On some properties of k-ary n-cubes

The k-ary n-cube has been used as the underlying topology for most practical multicomputers, and has been extensively studied in the past. We investigate some properties of this network. In particular, we study the problem of finding the number of nodes located i hops away from a given node (surface area) and the number of nodes located within i hops away from a given node (volume) in both the unidirectional and bidirectional k-ary n-cube, and have derived exact expressions calculating these numbers. These results are very useful when studying, for example, the spanning tree structure of the k-ary n-cube and the problem of resource placement in this network.