On the Problem of Determining which (n, k)-Star Graphs are Cayley Graphs

Abstract

In this paper we work to classify which of the (n, k)-star graphs, denoted by $$S_{n,k}$$Sn,k, are Cayley graphs. Although the complete classification is left open, we derive infinite and non-trivial classes of both Cayley and non-Cayley graphs. We give a complete classification of the case when $$k=2$$k=2, showing that $$S_{n,2}$$Sn,2 is Cayley if and only if n is a prime power. We also give a sufficient condition for $$S_{n,3}$$Sn,3 to be Cayley and study other structural properties, such as demonstrating that $$S_{n,k}$$Sn,k always has a uniform shortest path routing.