No abstract

In this paper we develop a method for determining the exact number of spanning trees in (directed or undirected) circulant graphs. Using this method we can, for any class of circulant graph, exhibit a recurrence relation for the number of its spanning trees. We describe the method and give examples of its use.

Kirchhoff 's Matrix Tree Theorem permits the calculation of the number of spanning trees in any given graph G through the evaluation of the determinant of an associated matrix. In the case of some special graphs Boesch and Prodinger [Graph Combin. 2 (1986) 191-200] have shown how to use properties of Chebyshev polynomials to evaluate the associated determinants and derive closed formulas for the number of spanning trees of graphs.In this paper, we extend this idea and describe how to use Chebyshev polynomials to evaluate the number of spanning trees in G when G belongs to one of three different classes of graphs: (i) when G is a circulant graph with fixed jumps (substantially simplifying earlier proofs), (ii) when G is a circulant graph with some non-fixed jumps and when (iii) G = K n ± C, where K n is the complete graph on n vertices and C is a circulant graph.