Unhooking Circulant Graphs: A Combinatorial Method for Counting Spanning Trees and Other Parameters

Abstract

It has long been known that the number of spanning trees in circulant graphs with fixed jumps and n nodes satisfies a recurrence relation in n. The proof of this fact was algebraic (relating the products of eigenvalues of the graphs’ adjacency matrices) and not combinatorial. In this paper we derive a straightforward combinatorial proof of this fact. Instead of trying to decompose a large circulant graph into smaller ones, our technique is to instead decompose a large circulant graph into different step graph cases and then construct a recurrence relation on the step graphs. We then generalize this technique to show that the numbers of Hamiltonian Cycles, Eulerian Cycles and Eulerian Orientations in circulant graphs also satisfy recurrence relations.