Let \(\Gamma \) be a distance-regular graph whose automorphism group \( \operatorname {Aut}(\Gamma )\) has rank at most 11 and admits a subgroup which is permutation equivalent to \(\operatorname {PSL}_2(q)\) acting on 2-subsets of \(\operatorname {PG}_{1}(q)\) . It is shown that either \(\operatorname {Aut}(\Gamma )\) does not preserve the edges of the Kneser graph \(K(q+1,2)\) or the graph \(\Gamma \) is one of: the Kneser graph \(K(q+1,2)\) , the Johnson graph \(J(q+1,2)\) , the line graph of the Petersen graph, the Coxeter graph or the line graph of the Tutte-Coxeter graph.
