Counting Spanning Trees and Other Structures in Non-constant-jump Circulant Graphs

Circulant graphs are an extremely well-studied subclass of regular graphs, partially because they model many practical computer network topologies. It has long been known that the number of spanning trees in n-node circulant graphs with constant jumps satisfies a recur- rence relation in n. For the non-constant-jump case, i.e., where some jump sizes can be functions of the graph size, only a few special cases such as the Mobius ladder had been studied but no general results were known. In this note we show how that the number of spanning trees for all classes of n node circulant graphs satisfies a recurrence relation in n even when the jumps are non-constant (but linear) in the graph size. The technique developed is very general and can be used to show that many other structures of these circulant graphs, e.g., number of Hamil- tonian Cycles, Eulerian Cycles, Eulerian Orientations, etc., also satisfy recurrence relations. The technique presented for deriving the recurrence relations is very mechanical and, for circulant graphs with small jump parameters, can easily be quickly implemented on a computer. We illustrate this by de- riving recurrence relations counting all of the structures listed above for various circulant graphs.