A remark on Mulderʹs conjecture about interval-regular graphs

Let Γ be a connected graph. Γ is said to beinterval-regular if |Γi−1(u) ∪ Γ(x)| = i holds for all vertices u and x ϵ Γi(u), i > 0. For u, v ϵΓ, let I(u,v) denote the set of all vertices on a shortest path connecting u,v. A subset W of V(Γ) is said to be convex if I(u,v) ⊂ W holds for each u, vϵ W. In (Mulder, 1982) Mulder conjectured that every interval I(u,v) in a interval-regular graph is convex. In this paper, we consider this problem for distance-regular graphs. We shall prove that, in a distance-regular interval-regular graph having a triangle, I(u,v) is convex for every vertices u,v at distance 3