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

Let = (X,R) be a distance-regular graph of diameter d . A parallelogram of length i is a 4-tuple xyzw consisting of vertices of 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 πi,j = | j (x)∩ Y | (i, j ∈ {0,1, . . . , d}) depend only on i = ∂(x,Y ) and j . A subset Y of X is said to be strongly closed if {x | ∂(u, x)≤ ∂(u, v), ∂(v, x)= 1} ⊂ Y, whenever u,v ∈ 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 be a parallelogram-free distanceregular graph of diameter d ≥ 4 such that every strongly closed subgraph of diameter two is completely regular. We show that 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, 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.