitive graph. The specified subgroup need not be the entire automorphism group.

Vertices are Lagrangian subspaces of a $2m$-dimensional symplectic vector space, joined when they intersect in codimension one. A pair with intersection dimension $m-r$ admits a symplectic basis in which $U=⟨e_1,\dots ,e_m⟩$ and $W=⟨f_1,\dots ,f_r,e_{r+1},\dots ,e_m⟩$. Replacing $e_i$ successively by $f_i$ gives an isotropic path of length $r$, while one edge can increase intersection dimension with a fixed target by at most one. Equal-distance ordered pairs have the same adapted form, so the symplectic group acts distance-transitively.