THE ISOPERIMETRIC NUMBER OF d–DIMENSIONAL k–ARY ARRAYS

The d–dimensional k-ary array is the d–fold Cartesian product graph of the path graph P_k with k vertices. We show that the (edge) isoperimetric number of is given by and identify the cardinalities and the structure of the isoperimetric sets. For odd k, the cardinalities of isoperimetric sets in are , whereas every isoperimetric set for k even has cardinality .