Parallelogram-free distance-regular graphs having completely regular strongly regular subgraphs

Let Γ=(X,R) be a distance-regular graph of diameter d. A parallelogram of length i is a 4-tuple xyzw consisting of vertices of Γ such that ?(x,y)=?(z,w)=1, ?(x,z)=i, and ?(x,w)=?(y,w)=?(y,z)=i?1. A subset Y of X is said to be a completely regular code if the numbers

$pi_{i,j}=|Gamma_{j}(x)cap Y|quad (i,jin {0,1,ldots,d})$

depend only on i=?(x,Y) and j. A subset Y of X is said to be strongly closed if

${xmid partial(u,x)leq partial(u,v),partial(v,x)=1}subset Y,mbox{ whenever }u,vin Y.$

Hamming graphs and dual polar graphs have strongly closed completely regular codes. In this paper, we study parallelogram-free distance-regular graphs having strongly closed completely regular codes. Let Γ be a parallelogram-free distance-regular graph of diameter d≥4 such that every strongly closed subgraph of diameter two is completely regular. We show that Γ has a strongly closed subgraph of diameter d?1 isomorphic to a Hamming graph or a dual polar graph. Moreover if the covering radius of the strongly closed subgraph of diameter two is d?2, Γ itself is isomorphic to a Hamming graph or a dual polar graph. We also give an algebraic characterization of the case when the covering radius is d?2.