Elsevier

Discrete Mathematics

Volume 298, Issues 1–3, 6 August 2005, Pages 334-364
Discrete Mathematics

Chebyshev polynomials and spanning tree formulas for circulant and related graphs

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Abstract

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=Kn±C, where Kn is the complete graph on n vertices and C is a circulant graph.

Keywords

Chebyshev polynomials
Spanning trees
Circulant graphs

Cited by (0)

This work was supported by Hong Kong CERG Grants 6082/97E, HKUST6137/98E, HKUST6162/00E, HKUST6082/01E and HKUST DAG98/99.EG23. An extended abstract of Sections 3 and 4 is in [14]. Some of the results in section 5 were presented as a poster session at the 14th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2002) FPSAC02.

1

Work done while at Hong Kong U. S. T.

2

Work done while at Hong Kong U. S. T.