In this chapter we investigate the relationship between primitivity and distance-transitivity. We shall prove that the automorphism group of a distance-transitive graph can act imprimitively in only two ways, both of which have simple characterizations in terms of the structure of the graph.

We begin by summarizing some terminology. If G is a group of permutations of a set X, a block B is a subset of X such that B and g(B) are either disjoint or identical, for each g in G. If G is transitive on X, then we say that the permutation group (X, G) is primitive if the only blocks are the trivial blocks, that is, those with cardinality 0, 1 or |X|. If B is a non-trivial block and G is transitive on X, then each g(B) is a block, and the distinct blocks g(B) form a partition of X which we refer to as a block system. Further, G acts transitively on these blocks.

A graph Г is said to be primitive or imprimitive according as the group G = Aut(Г) acting on VГ has the corresponding property. For example, the ladder graph L3 is imprimitive: there is a block system with two blocks, the vertices of the triangles in L3.

Proposition 22.1Let Г be a connected graph for which the group of automorphisms acts imprimitively and symmetrically (in the sense of Definition 15.5). Then a block system for the action of Aut(Г) on VГ must be a colour-partition of Г.