Weak-Geodetically Closed Subgraphs in Distance-Regular Graphs

Abstract.

 Let Γ=(X,R) denote a distance-regular graph with diameter D≥2 and distance function δ. A (vertex) subgraph Ω⊆X is said to be weak-geodetically closed whenever for all x,y∈Ω and all z∈X,

We show that if the intersection number c ₂>1 then any weak-geodetically closed subgraph of X is distance-regular. Γ is said to be i-bounded, whenever for all x,y∈X at distance δ(x,y)≤i,x,y are contained in a common weak-geodetically closed subgraph of Γ of diameter δ(x,y). By a parallelogram of length i, we mean a 4-tuple xyzw of vertices in X such that δ(x,y)=δ(z,w)=1, δ(x,w)=i, and δ(x,z)=δ(y,z)=δ(y,w)=i−1. We prove the following two theorems.

Theorem 1. LetΓdenote a distance-regular graph with diameter D≥2, and assume the intersection numbers c ₂>1, a ₁≠0. Then for each integer i (1≤i≤D), the following (i)–(ii) are equivalent.

(i)*Γis i-bounded.

(ii)*Γcontains no parallelogram of length≤i+1.

Restricting attention to the Q-polynomial case, we get the following stronger result.

Theorem 2. Let Γ denote a distance-regular graph with diameter D≥3, and assume the intersection numbers c ₂>1, a ₁≠0. Suppose Γ is Q-polynomial. Then the following (i)–(iii) are equivalent.

(i)*Γcontains no parallelogram of length 2 or 3.

(ii)*Γis D-bounded.

(iii)*Γhas classical parameters (D,b,α,β), and either b<−1, or elseΓis a dual polar graph or a Hamming graph.