Counting spanning trees in the graphs of Kleitman and Golden and a generalization. (English) Zbl 0569.05016

The number of trees in a graph G on \(n+1\) vertices is, by the celebrated matrix tree theorem, given by the determinant of an \(n\times n\) matrix whose (i-th entry is 1 when G has an edge connecting the i-th and j-th vertices (for i and j between 1 and n) and i-th diagonal element given by minus the degree of the i-th vertex. The authors here use properties of circulant matrices to evaluate this determinant in the case in which the \((n+1)\times (n+1)\) matrix of which this is a cofactor, is a circulant.
Reviewer: D.Kleitman


05C05 Trees
05C30 Enumeration in graph theory
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
Full Text: DOI


[1] Kleitman, D. J.; Golden, B., Counting trees in a certain class of graphs, Am. math. Monthly, Vol. 82, 40-44 (1975) · Zbl 0297.05123
[2] Harary, F., Graph Theory (1969), Addison-Wesley: Addison-Wesley Reading, Mass · Zbl 0797.05064
[3] Washington, L., Introduction to Cyclotomic Fields (1982), Springer: Springer New York, Heidelberg, Berlin · Zbl 0484.12001
[4] Marcus, D., Number Fields (1977), Springer: Springer New York, Heidelberg, Berlin · Zbl 0383.12001
[5] Bedrosian, S. D., The Fibonacci numbers via trigonometric expressions, J. Franklin Inst., Vol. 295, 175-177 (1973) · Zbl 0298.05104
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