Kirchhoff's theorem

On the number of spanning trees in a graph

In the mathematical field of graph theory, Kirchhoff's theorem or Kirchhoff's matrix tree theorem named after Gustav Kirchhoff is a theorem about the number of spanning trees in a graph, showing that this number can be computed in polynomial time from the determinant of a submatrix of the graph's Laplacian matrix; specifically, the number is equal to any cofactor of the Laplacian matrix. Kirchhoff's theorem is a generalization of Cayley's formula which provides the number of spanning trees in a complete graph.

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