= Symplectic dual polar graph
{title2=$d(U,W)=m-\dim(U\cap W)$}
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 $U=\langle e_1,\ldots,e_m\rangle$ and $W=\langle f_1,\ldots,f_r,e_{r+1},\ldots,e_m\rangle$. 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>.
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