Subconstituents of dual polar graph in finite classical space III

Abstract

Let F q be the finite field with q elements. Denote by Γ ( δ) the dual polar graph of (2 ν+ δ)-dimensional orthogonal space over F q , where δ=0, 1 or 2. For any vertex P of Γ ( δ) , all subconstituents Γ i (δ)(P) (1⩽i⩽ν) of Γ ( δ) are studied, and it is proved that Γ i ( δ) ( P) is isomorphic to ν i q·Λ (i,δ), where Λ ( i, δ) is a subgraph of the graph of i×( i+ δ) matrices over F q . 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 {q 2−1,q 2−q,1;1,q,q 2−1} and {q 3−1,q 3−q,q 3−q 2+1;1,q,q 2−1}, respectively.