Subconstituents of dual polar graph in finite classical space III Original Research Article

Let image be the finite field with q elements. Denote by Γ(δ) the dual polar graph of (2ν+δ)-dimensional orthogonal space over image, where δ=0, 1 or 2. For any vertex P of Γ(δ), all subconstituents Γi(δ)(P) (1less-than-or-equals, slantiless-than-or-equals, slantν) of Γ(δ) are studied, and it is proved that Γi(δ)(P) is isomorphic toimagewhere Λ(i,δ) is a subgraph of the graph of i×(i+δ) matrices over image. Moreover, some properties of the graph Λ(i,δ) are also studied. In particular, it is shown that Λ(i,δ) is edge-regular. Furthermore, both Λ(2,1) and Λ(3,1) are distance-regular with intersection arrays{q2−1,q2−q,1;1,q,q2−1} and {q3−1,q3−q,q3−q2+1;1,q,q2−1},respectively.