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Kirchhoff's theorem

In the mathematical field of graph theory, Kirchhoff's theorem or Kirchhoff's matrix tree theorem is a theorem about the number of spanning trees in a graph. It states that this number can be computed as any cofactor of the graph's Laplacian matrix. This shows in particular that the number of spanning trees can be computed from the graph data in…

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DimensionEigenvaluesMultigraphsLoopsIndeterminate
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Definitions and statement

TreeDegree matrixDiagonal matrixDegreesAdjacency matrix
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Proof outline

Cauchy–Binet formulaInductionOriented incidence matrixIncidence matrixTranspose
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Example

Diamond graph
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Kirchhoff's theorem

Nodes43
Edges42
Triples39
Avg. degree1.95
Density0.046512
Components1

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Kirchhoff's theorem

Top relations

related to Proof outline · 9
Kirchhoff's theorem → An, Binet, Cauchy, First, Kirchhoff's, Laplacian, Moore, The, Thus
related to Cayley's formula · 7
Kirchhoff's theorem → Alternatively, Cayley's, Kirchhoff's, Kn, Laplacian, The Laplacian, These
related to Explicit enumeration of spanning trees · 7
Kirchhoff's theorem → After, In, Kirchhoff, Kirchhoff's, Laplacian, Rather, The
related to Definitions and statement · 5
Kirchhoff's theorem → In, Kirchhoff's, Laplacian, Let, The Laplacian
related to Counting spanning k-component forests · 3
Kirchhoff's theorem → Given, Kirchhoff's, Then
related to Matroids · 3
Kirchhoff's theorem → Kirchhoff's, Maurer, The
related to Kirchhoff's theorem for directed multigraphs · 2
Kirchhoff's theorem → Kirchhoff's, The
related to Kirchhoff's theorem for multigraphs · 2
Kirchhoff's theorem → Kirchhoff's, The
is a · 1
Kirchhoff's theorem → generalization of Cayley's formula which provides the number of spanning trees in a complete graph

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Important terminology

matrix spanning number graph theorem trees laplacian kirchhoff's formula determinant tree eigenvalues row column edges cayley's cofactor vertex vertices example

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SubjectPredicateObjectConfidenceSrc
Kirchhoff's theoremis ageneralization of Cayley's formula which provides the number of spanning trees in a complete graph0.90text
Kirchhoff's theoremrelated to Cayley's formulaCayley's0.60section
Kirchhoff's theoremrelated to Cayley's formulaKirchhoff's0.60section
Kirchhoff's theoremrelated to Cayley's formulaLaplacian0.60section
Kirchhoff's theoremrelated to Cayley's formulaThese0.60section
Kirchhoff's theoremrelated to Cayley's formulaAlternatively0.60section
Kirchhoff's theoremrelated to Cayley's formulaKn0.60section
Kirchhoff's theoremrelated to Cayley's formulaThe Laplacian0.60section
Kirchhoff's theoremrelated to Counting spanning k-component forestsKirchhoff's0.60section
Kirchhoff's theoremrelated to Counting spanning k-component forestsGiven0.60section
Kirchhoff's theoremrelated to Counting spanning k-component forestsThen0.60section
Kirchhoff's theoremrelated to Definitions and statementLet0.60section

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