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Graph property
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 at distance j from v
Distance-regular_graph
Graph where any two nodes of equal distance are isomorphic
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 other two
Distance-transitive_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
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
Geometric graph with unit edge lengths
In mathematics, particularly geometric graph theory, a unit distance graph is a graph formed from a collection of points in the Euclidean plane by connecting
Unit_distance_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
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
Class of undirected graphs defined from systems of sets
distinct distance-regular graph; the intersection array of J ( 8 , 2 ) {\displaystyle J(8,2)} is shared with three other distance-regular graphs that are
Johnson_graph
Graph in which all ordered pairs of linked nodes are automorphic
Biggs–Smith graph. The ten distance-transitive graphs listed above, together with the Foster graph and the Biggs–Smith graph, are the only cubic distance-transitive
Symmetric_graph
Undirected graph with 14 vertices
It is a distance-transitive graph (see the Foster census) and therefore distance regular. There are 24 perfect matchings in the Heawood graph; for each
Heawood_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
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
Cubic graph with 10 vertices and 15 edges
Petersen graph can be transformed into every other such path by a symmetry of the graph. It is one of only 13 cubic distance-regular graphs. The automorphism
Petersen_graph
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
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
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
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
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
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
Vertices connected in pairs by edges
arc-transitive, and distance-transitive graphs; strongly regular graphs and their generalizations distance-regular graphs. Two vertices of a graph are called adjacent
Graph_(discrete_mathematics)
Cubic distance-regular graph with 102 nodes and 153 edges
3-vertex-connected graph and a 3-edge-connected graph. All the cubic distance-regular graphs are known. The Biggs–Smith graph is one of the 13 such graphs. The automorphism
Biggs–Smith_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
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
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
Bipartite, 3-regular undirected graph
configuration. All the cubic, distance-regular graphs are known; the Pappus graph is one of the 13 such graphs. The Pappus graph has rectilinear crossing number
Pappus_graph
Graph of the vertices and edges of a demihypercube
demihypercube, formed by connecting pairs of vertices at distance exactly two from each other in the hypercube graph. That is, it is the half-square of the hypercube
Halved_cube_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
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
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
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
Linear algebra aspects of graph theory
goes to 1. A pair of regular graphs are cospectral if and only if their complements are cospectral. A pair of distance-regular graphs are cospectral if and
Spectral_graph_theory
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
Longest distance between two vertices
In graph theory, the diameter of a connected undirected graph is the farthest distance between any two of its vertices. That is, it is the diameter of
Diameter_(graph_theory)
Graph of chess rook moves
distance in moves (making the graph distance-transitive). For rectangular chessboards whose width and height are relatively prime, the rook's graphs are
Rook's_graph
Bruijn graph Dense graph Dipole graph Directed acyclic graph Directed graph Distance regular graph Distance-transitive graph Edge-transitive graph Interval
List_of_graph_theory_topics
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
Area of discrete mathematics
vertex-transitive graphs, edge-transitive graphs, distance-transitive graphs, distance-regular graphs, and strongly regular graphs. Frucht's theorem states that every
Graph_theory
Bipartite 3-regular graph with 90 vertices and 135 edges
cubic symmetric graphs included this graph. The bipartite half of the Foster graph is a distance-regular graph and a locally linear graph. It is one of
Foster_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,
Matchstick_graph
The Wells graph is the unique distance-regular graph with intersection array ( 5 , 4 , 1 , 1 ; 1 , 1 , 4 , 5 ) . {\displaystyle (5,4,1,1;1,1,4,5).} Its
Wells_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
Undirected graph derived from a hypercube graph
folded cube graphs provide a class of triangle-free graphs with chromatic number four and arbitrarily large odd girth. As a distance-regular graph with odd
Folded_cube_graph
Solid with 12 equal pentagonal faces
polygon is replicated in the properties of this graph, which are distance-transitive, distance-regular, and symmetric. The automorphism group has order
Regular_dodecahedron
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
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
Square matrix used to represent a graph or network
studying strongly regular graphs and two-graphs. The distance matrix has in position (i, j) the distance between vertices vi and vj. The distance is the length
Adjacency_matrix
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
Family of symmetric graphs which generalize the Petersen graph
{\displaystyle O_{3}} is the familiar Petersen graph. The generalized odd graphs are defined as distance-regular graphs with diameter n − 1 {\displaystyle n-1}
Odd_graph
Strongly regular graph
Nina; Švob, Andrea (2021), Distance-regular graphs obtained from the Mathieu groups, arXiv:2101.02790 A.E. Brouwer's website: the Cameron graph v t e
Cameron_graph
Geometry theorem
the theorem itself. The Levi graph of the Pappus configuration is the Pappus graph, a bipartite distance-regular graph with 18 vertices and 27 edges
Pappus's_hexagon_theorem
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
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
Class of simple graphs defined from vector spaces
{n}{j-1}}_{q}\right)}} . Grassmannian Johnson graph Brouwer, Andries E. (1989). Distance-Regular Graphs. Cohen, Arjeh M., Neumaier, Arnold. Berlin, Heidelberg:
Grassmann_graph
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
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
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
Length of a shortest cycle contained in the graph
supplement to the book Distance-Regular Graphs (Brouwer, Cohen, and Neumaier 1989, Springer-Verlag). Erdős, Paul (1959), "Graph theory and probability"
Girth_(graph_theory)
24-vertex symmetric bipartite cubic graph
In the mathematical field of graph theory, the Nauru graph is a symmetric, bipartite, cubic graph with 24 vertices and 36 edges. It was named by David
Nauru_graph
Family of graphs with 2n nodes and n(n-1) edges
(2003) show, crown graphs are one of a small number of different types of graphs that can occur as distance-regular circulant graphs. Agarwal et al. (1994)
Crown_graph
In the mathematical field of graph theory, the Hoffman graph is a 4-regular graph with 16 vertices and 32 edges discovered by Alan Hoffman. Published in
Hoffman_graph
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
Study of graphs defined by geometric means
Geometric graph theory in the broader sense is a large and amorphous subfield of graph theory, concerned with graphs defined by geometric means. In a stricter
Geometric_graph_theory
Symmetric bipartite cubic graph with 16 vertices and 24 edges
In the mathematical field of graph theory, the Möbius–Kantor graph is a symmetric bipartite cubic graph with 16 vertices and 24 edges named after August
Möbius–Kantor_graph
Mapping a graph onto itself without changing edge-vertex connectivity
the same distance apart. A semi-symmetric graph is a graph that is edge-transitive but not vertex-transitive. A half-transitive graph is a graph that is
Graph_automorphism
Graph that is edge-transitive and regular but not vertex-transitive
graph theory, a semi-symmetric graph is an undirected graph that is edge-transitive and regular, but not vertex-transitive. In other words, a graph is
Semi-symmetric_graph
Periodic spatial graph
at distance 2 {\displaystyle {\sqrt {2}}} . It can also be defined, divorced from its geometry, as an abstract undirected graph, a covering graph of the
Laves_graph
graph theory, entitled Algebraic Graph Theory, with Gordon Royle, His earlier textbook on algebraic combinatorics discussed distance-regular graphs and
Chris_Godsil
Arrangement of 30 points and 12 lines
Cohen, A. M.; Neumaier, A. (1989), "Chapter 1: Special Regular Graphs", Distance-regular graphs, Results in Mathematics and Related Areas, vol. 18, Berlin:
Schläfli_double_six
Type of graph in graph theory
of graph theory, a half-transitive graph is a graph that is both vertex-transitive and edge-transitive, but not symmetric. In other words, a graph is
Half-transitive_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
Triangle-free graph requiring four colors
In the mathematical field of graph theory, the Grötzsch graph is a triangle-free graph with 11 vertices, 20 edges, chromatic number 4, and crossing number
Grötzsch_graph
Concept in incidence geometry
d {\displaystyle 2d} -gons whose point graph (also known as a collinearity graph) is a distance-regular graph. A generalized 2 d {\displaystyle 2d} -gon
Near_polygon
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
Graph whose nodes are one of the vertex sets of a bipartite graph
of the hypercube graph is the halved cube graph. When G is a distance-regular graph, its two bipartite halves are both distance-regular. For instance, the
Bipartite_half
the arcs of the graph. Therefore, the F26A graph is a symmetric graph (though not distance transitive). It has automorphisms that take any vertex to any
F26A_graph
In the mathematical field of graph theory, the Livingstone graph is a distance-transitive graph with 266 vertices and 1463 edges. Its intersection array
Livingstone_graph
Solid with six equal square faces
drawing a graph with vertices connected with an edge in a plane. Such a graph is called the cubical graph, a special case of the hypercube graph. The cube
Cube
Undirected unit-distance graph requiring four colors
colors. The method of construction of the Golomb graph as a unit distance graph, by drawing an outer regular polygon connected to an inner twisted polygon
Golomb_graph
Matrix representation of a graph
In the mathematical field of graph theory, the Laplacian matrix, also called the graph Laplacian, admittance matrix, Kirchhoff matrix, or discrete Laplacian
Laplacian_matrix
Segment in a circle or sphere from its center to its perimeter or surface
modern usage, it is also their length. The radius of a regular polygon is the line segment or distance from its center to any of its vertices. The name comes
Radius
mathematical field of graph theory, the Gray graph is an undirected bipartite graph with 54 vertices and 81 edges. It is a cubic graph: every vertex touches
Gray_graph
Mathematical problem
plane so that no two points at unit distance are the same color? More unsolved problems in mathematics In geometric graph theory, the Hadwiger–Nelson problem
Hadwiger–Nelson_problem
-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
Type of graph in mathematics
In graph theory, a branch of mathematics, a half graph is a special type of bipartite graph. These graphs are called the half graphs because they have
Half_graph
Undirected, connected, and acyclic graph
In graph theory, a tree is an undirected graph in which every pair of distinct vertices is connected by exactly one path, or equivalently, a connected
Tree_(graph_theory)
Geometric model of the planar projection of the physical universe
Such a drawing is called a plane graph or planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to
Euclidean_plane
numerical properties of a distance regular graph. A.E. Brouwer, A.M. Cohen, and A. Neumaier (1989), Distance-Regular Graphs, Springer-Verlag, Berlin.
Equitable_partition
Solid with twenty equal triangular faces
icosahedral graph represents the skeleton of a regular icosahedron. Many polyhedra and other related figures are constructed from the regular icosahedron
Regular_icosahedron
Graph used in computational complexity theory and graph theory
specified even distance from each other. The graphs of this type are parameterized by the dimension of the hypercube and by the distance between adjacent
Frankl–Rödl_graph
Neumaier, A. Distance-Regular Graphs. Springer-Verlag, Berlin, 1989. Section 3.8. Godsil, Chris; Royle, Gordon (2001), Algebraic Graph Theory, Graduate
Equiangular_lines
Topic in algebraic graph theory
{\displaystyle K_{2}} . The only cubic distance-regular graph that admits perfect state transfer is the cubical graph. Farhi, Edward; Gutmann, Sam (1 August
Continuous-time_quantum_walk
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
{\displaystyle k(n-k)} -connected. It has graph diameter ⌊ 3 k / 2 ⌋ {\displaystyle \lfloor 3k/2\rfloor } and average distance H k + k ( k − 2 ) / n {\displaystyle
Arrangement_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
Dutch mathematician and computer programmer (born 1951)
Brouwer, Andries; Arjeh Cohen; Arnold Neumaier (August 1989). Distance Regular Graphs. Ergebnisse der Mathematik und ihrer Grenzgebiete 3.19. Springer-Verlag
Andries_Brouwer
In polytope theory, the edge graph (also known as vertex-edge graph or just graph) of a polytope is a combinatorial graph whose vertices and edges correspond
Graph_of_a_polytope
field of graph theory, the triangle graph is a planar undirected graph with 3 vertices and 3 edges, in the form of a triangle. The triangle graph is also
Triangle_graph
Graph with 24 vertices and 36 edges
mathematical field of graph theory, the McGee graph or the (3-7)-cage is a 3-regular graph with 24 vertices and 36 edges. The McGee graph is the unique (3
McGee_graph
Graph representing faces of another graph
mathematical discipline of graph theory, the dual graph of a planar graph G is a graph that has a vertex for each face of G. The dual graph has an edge for each
Dual_graph
Cycle graph with all opposite nodes linked
are an even distance apart in the initial cycle, so adding each rung creates an odd cycle. In this case, because the graph is 3-regular but not bipartite
Möbius_ladder
DISTANCE REGULAR-GRAPH
DISTANCE REGULAR-GRAPH
Boy/Male
Indian, Modern
Full of Light
Boy/Male
Arabic
Distance
Girl/Female
American, British, English, French, Greek
Fate; Certain Fortune; The Mythological Greek God of Fate
Boy/Male
Indian
Distance
Female
French
French form of Latin Constantia, CUSTANCE means "steadfast."Â
Girl/Female
Arabic, Muslim, Sindhi
Some Distance
Girl/Female
Muslim
Some distance
Girl/Female
Muslim
Some distance
Girl/Female
Indian
Some distance
Girl/Female
Arabic, Muslim, Sindhi
Some Distance
Girl/Female
Muslim
Some distance
Girl/Female
English French
Certain fortune; fate. The mythological Greek god of fate.
Boy/Male
Hindu, Indian, Tamil
Regular Winner
Girl/Female
Muslim/Islamic
Some distance
Girl/Female
Muslim/Islamic
Some distance
Boy/Male
Gujarati, Haryanvi, Hindu, Indian, Kannada, Marathi, Telugu
Regular; Ethical; Good in Nature
Girl/Female
Muslim
Some distance
Girl/Female
Hebrew
Precious.
Girl/Female
Indian
Some distance
Boy/Male
Hindu, Indian, Traditional
Conduct; Regular Performance of Worship
DISTANCE REGULAR-GRAPH
DISTANCE REGULAR-GRAPH
Boy/Male
Indian, Sanskrit
With a Sound Mind
Boy/Male
Indian, Malay, Nigerian, Tamil
Young
Biblical
exaltation of help
Girl/Female
Muslim/Islamic
Purity
Boy/Male
Hebrew
Steady; strong.
Boy/Male
Arabic, Muslim
Decorated Throne
Girl/Female
Arabic, Farsi, Iranian, Muslim, Parsi
Heaven; Paradise
Girl/Female
Indian
Celestial maiden, Divine damsel
Surname or Lastname
English
English : from Middle English keech ‘lump’, ‘fat’, hence an unflattering nickname for a fat, lumpish person.
Girl/Female
Hindu
Wife of Lord Shiva, Goddess Parvati (Wife of Lord Shiva)
DISTANCE REGULAR-GRAPH
DISTANCE REGULAR-GRAPH
DISTANCE REGULAR-GRAPH
DISTANCE REGULAR-GRAPH
DISTANCE REGULAR-GRAPH
a.
Not regular; not bound by monastic vows or rules; not confined to a monastery, or subject to the rules of a religious community; as, a secular priest.
a.
Indistinct; faint; obscure, as from distance.
a.
Not regular; not conforming to a law, method, or usage recognized as the general rule; not according to common form; not conformable to nature, to the rules of moral rectitude, or to established principles; not normal; unnatural; immethodical; unsymmetrical; erratic; no straight; not uniform; as, an irregular line; an irregular figure; an irregular verse; an irregular physician; an irregular proceeding; irregular motion; irregular conduct, etc. Cf. Regular.
n.
The interval between two notes; as, the distance of a fourth or seventh.
v. t.
To place at a distance or remotely.
v. t.
To outstrip by as much as a distance (see Distance, n., 3); to leave far behind; to surpass greatly.
v. t.
To cause to appear as if at a distance; to make seem remote.
pl.
of Regulus
a.
Governed by rule or rules; steady or uniform in course, practice, or occurence; not subject to unexplained or irrational variation; returning at stated intervals; steadily pursued; orderlly; methodical; as, the regular succession of day and night; regular habits.
a.
Belonging to a monastic order or community; as, regular clergy, in distinction dfrom the secular clergy.
n.
Distance.
a.
Thorough; complete; unmitigated; as, a regular humbug.
n.
Remoteness in succession or relation; as, the distance between a descendant and his ancestor.
imp. & p. p.
of Distance
a.
Constituted, selected, or conducted in conformity with established usages, rules, or discipline; duly authorized; permanently organized; as, a regular meeting; a regular physican; a regular nomination; regular troops.
a.
Having all the parts of the same kind alike in size and shape; as, a regular flower; a regular sea urchin.
n.
One who is not regular; especially, a soldier not in regular service.
a.
Conformed to a rule; agreeable to an established rule, law, principle, or type, or to established customary forms; normal; symmetrical; as, a regular verse in poetry; a regular piece of music; a regular verb; regular practice of law or medicine; a regular building.
pl.
of Tegula
a.
Measured by an angle; as, angular distance.