2-Walk-regular graphs with a small number of vertices compared to the valency

In 2013, it was shown that, for a given real number α>2α>2, there are only finitely many distance-regular graphs Γ with valency k   and diameter D≥3D≥3 having at most αk vertices, except for the following two cases: (i  ) D=3D=3 and Γ is imprimitive; (ii)(ii)D=4D=4 and Γ is antipodal and bipartite. In this paper, we will generalize this result to 2-walk-regular graphs. In this case, also incidence graphs of certain group divisible designs appear.