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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
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
r-regular graph is a graph selected from G n , r {\displaystyle {\mathcal {G}}_{n,r}} , which denotes the probability space of all r-regular graphs on
Random_regular_graph
4-regular undirected graph in mathematics
In the mathematical field of graph theory, the Robertson graph or (4,5)-cage, is a 4-regular undirected graph with 19 vertices and 38 edges named after
Robertson_graph
Cubic graph with 10 vertices and 15 edges
bridgeless graph has a cycle-continuous mapping to the Petersen graph. More unsolved problems in mathematics In the mathematical field of graph theory, the
Petersen_graph
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
Decomposition of a graph into hamiltonion cycles
decomposition to exist in an undirected graph, the graph must be connected and regular of even degree. A directed graph with such a decomposition must be strongly
Hamiltonian_decomposition
Graph with nodes connected in a closed chain
In graph theory, a cycle graph or circular graph is a graph that consists of a single cycle, or in other words, some number of vertices (at least 3, if
Cycle_graph
Set of unordered triples from a vertex set
the two-graph. A regular two-graph has the property that every pair of vertices lies in the same number of triples of the two-graph. Two-graphs have been
Two-graph
Vertices connected in pairs by edges
In discrete mathematics, particularly in graph theory, a graph is a structure consisting of a set of objects where some pairs of the objects are in some
Graph_(discrete_mathematics)
Graph of chess rook moves
In graph theory, a rook's graph is an undirected graph that represents all legal moves of the rook chess piece on a chessboard. Each vertex of a rook's
Rook's_graph
Graph of numbers differing by a square
Paley graphs form an infinite family of conference graphs, which yield an infinite family of symmetric conference matrices. Paley graphs allow graph-theoretic
Paley_graph
Symmetric tessellation of a closed surface
lines. Topological graph theory Abstract polytope Planar graph Toroidal graph Graph embedding Regular tiling Platonic solid Platonic graph Nedela (2007) Coxeter
Regular_map_(graph_theory)
On existence of a strongly regular graph
exist a strongly regular graph with parameters (99,14,1,2)? More unsolved problems in mathematics In graph theory, Conway's 99-graph problem is an unsolved
Conway's_99-graph_problem
Appendix:Glossary of graph theory in Wiktionary, the free dictionary. This is a glossary of graph theory. Graph theory is the study of graphs, systems of nodes
Glossary_of_graph_theory
Regular graph with girth more than twice its diameter
Does a Moore graph with girth 5 and degree 57 exist? More unsolved problems in mathematics In graph theory, a Moore graph is a regular graph whose girth
Moore_graph
Mathematical Graph
In graph theory, a walk-regular graph is a simple graph where the number of closed walks of any length ℓ {\displaystyle \ell } from a vertex to itself
Walk-regular_graph
Area of discrete mathematics
computer science, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context
Graph_theory
Sylvester graph Tutte's fragment Tutte graph Young–Fibonacci graph Wagner graph Wells graph Wiener–Araya graph Windmill graph The strongly regular graph on v
List_of_graphs
Solid with eight equal triangular faces
edges of a regular octahedron give rise to a graph, a discrete structure drawn in a plane. The name is octahedral graph. The octahedral graph is an example
Regular_octahedron
One of two different regular graphs with 16 vertices
field of graph theory, the Clebsch graph is either of two complementary graphs on 16 vertices, a 5-regular graph with 40 edges and a 10-regular graph with
Clebsch_graph
Solid with 12 equal pentagonal faces
skeleton of a regular dodecahedron can be represented as a graph, and it is called the dodecahedral graph, a Platonic graph. This graph can also be constructed
Regular_dodecahedron
Bipartite non-Hamiltonian polyhedral graph
In graph theory, a branch of mathematics, the Herschel graph is a bipartite undirected graph with 11 vertices and 18 edges. It is a polyhedral graph (the
Herschel_graph
Graph whose embedding in a Euclidean space forms a regular tiling
In graph theory, a lattice graph, mesh graph, or grid graph is a graph whose drawing, embedded in some Euclidean space R n {\displaystyle \mathbb {R}
Lattice_graph
field of graph theory, the Brouwer–Haemers graph is a 20-regular undirected graph with 81 vertices and 810 edges. It is a strongly regular graph, a distance-transitive
Brouwer–Haemers_graph
Spectral graph theory concept
spectral graph theory, a Ramanujan graph is a regular graph whose spectral gap is almost as large as possible (see extremal graph theory). Such graphs are
Ramanujan_graph
Partition of a graph into spanning subgraphs
1-factorable then it has to be a regular graph. However, not all regular graphs are 1-factorable. A k-regular graph is 1-factorable if it has chromatic
Graph_factorization
Graph where every edge is in one triangle
Examples of locally linear graphs include the triangular cactus graphs, the line graphs of 3-regular triangle-free graphs, and the Cartesian products
Locally_linear_graph
Regular graph with fewest possible nodes for its girth
of graph theory, a cage is a regular graph that has as few vertices as possible for its girth. Formally, an (r, g)-graph is defined to be a graph in which
Cage_(graph_theory)
Linear algebra aspects of graph theory
In mathematics, spectral graph theory is the study of the properties of a graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors
Spectral_graph_theory
Order-zero graph or any edgeless graph
has no edges. Thus the null graph is a regular graph of degree zero. Some authors exclude K0 from consideration as a graph (either by definition, or more
Null_graph
In the mathematical field of graph theory, the Brinkmann graph is a 4-regular graph with 21 vertices and 42 edges discovered by Gunnar Brinkmann in 1992
Brinkmann_graph
Sparse graph with strong connectivity
In graph theory, an expander graph is a sparse graph that has strong connectivity properties, quantified using vertex, edge or spectral expansion. Expander
Expander_graph
Branch of mathematics
of graphs based on symmetry (such as symmetric graphs, vertex-transitive graphs, edge-transitive graphs, distance-transitive graphs, distance-regular graphs
Algebraic_graph_theory
Class of undirected graphs defined from systems of sets
mathematics, Johnson graphs are a special class of undirected graphs defined from systems of sets. The vertices of the Johnson graph J ( n , k ) {\displaystyle
Johnson_graph
Distance-regular graph with 56 vertices
The Gosset graph, named after Thorold Gosset, is a distance-regular graph with 56 vertices and valency 27. It is the 1-skeleton of the 7-dimensional 321
Gosset_graph
Graph in which every two vertices are adjacent
Kuratowski to graph theory. Kn has n(n − 1)/2 edges (a triangular number), and is a regular graph of degree n − 1. All complete graphs are their own maximal
Complete_graph
Undirected graph named after S. S. Shrikhande
mathematical field of graph theory, the Shrikhande graph is a graph discovered by S. S. Shrikhande in 1959. It is a strongly regular graph with 16 vertices
Shrikhande_graph
Writing paper with a grid
Graph paper, coordinate paper, grid paper, or squared paper is writing paper that is printed with fine lines making up a regular grid. It is available
Graph_paper
Graph with an antiprism as its skeleton
with regular-polygon bases include the prism graphs (graphs of prisms) and wheel graphs (graphs of pyramids). Other vertex-transitive polyhedral graphs include
Antiprism_graph
Binary operation combining the vertex and edge sets of two graphs
cluster graphs are the disjoint unions of complete graphs. The 2-regular graphs are the disjoint unions of cycle graphs. More generally, every graph is the
Disjoint_union_of_graphs
Number of edges touching a vertex in a graph
degree is 0. In a regular graph, every vertex has the same degree, and so we can speak of the degree of the graph. A complete graph (denoted K n {\displaystyle
Degree_(graph_theory)
The Sylvester graph is the unique distance-regular graph with intersection array { 5 , 4 , 2 ; 1 , 1 , 4 } {\displaystyle \{5,4,2;1,1,4\}} . It is a subgraph
Sylvester_graph
Graph representing edges of another graph
In the mathematical discipline of graph theory, the line graph of an undirected graph G is another graph L(G) that represents the adjacencies between edges
Line_graph
Balanced complete multipartite graph
The Turán graph, denoted by T ( n , r ) {\displaystyle T(n,r)} , is a complete multipartite graph; it is formed by partitioning a set of n {\displaystyle
Turán_graph
Graph with an Archimedean solid as its skeleton
all of them are regular, polyhedral (and therefore by necessity also 3-vertex-connected planar graphs), and also Hamiltonian graphs. Along with the 13
Archimedean_graph
-minor-free graph is an apex graph Does a Moore graph with girth 5 and degree 57 exist? Do there exist infinitely many strongly regular geodetic graphs, or any
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Graph with a prism as its skeleton
vertex-transitive graphs, the prism graphs may also be constructed as Cayley graphs. The order-n dihedral group is the group of symmetries of a regular n-gon in
Prism_graph
Cubic distance-regular graph with 102 nodes and 153 edges
In the mathematical field of graph theory, the Biggs–Smith graph is a 3-regular graph with 102 vertices and 153 edges. It has chromatic number 3, chromatic
Biggs–Smith_graph
Two special graphs in graph theory
In the mathematical field of graph theory, the Klein graphs are two different but related regular graphs, each with 84 edges. Each can be embedded in
Klein_graphs
triangle-free, 4-regular, and 4-chromatic. The Chvátal graph is triangle-free: its girth (the length of its shortest cycle) is four. It is 4-regular: each vertex
Chvátal_graph
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
7-regular undirected graph with 50 nodes and 175 edges
of graph theory, the Hoffman–Singleton graph is a 7-regular undirected graph with 50 vertices and 175 edges. It is the unique strongly regular graph with
Hoffman–Singleton_graph
Graph with all vertices of degree 3
of graph theory, a cubic graph is a graph in which all vertices have degree three. In other words, a cubic graph is a 3-regular graph. Cubic graphs are
Cubic_graph
In graph theory, the Games graph is the largest known locally linear strongly regular graph. Its parameters as a strongly regular graph are (729,112,1
Games_graph
Graph where all pairs of vertices are automorphic
regular graphs are vertex-transitive (for example, the Frucht graph and Tietze's graph). Finite vertex-transitive graphs include the symmetric graphs
Vertex-transitive_graph
The line graph of a regular integral graph is again integral. For instance, as the line graph of K 4 {\displaystyle K_{4}} , the octahedral graph is integral
Integral_graph
6-regular graph with 57 vertices and 171 edges
the Perkel graph, named after Manley Perkel, is a 6-regular graph with 57 vertices and 171 edges. It is the unique distance-regular graph with intersection
Perkel_graph
Mathematical finite graph-associated function
reinterpreted graph-theoretically. It was Toshikazu Sunada who put this suggestion into practice in 1985. As observed by Sunada, a regular graph is a Ramanujan
Ihara_zeta_function
Family of cubic graphs formed from regular and star polygons
In graph theory, the generalized Petersen graphs are a family of cubic graphs formed by connecting the vertices of a regular polygon to the corresponding
Generalized_Petersen_graph
mathematical field of graph theory, the McLaughlin graph is a strongly regular graph with parameters (275, 112, 30, 56) and is the only such graph. The group theorist
McLaughlin_graph
field of graph theory, the Chang graphs are three 12-regular undirected graphs, each with 28 vertices and 168 edges. They are strongly regular, with the
Chang_graphs
Cubic graph with 28 vertices and 42 edges
field of graph theory, the Coxeter graph is a 3-regular graph with 28 vertices and 42 edges. It is one of the 13 known cubic distance-regular graphs. It is
Coxeter_graph
Special case of a strongly regular graph
of graph theory, a conference graph is a strongly regular graph with parameters v, k = (v − 1)/2, λ = (v − 5)/4, and μ = (v − 1)/4. It is the graph associated
Conference_graph
16-regular graph with 27 vertices and 216 edges
the mathematical field of graph theory, the Schläfli graph, named after Ludwig Schläfli, is a 16-regular undirected graph with 27 vertices and 216 edges
Schläfli_graph
On graphs with given symmetry groups
of the graph of the lattice, a median graph. It is possible to realize every finite group as the group of symmetries of a strongly regular graph. Every
Frucht's_theorem
Undirected graph with 14 vertices
mathematical field of graph theory, the Heawood graph is an undirected graph with 14 vertices and 21 edges, named after Percy John Heawood. The graph is cubic, and
Heawood_graph
Unsolved problem in computational complexity theory
bipartite Eulerian graphs bipartite regular graphs line graphs split graphs chordal graphs regular self-complementary graphs polytopal graphs of general, simple
Graph_isomorphism_problem
Points with no three in a line
The Games graph is a strongly regular graph with 729 vertices. Every edge belongs to a unique triangle, so it is a locally linear graph, the largest
Cap_set
Mathematical graph of a Sudoku
and is 7-regular. For the most common form of Sudoku, on a 9 × 9 {\displaystyle 9\times 9} board, the Sudoku graph is a 20-regular graph with 81 vertices
Sudoku_graph
Every graph has evenly many odd vertices
In graph theory, the handshaking lemma is the statement that, in every finite undirected graph, the number of vertices that touch an odd number of edges
Handshaking_lemma
Graph whose vertices correspond to combinations of a set of n elements
The Kneser graph is vertex transitive and arc transitive. When k = 2 {\displaystyle k=2} , the Kneser graph is a strongly regular graph, with parameters
Kneser_graph
Graph generated by a random process
In mathematics, random graph is the general term to refer to probability distributions over graphs. Random graphs may be described simply by a probability
Random_graph
Graph with all vertices of degree 4
mathematical field of graph theory, a quartic graph is a graph where all vertices have degree 4. In other words, a quartic graph is a 4-regular graph. Several well-known
Quartic_graph
Gewirtz graph is a strongly regular graph with 56 vertices and valency 10. It is named after the mathematician Allan Gewirtz, who described the graph in his
Gewirtz_graph
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
Graphs formed by a hypercube's edges and vertices
{\displaystyle 2^{n-1}n} edges, and is a regular graph with n {\displaystyle n} edges touching each vertex. The hypercube graph Q n {\displaystyle Q_{n}} may also
Hypercube_graph
mathematical graph theory, the Higman–Sims graph is a 22-regular undirected graph with 100 vertices and 1100 edges. It is the unique strongly regular graph srg(100
Higman–Sims_graph
Bivariegated graph Cage (graph theory) Cayley graph Circle graph Clique graph Cograph Common graph Complement of a graph Complete graph Cubic graph Cycle graph De
List_of_graph_theory_topics
3-regular graph with 30 vertices and 45 edges
mathematical field of graph theory, the Tutte–Coxeter graph or Tutte eight-cage or Cremona–Richmond graph is a 3-regular graph with 30 vertices and 45
Tutte–Coxeter_graph
Topics referred to by the same term
Regular graph, a graph such that all the degrees of the vertices are equal Szemerédi regularity lemma, some random behaviors in large graphs Regular language
Regular
Bipartite 3-regular graph with 90 vertices and 135 edges
mathematical field of graph theory, the Foster graph is a bipartite 3-regular graph with 90 vertices and 135 edges. The Foster graph is Hamiltonian and has
Foster_graph
Cartesian product of complete graphs
complete graphs that may be of varying sizes. Unlike the Hamming graphs H(d,q), the graphs in this more general class are not necessarily distance-regular, but
Hamming_graph
Bipartite 4-regular graph with 20 nodes and 40 edges
mathematical field of graph theory, the Folkman graph is a 4-regular graph with 20 vertices and 40 edges. It is a regular bipartite graph with symmetries taking
Folkman_graph
Graph with edges of length one, able to be drawn without crossings
unit-distance graphs but are not matchstick graphs. An example is the Dürer graph. Much of the research on matchstick graphs has concerned regular graphs, in which
Matchstick_graph
Concept in graph theory
problem of obtaining the diameter of the pancake graph. The pancake graph of dimension n, Pn, is a regular graph with n ! {\displaystyle n!} vertices. Its degree
Pancake_graph
Cubic graph with 8 vertices and 12 edges
mathematical field of graph theory, the Wagner graph is a 3-regular graph with 8 vertices and 12 edges. It is the 8-vertex Möbius ladder graph. As a Möbius ladder
Wagner_graph
Constructs with triply-connected vertices
The connected 3-regular (cubic) simple graphs are listed for small vertex numbers. The number of connected simple cubic graphs on 4, 6, 8, 10, ... vertices
Table_of_simple_cubic_graphs
The Suzuki graph is a strongly regular graph with parameters ( 1782 , 416 , 100 , 96 ) {\displaystyle (1782,416,100,96)} . Its automorphism group has
Suzuki_graph
Undirected graph
In graph theory, a critical graph is an undirected graph all of whose proper subgraphs have smaller chromatic number. In such a graph, every vertex or
Critical_graph
Undirected graph acted on by a vertex-transitive cyclic group of symmetries
In graph theory, a circulant graph is an undirected graph acted on by a cyclic group of symmetries which takes any vertex to any other vertex. It is sometimes
Circulant_graph
Computer science algorithm
computer science, graph traversal (also known as graph search) refers to the process of visiting (checking and/or updating) each vertex in a graph. Such traversals
Graph_traversal
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
Strongly regular graph
The Cameron graph is a strongly regular graph of parameters ( 231 , 30 , 9 , 3 ) {\displaystyle (231,30,9,3)} . This means that it has 231 vertices, 30
Cameron_graph
3-regular graph with no 3-edge-coloring
In the mathematical field of graph theory, a snark is an undirected graph with exactly three edges per vertex whose edges cannot be colored with only three
Snark_(graph_theory)
In graph theory, the Berlekamp–Van Lint–Seidel graph is a locally linear strongly regular graph with parameters ( 243 , 22 , 1 , 2 ) {\displaystyle (243
Berlekamp–Van Lint–Seidel graph
Berlekamp–Van_Lint–Seidel_graph
Assignment of colors to edges of a graph
In graph theory, a proper edge coloring of a graph is an assignment of "colors" to the edges of the graph so that no two incident edges have the same color
Edge_coloring
Theorem in graph theory
graph theory. It can be stated as follows: Let G {\displaystyle G} be a regular graph whose degree is an even number, 2 k {\displaystyle 2k} . Then the edges
2-factor_theorem
In the mathematical field of graph theory, the Dyck graph is a 3-regular graph with 32 vertices and 48 edges, named after Walther von Dyck. It is Hamiltonian
Dyck_graph
Distance-transitive cubic graph with 20 nodes and 30 edges
In the mathematical field of graph theory, the Desargues graph is a distance-transitive, cubic graph with 20 vertices and 30 edges. It is named after
Desargues_graph
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