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For   $n\geq 2$  , 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 ordinary 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.

Multi-Way Expanders and Imprimitive Group Actions on Graphs