matrix-tree theorem
E824090
The matrix-tree theorem is a fundamental result in algebraic graph theory that expresses the number of spanning trees of a graph as a determinant of a matrix derived from the graph’s Laplacian.
All labels observed (1)
| Label | Occurrences |
|---|---|
| matrix-tree theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T9838898 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: matrix-tree theorem Context triple: [de Bruijn–van Aardenne–Ehrenfest theorem, relatedTo, matrix-tree theorem]
-
A.
Sylvester determinant
The Sylvester determinant is a mathematical construct introduced by James Joseph Sylvester, typically referring to a determinant associated with resultants and elimination theory in algebra.
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B.
Cauchy determinant
The Cauchy determinant is a classical determinant formula in linear algebra that gives a closed-form expression for matrices with entries of the form 1/(x_i + y_j), named after the French mathematician Augustin-Louis Cauchy.
-
C.
Cayley–Hamilton theorem
The Cayley–Hamilton theorem is a fundamental result in linear algebra stating that every square matrix satisfies its own characteristic polynomial.
-
D.
Cauchy matrix
A Cauchy matrix is a structured matrix whose entries are defined by the reciprocals of pairwise differences of two sequences, widely used in numerical analysis, interpolation, and algebra.
-
E.
Pólya enumeration theorem
The Pólya enumeration theorem is a fundamental result in combinatorics that counts distinct configurations of objects under group actions by using cycle index polynomials and generating functions.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: matrix-tree theorem Target entity description: The matrix-tree theorem is a fundamental result in algebraic graph theory that expresses the number of spanning trees of a graph as a determinant of a matrix derived from the graph’s Laplacian.
-
A.
Sylvester determinant
The Sylvester determinant is a mathematical construct introduced by James Joseph Sylvester, typically referring to a determinant associated with resultants and elimination theory in algebra.
-
B.
Cauchy determinant
The Cauchy determinant is a classical determinant formula in linear algebra that gives a closed-form expression for matrices with entries of the form 1/(x_i + y_j), named after the French mathematician Augustin-Louis Cauchy.
-
C.
Cayley–Hamilton theorem
The Cayley–Hamilton theorem is a fundamental result in linear algebra stating that every square matrix satisfies its own characteristic polynomial.
-
D.
Cauchy matrix
A Cauchy matrix is a structured matrix whose entries are defined by the reciprocals of pairwise differences of two sequences, widely used in numerical analysis, interpolation, and algebra.
-
E.
Pólya enumeration theorem
The Pólya enumeration theorem is a fundamental result in combinatorics that counts distinct configurations of objects under group actions by using cycle index polynomials and generating functions.
- F. None of above. chosen
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
result in algebraic graph theory
ⓘ
theorem ⓘ |
| alsoKnownAs |
Kirchhoff’s matrix-tree theorem
NERFINISHED
ⓘ
Kirchhoff’s theorem on trees NERFINISHED ⓘ |
| appearsIn |
textbooks on algebraic graph theory
ⓘ
textbooks on spectral graph theory ⓘ |
| appliesTo |
finite graph
ⓘ
multigraph ⓘ simple graph ⓘ |
| assumes | graph is connected for a positive number of spanning trees ⓘ |
| coreClaim |
any cofactor of the Laplacian matrix equals the number of spanning trees of the graph
ⓘ
deleting any one row and any one column from the Laplacian and taking the determinant yields the number of spanning trees ⓘ |
| field |
algebraic graph theory
ⓘ
graph theory ⓘ |
| generalizationOf | Cayley’s formula for the number of labeled trees NERFINISHED ⓘ |
| gives | number of spanning trees of a graph ⓘ |
| hasVariant |
all-minors matrix-tree theorem
NERFINISHED
ⓘ
directed matrix-tree theorem NERFINISHED ⓘ weighted matrix-tree theorem NERFINISHED ⓘ |
| historicalPeriod | 19th century ⓘ |
| implies |
Laplacian matrix has one zero eigenvalue for a connected graph
ⓘ
Laplacian matrix of a connected graph has rank n-1 ⓘ |
| importance |
central tool for counting spanning trees
ⓘ
fundamental theorem in graph enumeration ⓘ |
| namedAfter | Gustav Kirchhoff NERFINISHED ⓘ |
| proofTechniques |
Cauchy–Binet formula
NERFINISHED
ⓘ
combinatorial arguments ⓘ linear algebra ⓘ |
| relatedTo |
Kirchhoff’s circuit laws
NERFINISHED
ⓘ
Laplacian eigenvalues ⓘ Matrix-Tree theorem for directed graphs NERFINISHED ⓘ |
| relatesConcept |
Kirchhoff matrix
NERFINISHED
ⓘ
cofactor ⓘ determinant ⓘ graph Laplacian NERFINISHED ⓘ spanning tree ⓘ |
| statementForm | determinant formula ⓘ |
| usedIn |
electrical network theory
ⓘ
enumeration of spanning trees ⓘ network reliability ⓘ probability on graphs ⓘ random spanning tree algorithms ⓘ spectral graph theory ⓘ |
| usesMatrix |
Laplacian matrix of a graph
ⓘ
combinatorial Laplacian ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: matrix-tree theorem Description of subject: The matrix-tree theorem is a fundamental result in algebraic graph theory that expresses the number of spanning trees of a graph as a determinant of a matrix derived from the graph’s Laplacian.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.