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Graph defined from a mathematical group
In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group, is a graph that encodes the abstract
Cayley_graph
Cubic graph with 10 vertices and 15 edges
symmetry, the Petersen graph is not a Cayley graph. It is the smallest vertex-transitive graph that is not a Cayley graph. The Petersen graph has a Hamiltonian
Petersen_graph
Graph where all pairs of vertices are automorphic
graphs (such as the Petersen graph, the Heawood graph and the vertices and edges of the Platonic solids). The finite Cayley graphs (such as cube-connected
Vertex-transitive_graph
Group whose Cayley graph is an initially subamenable graph
In mathematics, a sofic group is a group whose Cayley graph is an initially subamenable graph, or equivalently a subgroup of an ultraproduct of finite-rank
Sofic_group
Mathematical group that can be generated as the set of powers of a single element
Cayley graph is a cycle graph, and for an infinite cyclic group with its generator the Cayley graph is a doubly infinite path graph. However, Cayley graphs
Cyclic_group
Problem in graph theory
directed Cayley graph of an abelian group has a Hamiltonian path; however, every cyclic group whose order is not a prime power has a directed Cayley graph that
Lovász_conjecture
Branch of mathematics
graph, such as the Petersen graph, has few distinct values (the Petersen graph has 3, which is the minimum possible, given its diameter). For Cayley graphs
Algebraic_graph_theory
One of two different regular graphs with 16 vertices
characteristic polynomial, making it a graph determined by its spectrum. The 5-regular Clebsch graph is a Cayley graph with an automorphism group of order
Clebsch_graph
English mathematician (1821–1895)
theory, Cayley tables, Cayley graphs, and Cayley's theorem are named in his honour, as well as Cayley's formula in combinatorics. Arthur Cayley was born
Arthur_Cayley
Symmetric bipartite cubic graph with 16 vertices and 24 edges
group with genus two. The Cayley graph on 96 vertices is a flag graph of the genus 2 regular map having Möbius–Kantor graph as a skeleton. This means
Möbius–Kantor_graph
10316v1 [math.CO]. Abdollahi A., Zallaghi M. (2015). "Character sums for Cayley graphs". Communications in Algebra. 43 (12): 5159–5167. doi:10.1080/00927872
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Mathematics concept
rank (given by 1 plus the Euler characteristic of the quotient graph). The Cayley graph of a free group of finite rank, with respect to a free generating
Free_group
Polytope whose vertices represent permutations
vertices of the Cayley graph are the inverse permutations of those in the permutohedron. The image on the right shows the Cayley graph of S4. Its edge
Permutohedron
Mathematical graph of a Sudoku
extension on this graph. It is an integral Cayley graph. On a Sudoku board of size n 2 × n 2 {\displaystyle n^{2}\times n^{2}} , the Sudoku graph has n 4 {\displaystyle
Sudoku_graph
Seidel [de] as the coset graph of the ternary Golay code. This graph is the Cayley graph of an abelian group. Among abelian Cayley graphs that are strongly regular
Berlekamp–Van Lint–Seidel graph
Berlekamp–Van_Lint–Seidel_graph
Decomposition of a graph into hamiltonion cycles
the group. Infinitely many 6-regular Cayley graphs have no Hamiltonian decomposition, and there exist Cayley graphs of arbitrarily large even degree with
Hamiltonian_decomposition
Group of symmetries of the square
products of powers of a and b. This group of order 8 has the following Cayley table: For any two elements in the group, the table records what their composition
Dihedral_group_of_order_8
Graph able to be embedded on a torus
only if it has none of these graphs as a topological minor. Two isomorphic Cayley graphs of the quaternion group. Cayley graph of the quaternion group embedded
Toroidal_graph
Formula for the "volume" of an n-simplex
In linear algebra, geometry, and trigonometry, the Cayley–Menger determinant is a formula for the content, i.e. the higher-dimensional volume, of an n
Cayley–Menger_determinant
Graph with nodes connected in a closed chain
vertex set. A directed cycle graph has uniform in-degree 1 and uniform out-degree 1. Directed cycle graphs are Cayley graphs for cyclic groups (see e.g
Cycle_graph
Spectral graph theory concept
S=-S} . Then the Cayley graph for F q {\displaystyle \mathbb {F} _{q}} with generators from S {\displaystyle S} is a Ramanujan graph. Mathematicians are
Ramanujan_graph
Graph with a prism as its skeleton
single edge. As with many vertex-transitive graphs, the prism graphs may also be constructed as Cayley graphs. The order-n dihedral group is the group of
Prism_graph
Undirected, connected, and acyclic graph
was coined in 1857 by the British mathematician Arthur Cayley. A tree is an undirected graph G that satisfies any of the following equivalent conditions:
Tree_(graph_theory)
Longest distance between two vertices
an exponent depending on the graph family. Triameter (graph theory) Diameter (group theory), the diameter of a Cayley graph of the group, for generators
Diameter_(graph_theory)
Path in a graph that visits each vertex exactly once
as a graph, is Hamiltonian The Cayley graph of a finite Coxeter group is Hamiltonian (see Lovász conjecture for a more general claim) Cayley graphs on nilpotent
Hamiltonian_path
Concept in graph theory
In graph theory, a strongly regular graph (SRG) is a regular graph G = (V, E) with v vertices and degree k such that for some given integers λ , μ ≥ 0
Strongly_regular_graph
Four-dimensional number system
numbers. From this perspective, quaternions are the result of applying the Cayley–Dickson construction to the complex numbers. This is a generalization of
Quaternion
Basic concept of graph theory
graph of degree d, we have: 2(d + 1)/3 ≤ κ(G) ≤ λ(G) = d. For a vertex-transitive graph of degree d ≤ 4, or for any (undirected) minimal Cayley graph
Connectivity_(graph_theory)
Mapping a graph onto itself without changing edge-vertex connectivity
In the mathematical field of graph theory, an automorphism of a graph is a form of symmetry in which the graph is mapped onto itself while preserving
Graph_automorphism
Regular infinite tree structure used in statistical mechanics
result in the study of (n,d,λ)-graphs. A Bethe graph of even coordination number 2n is isomorphic to the unoriented Cayley graph of a free group of rank n
Bethe_lattice
were George Pólya, Arthur Cayley and J. Howard Redfield. In some graphical enumeration problems, the vertices of the graph are considered to be labeled
Graph_enumeration
Polyhedron with 54 faces
total of 120 different points. The truncated dodecadodecahedron forms a Cayley graph for the symmetric group on five elements, as generated by two group members:
Truncated_dodecadodecahedron
Area of discrete mathematics
concern the enumeration of graphs with particular properties. Enumerative graph theory then arose from the results of Cayley and the fundamental results
Graph_theory
24-vertex symmetric bipartite cubic graph
{\displaystyle G(10,2)} and the Desargues graph G ( 10 , 3 ) {\displaystyle G(10,3)} . The Nauru graph is a Cayley graph of S4, the symmetric group of permutations
Nauru_graph
Sparse graph with strong connectivity
examples of highly expanding graphs. Algebraic constructions based on Cayley graphs are known for various variants of expander graphs. The following construction
Expander_graph
Group with no large product-free subset
their connection to graph theory: bipartite Cayley graphs over any subset of a quasirandom group are always bipartite quasirandom graphs. The notion of quasirandom
Quasirandom_group
more general classes of algebraic groups G, is that the sequence of Cayley graphs for reductions Γp modulo prime numbers p, with respect to any fixed
Superstrong_approximation
Number of spanning trees of a complete graph
In mathematics, Cayley's formula is a result in graph theory named after Arthur Cayley. It states that for every positive integer n {\displaystyle n}
Cayley's_formula
engineering Cayley graph Cayley numbers Cayley plane Cayley table Cayley transform Cayleyan Cayley–Bacharach theorem Cayley–Dickson construction Cayley–Hamilton
List of things named after Arthur Cayley
List_of_things_named_after_Arthur_Cayley
Undirected graph acted on by a vertex-transitive cyclic group of symmetries
symmetry of the drawing. The graph is a Cayley graph of a cyclic group. Every cycle graph is a circulant graph, as is every crown graph with number of vertices
Circulant_graph
Vertices connected in pairs by edges
regular examples of directed graphs are given by the Cayley graphs of finitely-generated groups, as well as Schreier coset graphs In category theory, every
Graph_(discrete_mathematics)
off Cayley graph Γ ^ ( G , H ) {\displaystyle {\hat {\Gamma }}(G,H)} as follows: For each left coset gH, add a vertex v(gH) to the Cayley graph Γ(G)
Relatively_hyperbolic_group
Mathematical concept
be its Cayley graph with respect to some finite set S {\displaystyle S} of generators. The set X {\displaystyle X} is endowed with its graph metric (in
Hyperbolic_group
finite connected vertex-transitive graph and every finite Cayley graph is Hamiltonian. Semi-symmetric graph, graphs that have symmetries between every
Zero-symmetric_graph
Branch of mathematics that studies the properties of groups
objects a group acts on. The first idea is made precise by means of the Cayley graph, whose vertices correspond to group elements and edges correspond to
Group_theory
Theorem in group theory
{\displaystyle G} and let Γ ( G , S ) {\displaystyle \Gamma (G,S)} be the Cayley graph of G {\displaystyle G} with respect to S {\displaystyle S} . The number
Stallings theorem about ends of groups
Stallings_theorem_about_ends_of_groups
Graph in which all ordered pairs of linked nodes are automorphic
In the mathematical field of graph theory, a graph G is symmetric or arc-transitive if, given any two ordered pairs of adjacent vertices ( u 1 , v 1 )
Symmetric_graph
{\displaystyle G} acts properly discontinuously and cocompactly (for instance its Cayley graph). This is well-defined as a topological space by the invariance under
Gromov_boundary
Graph where each vertex has the same number of neighbors
In graph theory, a regular graph is a graph where each vertex has the same number of neighbors; i.e. every vertex has the same degree or valency. A regular
Regular_graph
Construction in combinatorial group theory
a pointed graph. The Cayley graph of the group G itself is the Schreier coset graph for H = {1G}. A spanning tree of a Schreier coset graph corresponds
Schreier_coset_graph
Trail in a graph that visits each edge once
for instance, the infinite Cayley graph shown, with all vertex degrees equal to four, has no Eulerian line. The infinite graphs that contain Eulerian lines
Eulerian_path
Representation of molecules in terms of graph theory
Euclidean graphs, in particular, crystals as periodic graphs. Arthur Cayley was probably the first to publish results that consider molecular graphs as early
Molecular_graph
{\displaystyle \Gamma =\operatorname {Cay} (G,S)} be the Cayley graph (or directed Cayley graph) corresponding to a generating subset S {\displaystyle S}
Babai's_problem
Type of topological space
universal cover is an infinite tree, which can be identified with the Cayley graph of the free group. (This is a special case of the presentation complex
Rose_(topology)
universal cover of the presentation complex is a Cayley complex for G, whose 1-skeleton is the Cayley graph of G. Any presentation complex for G is the 2-skeleton
Presentation_complex
Undirected graph named after S. S. Shrikhande
or not the pair of nodes is connected. The Shrikhande graph can be constructed as a Cayley graph. The vertex set is Z 4 × Z 4 {\displaystyle \mathbb {Z}
Shrikhande_graph
Undirected cubic graph derived from a hypercube graph
the Cayley graph of a group that acts on binary words of length n by rotation and flipping bits of the word. The generators used to form this Cayley graph
Cube-connected_cycles
Topics referred to by the same term
form a regular language Sofic group, a group whose Cayley graph is an initially subamenable graph With other meanings, it may also refer to: Vedran Sofic
Sofic
In graph theory, the Holt graph or Doyle graph is the smallest half-transitive graph, that is, the smallest example of a vertex-transitive and edge-transitive
Holt_graph
On the number of spanning trees in a graph
Kirchhoff's theorem is a generalization of Cayley's formula which provides the number of spanning trees in a complete graph. The theorem is named after the German
Kirchhoff's_theorem
equipped with several finite-state automata. These automata represent the Cayley graph of the group. That is, they can tell whether a given word representation
Automatic_group
Branch of mathematics
Perelman geometrization with cubulation techniques. Group actions on their Cayley graphs are foundational examples of isometric group actions. Other major topics
Geometry
Non-commutative group with 6 elements
this group. We can then summarize the group operations in the form of a Cayley table: Note that non-equal non-identity elements only commute if they are
Dihedral_group_of_order_6
Graph representing edges of another graph
generate families of graphs that (like the Petersen graph) are vertex-transitive but are not Cayley graphs: if G is an edge-transitive graph that has at least
Line_graph
{\displaystyle H(2,3)} . The Bruhat graph is the edge graph of the permutahedron. More generally, the Cayley graph of a finite Coxeter group (with the
Graph_of_a_polytope
topological spaces associated with the graph. Ends of graphs may be used (via Cayley graphs) to define ends of finitely generated groups. Finitely generated infinite
End_(graph_theory)
Family of cubic graphs formed from regular and star polygons
k {\displaystyle k} is odd. G ( n , k ) {\displaystyle G(n,k)} is a Cayley graph if and only if k 2 ≡ 1 ( m o d n ) {\displaystyle k^{2}\equiv 1\
Generalized_Petersen_graph
of odd order into odd cycles. Their proof relies on Cayley graphs, in particular, circulant graphs, and many of their decompositions come from the action
Cycle decomposition (graph theory)
Cycle_decomposition_(graph_theory)
Concept in group theory
group of order s, the Cayley graph for a generating set with one generator is an s-vertex cycle graph. The diameter of this graph, and of the group, is
Diameter_(group_theory)
Area in mathematics devoted to the study of finitely generated groups
objects. This is usually done by studying the Cayley graphs of groups, which, in addition to the graph structure, are endowed with the structure of a
Geometric_group_theory
Tiling of the hyperbolic plane
by Tōsaku Mizuhashi, Phillip Hagar Smith, and Amiel R. Volpert. The Cayley graph of the Baumslag–Solitar group B S ( 1 , 2 ) {\displaystyle BS(1,2)}
Binary_tiling
Function between two metric spaces that only respects their large-scale geometry
finitely generated group G, we can form the corresponding Cayley graph of S and G. This graph becomes a metric space if we declare the length of each edge
Quasi-isometry
very closely related to the Cayley graph of G: the word metric measures the length of the shortest path in the Cayley graph between two elements of G.
Word_metric
Square matrix used to represent a graph or network
In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether
Adjacency_matrix
Archimedean solid with 14 faces
labeling, the edges and vertices of the truncated octahedron form the Cayley graph of the symmetric group S 4 {\displaystyle S_{4}} , the group of four-element
Truncated_octahedron
Graph property
In the mathematical field of graph theory, a distance-regular graph is a regular graph such that for any two vertices v and w, the number of vertices
Distance-regular_graph
Undirected graph with no non-trivial symmetries
In graph theory, a branch of mathematics, an undirected graph is called an asymmetric graph if it has no nontrivial symmetries. Formally, an automorphism
Asymmetric_graph
Group that admits a formal description in terms of reflections
group element; this is precisely the length in the word metric in the Cayley graph. An expression for v using ℓ(v) generators is a reduced word. For example
Coxeter_group
Method for producing composition algebras
In mathematics, the Cayley–Dickson construction, sometimes also known as the Cayley–Dickson process or the Cayley–Dickson procedure produces a sequence
Cayley–Dickson_construction
In graph-theoretic mathematics, a biregular graph or semiregular bipartite graph is a bipartite graph G = ( U , V , E ) {\displaystyle G=(U,V,E)} for which
Biregular_graph
Methodic assignment of colors to elements of a graph
In graph theory, graph coloring is a methodic assignment of labels traditionally called "colors" to elements of a graph. The assignment is subject to certain
Graph_coloring
Concept in mathematics
quasi-isometric to the Euclidean plane). It is the Cayley graph of the fundamental group of the torus; the Cayley graphs of the fundamental groups of a surface of
Hyperbolic_metric_space
Branch of discrete mathematics
provided examples of what is now known as Hamiltonian cycles in certain Cayley graphs on permutations. During the Renaissance, together with the rest of mathematics
Combinatorics
turns is 26. These numbers are also the diameters of the corresponding Cayley graphs of the Rubik's Cube group. In STM (slice turn metric, which counts turning
Optimal solutions for the Rubik's Cube
Optimal_solutions_for_the_Rubik's_Cube
said to be thin if there is a finite upper bound on the girth of the Cayley graph induced by any finite generating set. The group is called fat if it is
Thin group (combinatorial group theory)
Thin_group_(combinatorial_group_theory)
Theorem in set-theoretic geometry
proved that most of the classical paradoxes are an easy consequence of a graph theoretical result and the fact that the groups in question are rich enough
Banach–Tarski_paradox
Theorem in algebra
language W ( G , X ) {\displaystyle {\mathcal {W}}(G,X)} , that the Cayley graph Γ ( G , X ) {\displaystyle \Gamma (G,X)} of G with respect to X is K-triangulable
Muller–Schupp_theorem
Mathematical space with a notion of distance
distance. In geometric group theory this construction is applied to the Cayley graph of a (typically infinite) finitely-generated group, yielding the word
Metric_space
Combinatorial algorithm
defined by its vertex coordinates, the resulting labeling describes a Cayley graph of the symmetric group of permutations on n {\displaystyle n} items,
Steinhaus–Johnson–Trotter algorithm
Steinhaus–Johnson–Trotter_algorithm
String in combinatorial math
construction by Aaron Williams for constructing Hamiltonian paths through the Cayley graph of the symmetric group, science fiction author and mathematician Greg
Superpermutation
{\displaystyle x} and y {\displaystyle y} is the length of shortest path in the Cayley graph (which is circular since the group is cyclic) between them. More generally
Lee_distance
Largest distance between two points
arbitrary graphs and in special classes of graphs. Special cases of graph diameter include the diameter of a group, defined using a Cayley graph with the
Diameter_of_a_set
Graph where any two nodes of equal distance are isomorphic
In the mathematical field of graph theory, a distance-transitive graph is a graph such that, given any two vertices v and w at any distance i, and any
Distance-transitive_graph
Isomorphism of an object to itself
{\displaystyle \mathbb {O} } ) is the exceptional Lie group G2. In graph theory an automorphism of a graph is a permutation of the nodes that preserves edges and
Automorphism
Group of transformations under which the object is invariant
graph: a graph symmetry is a permutation of the vertices which takes edges to edges. Any finitely presented group is the symmetry group of its Cayley
Symmetry_group
Process forming a path from many random steps
Laplace's equation. A significant portion of this research was focused on Cayley graphs of finitely generated groups. In many cases these discrete results carry
Random_walk
Term in group theory mathematics
factor group" J.Lond.Math.Soc.26 (1951), pp. 59–61. John Meier, "Groups, Graphs and Trees", Cambridge University Press, (2008), page 100. D.J. Collins (2001)
Baumslag–Solitar_group
Length of a shortest cycle contained in the graph
coloring. Explicit, though large, graphs with high girth and chromatic number can be constructed as certain Cayley graphs of linear groups over finite fields
Girth_(graph_theory)
Mathematical group based upon a finite number of elements
many subgroups of a given order are contained in G. Cayley's theorem, named in honour of Arthur Cayley, states that every group G is isomorphic to a subgroup
Finite_group
Graph where all pairs of edges are automorphic
In the mathematical field of graph theory, an edge-transitive graph is a graph G such that, given any two edges e1 and e2 of G, there is an automorphism
Edge-transitive_graph
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