On top locally-s-distance-transitive graphs

A graph with some graph symmetry property is top, if it cannot be viewed as a nontrivial normal quotient of some other graph with the same graph symmetry property. Therefore, a graph being top implies that it has no nontrivial normal multicovers, including normal covers. John Conway proved that every s-arc-transitive graph has a nontrivial s-arc-transitive normal cover, so there is no top s-arc-transitive graph. However, there exist top locally-s-distance-transitive graphs, and complete multipartite graphs are examples of this. In this paper, we give a generic condition for locally-s-distance-transitive graphs to be top. Also, examples and characterizations of graphs that admit this condition are given.