On 2-walk-regular graphs with a large intersection number c₂

For a connected amply regular graph with parameters (v,k,λ,μ) satisfying λ≤μ, it is known that its diameter is bounded by k. This was generalized by Terwilliger to (s,c,a,k)-graphs satisfying c≥max{a,2}. It follows from Terwilliger that a connected amply regular graph with parameters (v,k,λ,μ) satisfying μ> [Formula presented] ≥1 and μ≥λ has diameter at most 7. In this paper we will classify the 2-walk-regular graphs with valency k≥3 and diameter at least 4 such that its intersection number c₂ satisfies c₂> [Formula presented]. This result generalizes a result of Koolen and Park for distance-regular graphs. And we show that if such a 2-walk-regular graph is not distance-regular, then it is the incidence graph of a group divisible design with the dual property with parameters (2,m;k;0,λ₂).