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Spanning tree formulas and Chebyshev polynomials. (English) Zbl 0651.05028
Summary: The Kirchhoff matrix tree theorem provides an efficient algorithm for determining t(G), the number of spanning trees of any graph G, in terms of a determinant. However for many special classes of graphs, one can avoid the evaluation of a determinant, as there are simple, explicit formulas that give the value of t(G). In this work we show that many of these formulas can be simply derived from known properties of Chebyshev polynomials. This is demonstrated for wheels, fans, ladders, Moebius ladders, and squares of cycles. The method is then used to derive a new spanning tree formula for the complete prism \(R_ n(m)=K_ m\times C_ n\). It is shown that \[ t(R_ n(m))\quad =\quad \frac{n}{m}2^{m- 1}[T_ n(1\quad +\quad \frac{m}{2})_{-1}]^{m-1} \] where \(T_ n(x)\) is the nth order Chebyshev polynomial of the first kind.

05C05 Trees
05C30 Enumeration in graph theory
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
Full Text: DOI
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