A recurrence for the surface area of the (n,k)$$ \left(n,k\right) $$‐star graph

Abstract

SummaryWe present a simple recurrence for the surface area of the ‐star graph, , that is, the number of nodes at a certain distance from the identity node in the graph, an important parameter for interconnection networks in parallel computing. The family of the ‐star graphs includes several popular interconnection networks such as the star graph and the alternating group network. Previously, a surface area recurrence has been obtained for a special case, for example, when , in the family of ‐star. Our recurrence gives one single recurrence for all graphs in the family, thus completely solving the surface area of ‐star for all . Compared to explicit surface area formulas previously obtained through complicated and involved combinatorial analysis and generating function approach, our derivation is more elementary and our recurrence gives a way to compute the surface area of the ‐star efficiently.