## 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

### MSC:

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

 [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 Reading, Mass · Zbl 0797.05064 [3] Washington, L., Introduction to cyclotomic fields, (1982), Springer New York, Heidelberg, Berlin · Zbl 0484.12001 [4] Marcus, D., Number fields, (1977), Springer New York, Heidelberg, Berlin [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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