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Chebyshev polynomials and spanning tree formulas for circulant and related graphs. (English) Zbl 1070.05029

Summary: 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 F. T. Boesch and H. Prodinger [Graphs Comb. 2, 191–200 (1986; Zbl 0651.05028)] 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) \(G=K_{n}\pm C\), where \(K_n\) is the complete graph on \(n\) vertices and \(C\) is a circulant graph.

MSC:

05C05 Trees
05C30 Enumeration in graph theory
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)

Citations:

Zbl 0651.05028
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References:

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