Math 2.0

Combinatorics Definition

Cayley graph

Builds on Subgroup generated by a subsetGroup action

Definition

Let GG be a group and let SS be a generating set of GG. The Cayley graph Γ(G,S)\Gamma(G, S) is the directed graph with vertex set GG. For each g∈Gg \in G and s∈Ss \in S, it has an edge with colour ss from gg to gsg s.

Remarks

For G=ℤG = \mathbb{Z} and S={1,−1}S = \{1, -1\}, the Cayley graph is an infinite path. For ℤ/nℤ\mathbb{Z}/n\mathbb{Z} with a generator and its inverse, it is the cycle CnC_n. The Cayley graph of a free group with respect to a basis is a tree. The left multiplication action of GG on itself acts on Γ(G,S)\Gamma(G, S), so each Cayley graph is vertex-transitive. Arthur Cayley first considered these graphs for finite groups in 1878.

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

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Group, concept, no evidence levelGroupconceptGroup homomorphism, definition, no evidence levelGroup homomorphismdefinitionSubgroup, definition, no evidence levelSubgroupdefinitionSymmetric group, definition, no evidence levelSymmetric groupdefinitionGroup action, definition, no evidence levelGroup actiondefinitionSubgroup criterion, statement, no evidence levelSubgroup criterionstatementIntersection of subgroups is a subgroup, statement, no evidence levelIntersection of subgr…statementSubgroup generated by a subset, definition, no evidence levelSubgroup generated by…definitionCayley graph, definition, no evidence levelCayley graphdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Group concept
  2. Group homomorphism definition
  3. Subgroup definition
  4. Symmetric group definition
  5. Group action definition
  6. Subgroup criterion statement
  7. Intersection of subgroups is a subgroup statement
  8. Subgroup generated by a subset definition
  9. Cayley graph definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Cayley_graph, Definition
  2. Web https://en.wikipedia.org/wiki/Cayley_graph, Examples
  3. Web https://en.wikipedia.org/wiki/Cayley_graph, Elementary properties
  4. Web https://en.wikipedia.org/wiki/Cayley_graph, History
About this page
Tag0YE597NHGB
URIm2://0YE597NHGB
Aliasescayley-diagram
Kinddefinition
WritingW1 stated
MSC05C25, 20F65
DefinesCayley graph
UsesSubgroup generated by a subset, Group action

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