Locally s-arc-transitive graphs related to sporadic simple groups

We show that, up to isomorphism, there are ten connected locally (M11,2)-arc-transitive graphs. We compute, up to conjugacy, the connected locally (G,s)-arc-transitive graphs for G=M11, M12, M22, M23, M24, J1, J2, J3, HS, McL, He, Ru, Suz, Co3, and we obtain a partial classification for O′N. The classification provides locally (G,7)- and (G,9)-arc-transitive graphs for He and Ru, and locally (G,7)-arc-transitive graphs for J3 and O′N. We also obtain new cubic 5-arc-transitive graphs for He and Ru.