Multi-way expanders and imprimitive group actions on graphs

For n ≥ 2, the concept of n-way expanders was defined by many researchers. Bigger n gives a weaker notion in general, and 2-way expanders coincide with expanders in usual sense. Koji Fujiwara asked whether these concepts are equivalent to that of ordi-nary expanders for all n for a sequence of Cayley graphs. In this paper, we answer his question in the affirmative. Furthermore, we obtain universal inequalities on multi-way isoperimetric constants on any finite connected vertex-transitive graph, and show that gaps between these constants imply the imprimitivity of the group action on the graph.