Cayley graph
E911229
A Cayley graph is a graphical representation of a group where vertices correspond to group elements and edges represent multiplication by chosen generators, widely used in group theory and geometric group theory.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Cayley graph canonical | 5 |
| Cayley digraph | 1 |
| Cayley graphs | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11215182 — 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: Cayley graph Context triple: [Dehn complex, relatedTo, Cayley graph]
-
A.
Coxeter–Dynkin diagrams
Coxeter–Dynkin diagrams are graphical representations that encode the structure of reflection groups and root systems, widely used in the classification of regular polytopes, Lie algebras, and symmetries.
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B.
Conway's 99-graph problem
Conway's 99-graph problem is an unsolved combinatorial question in graph theory, posed by John H. Conway, concerning the existence and properties of a hypothetical 99-vertex graph with highly constrained adjacency conditions.
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C.
Graham–Pollak theorem
The Graham–Pollak theorem is a result in graph theory that states the edges of a complete graph on n vertices cannot be partitioned into fewer than n−1 complete bipartite subgraphs.
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D.
Introduction to Graph Theory
Introduction to Graph Theory is a widely used textbook that provides a clear and accessible introduction to the fundamental concepts and techniques of graph theory.
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E.
Penrose graphical notation
Penrose graphical notation is a diagrammatic method for representing and manipulating tensors using networks of shapes and lines, widely used in mathematics and theoretical physics.
- 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: Cayley graph Target entity description: A Cayley graph is a graphical representation of a group where vertices correspond to group elements and edges represent multiplication by chosen generators, widely used in group theory and geometric group theory.
-
A.
Coxeter–Dynkin diagrams
Coxeter–Dynkin diagrams are graphical representations that encode the structure of reflection groups and root systems, widely used in the classification of regular polytopes, Lie algebras, and symmetries.
-
B.
Conway's 99-graph problem
Conway's 99-graph problem is an unsolved combinatorial question in graph theory, posed by John H. Conway, concerning the existence and properties of a hypothetical 99-vertex graph with highly constrained adjacency conditions.
-
C.
Graham–Pollak theorem
The Graham–Pollak theorem is a result in graph theory that states the edges of a complete graph on n vertices cannot be partitioned into fewer than n−1 complete bipartite subgraphs.
-
D.
Introduction to Graph Theory
Introduction to Graph Theory is a widely used textbook that provides a clear and accessible introduction to the fundamental concepts and techniques of graph theory.
-
E.
Penrose graphical notation
Penrose graphical notation is a diagrammatic method for representing and manipulating tensors using networks of shapes and lines, widely used in mathematics and theoretical physics.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
construction in group theory
ⓘ
graph theory concept ⓘ mathematical object ⓘ |
| appliedIn |
combinatorial group theory
ⓘ
geometric group theory ⓘ interconnection networks in parallel computing ⓘ network design ⓘ |
| constructionOf |
finite group
ⓘ
finitely generated group ⓘ infinite group ⓘ |
| dependsOn | choice of generating set ⓘ |
| edgeCondition | edge from g to gs for g in G and s in S ⓘ |
| field |
geometric group theory
ⓘ
graph theory ⓘ group theory ⓘ |
| generalizationOf |
cycle graph of cyclic group
ⓘ
dihedral group graphs ⓘ hypercube graph of (Z2)^n ⓘ |
| hasAutomorphismGroup | contains left-regular representation of group ⓘ |
| hasEdgeDefinition | edges connect elements differing by a generator ⓘ |
| hasNotation | Cay(G,S) ⓘ |
| hasProperty |
can be colored by generators
ⓘ
can be directed or undirected ⓘ connected if generating set generates the group ⓘ edge-colorable by generators ⓘ encodes word metric of group ⓘ locally finite if generating set is finite ⓘ quasi-isometry invariant up to generating set choice ⓘ regular graph ⓘ vertex-transitive graph ⓘ |
| hasVertexSet | underlying group ⓘ |
| isUndirectedIf | generating set is symmetric ⓘ |
| namedAfter | Arthur Cayley ⓘ |
| parameter |
generating set S
ⓘ
group G ⓘ |
| relatedTo |
Cayley complex
ⓘ
linked to:
Dehn complex
Cayley digraph ⓘ
linked to:
Cayley graph
Cayley table ⓘ Schreier graph ⓘ |
| specialCaseOf | vertex-transitive graph ⓘ |
| usedFor |
studying automorphism groups of graphs
ⓘ
studying expander graphs ⓘ studying finitely generated groups ⓘ studying growth of groups ⓘ studying random walks on groups ⓘ studying spectral properties of groups ⓘ studying symmetry ⓘ studying word metrics on groups ⓘ visualizing group structure ⓘ |
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: Cayley graph Description of subject: A Cayley graph is a graphical representation of a group where vertices correspond to group elements and edges represent multiplication by chosen generators, widely used in group theory and geometric group theory.
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.