Valency of Distance-regular Antipodal Graphs with Diameter 4

Let G be a non-bipartite strongly regular graph on n vertices of valency k. We prove that if G has a distance-regular antipodal cover of diameter 4, then k ≤ ⌊2(n + 1) / 5⌋, unless G is the complement of triangular graph T(7), the folded Johnson graph J(8, 4) or the folded halved 8-cube. However, for these three graphs the bound k ≤ ⌊(n − 1) / 2⌋ holds. This result implies that only one of a complementary pair of strongly regular graphs can be the antipodal quotient of an antipodal distance-regular graph.