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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi_{i,j}=|\Gamma_{j}(x)\cap Y|\quad (i,j\in \{0,1,\ldots,d\})$$\end{document} depend only on i=∂(x,Y) and j. A subset Y of X is said to be strongly closed if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{x\mid \partial(u,x)\leq \partial(u,v),\partial(v,x)=1\}\subset Y,\mbox{ whenever }u,v\in Y.$$\end{document} 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.