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

Abstract

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 πi,j}=|Γj}(x)∩ Y|(i,j ε {0,1,}) 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, ε 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. © 2009 Springer Science+Business Media, LLC.