THE ISOPERIMETRIC NUMBER OF d–DIMENSIONAL k–ARY ARRAYS

Abstract

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