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Distance-transitive cubic graph with 20 nodes and 30 edges
field of graph theory, the Desargues graph is a distance-transitive, cubic graph with 20 vertices and 30 edges. It is named after Girard Desargues, arises
Desargues_graph
French mathematician and engineer (1591–1661)
geometry. Desargues's theorem, the Desargues graph, and the crater Desargues on the Moon are named in his honour. Born in Lyon, Desargues came from a
Girard_Desargues
Geometric configuration of ten points and lines
after Girard Desargues. The Desargues configuration can be constructed in two dimensions from the points and lines occurring in Desargues's theorem, in
Desargues_configuration
Cubic graph with 10 vertices and 15 edges
Petersen graphs also include the n-prism G(n,1), the Dürer graph G(6,2), the Möbius–Kantor graph G(8,3), the dodecahedron G(10,2), the Desargues graph G(10
Petersen_graph
Graph representing incident points and lines
space. For every Levi graph, there is an equivalent hypergraph, and vice versa. The Desargues graph is the Levi graph of the Desargues configuration, composed
Levi_graph
Family of cubic graphs formed from regular and star polygons
the Möbius-Kantor graph G ( 8 , 3 ) {\displaystyle G(8,3)} , the dodecahedron G ( 10 , 2 ) {\displaystyle G(10,2)} , the Desargues graph G ( 10 , 3 ) {\displaystyle
Generalized_Petersen_graph
Graph whose vertices correspond to combinations of a set of n elements
vertices. The bipartite Kneser graph H(5, 2) is the Desargues graph and the bipartite Kneser graph H(n, 1) is a crown graph. Watkins (1970). Lovász (1978)
Kneser_graph
3-regular graphs. Every strongly regular graph is symmetric, but not vice versa. Heawood graph Möbius–Kantor graph Pappus graph Desargues graph Nauru graph Coxeter
List_of_graphs
Nauru graph and the Desargues graph are integral. The Higman–Sims graph, the Hall–Janko graph, the Clebsch graph, the Hoffman–Singleton graph, the Shrikhande
Integral_graph
Fewest edge crossings in drawing of a graph
The smallest 6-crossing cubic graph is the Desargues graph, with 20 vertices. None of the four 7-crossing cubic graphs, with 22 vertices, are well known
Crossing number (graph theory)
Crossing_number_(graph_theory)
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
Partition of a graph into spanning subgraphs
mathematics In graph theory, a factor of a graph G is a spanning subgraph, i.e., a subgraph that has the same vertex set as G. A k-factor of a graph is a spanning
Graph_factorization
Graph property
Cubical graph, the Heawood graph, the Pappus graph, the Coxeter graph, the Tutte–Coxeter graph, the Dodecahedral graph, the Desargues graph, Tutte 12-cage
Distance-regular_graph
Symmetric bipartite cubic graph with 16 vertices and 24 edges
10 , 2 ) {\displaystyle G(10,2)} , the Desargues graph G ( 10 , 3 ) {\displaystyle G(10,3)} and the Nauru graph G ( 12 , 5 ) {\displaystyle G(12,5)} .
Möbius–Kantor_graph
Operation in graph theory
is the Desargues graph: K2 × G(5,2) = G(10,3). The bipartite double cover of a complete graph Kn is a crown graph (a complete bipartite graph Kn,n minus
Tensor_product_of_graphs
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
Graph with all vertices of degree 3
graph, the Desargues graph, the Nauru graph, the Coxeter graph, the Tutte–Coxeter graph, the Dyck graph, the Foster graph and the Biggs–Smith graph. W. T.
Cubic_graph
Bipartite, 3-regular undirected graph
4230/LIPIcs.GD.2025.14, ISBN 978-3-95977-403-1. Kagno, I. N. (1947), "Desargues' and Pappus' graphs and their groups", American Journal of Mathematics, 69 (4),
Pappus_graph
graph and the Desargues graph. Any hypercube graph, such as the four-dimensional hypercube shown below, is also bivariegated. However, the graph shown below
Bivariegated_graph
Derived bipartite graph with twice as many nodes as the original graph
graph is the Desargues graph: K2 × G(5,2) = G(10,3). The bipartite double cover of a complete graph Kn is a crown graph (a complete bipartite graph Kn
Bipartite_double_cover
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
the graph is planar and F indicates that the graph is not planar. Wikimedia Commons has media related to Graphs by number of vertices. See also Graph theory
List of graphs by edges and vertices
List_of_graphs_by_edges_and_vertices
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
trivial configuration (11), DCD(2) is the trilateral (32) and DCD(3) is the Desargues configuration (103). In configurations DCD(n) were further generalized
Danzer's_configuration
Geometric concept of a 2D space with "points at infinity" adjoined
projective spaces; such embeddability is a consequence of a property known as Desargues' theorem, not shared by all projective planes. A projective plane is a
Projective_plane
Argentine-born American mathematician
distance-transitive graphs into C-UH graphs that yielded the above-mentioned paper and also allowed to confront, as digraphs, the Pappus graph to the Desargues graph. These
Italo_Jose_Dejter
Representation of cubic graphs
In the mathematical field of graph theory, LCF notation or LCF code is a notation devised by Joshua Lederberg, and extended by H. S. M. Coxeter and Robert
LCF_notation
2D surface which extends indefinitely
plane is the real projective plane provided with a metric. Kepler and Desargues used the gnomonic projection to relate a plane σ to points on a hemisphere
Plane_(mathematics)
Geometry with 7 points and 7 lines
plane, even though the plane is too small to contain a non-degenerate Desargues configuration (which requires 10 points and 10 lines). The lines of the
Fano_plane
Points and lines with equal incidences
edition of his book Geometrie der Lage, in the context of a discussion of Desargues' theorem. Ernst Steinitz wrote his dissertation on the subject in 1894
Configuration_(geometry)
Isometric subgraph of a hypercube
resulting graph is a bipartite Kneser graph; the graph formed in this way with n = 2 has 20 vertices and 30 edges, and is called the Desargues graph. All median
Partial_cube
algebra is mostly done using graphs, using points, lines, areas etc. to represent various parameters of problems. Graphing calculators can take inputs
Mathematical_visualization
General concept and operation in mathematics
sometimes have fixed points, so that the dual of A is A itself. For example, Desargues' theorem is self-dual in this sense under the standard duality in projective
Duality_(mathematics)
Geometry without using coordinates
incidence of lines in geometric configurations. David Hilbert showed that the Desargues configuration played a special role. Further work was done by Ruth Moufang
Synthetic_geometry
Geometry theorem
disregarded the possibility that some additional incidences could occur in the Desargues configuration. A complete proof is provided by Cronheim 1953. W. Blaschke:
Pappus's_hexagon_theorem
Function that is its own inverse
Archive J. V. Field and J. J. Gray (1987) The Geometrical Work of Girard Desargues, (New York: Springer), p. 54 Ivor Thomas (editor) (1980) Selections Illustrating
Involution_(mathematics)
Geometric configuration of 9 points and 9 lines
produces the Hesse configuration. Like the Pappus configuration, the Desargues configuration can be defined in terms of perspective triangles, and the
Pappus_configuration
Pseudolines arranged largely to study arrangements of lines
triangle flips. In other words, approaching arrangements have a connected flip graph. Each rank-3 oriented matroid is equivalent to an arrangement of pseudolines
Arrangement_of_pseudolines
Field of mathematics dealing with three-dimensional Euclidean spaces
include: projective geometry of three dimensions (leading to a proof of Desargues' theorem by using an extra dimension) further polyhedra descriptive geometry
Solid_geometry
Irrational system of points and lines
additional applications as a counterexample in the theory of visibility graphs and in graph drawing. One way of constructing the Perles configuration is to start
Perles_configuration
Geometric structure of 8 points and 8 lines
particular, one possible solution for p = 5 {\displaystyle p=5} is the Desargues configuration, a set of ten points and ten lines, three points per line
Möbius–Kantor_configuration
(combinatorics) Graph structure theorem (graph theory) Grinberg's theorem (graph theory) Grötzsch's theorem (graph theory) Hajnal–Szemerédi theorem (graph theory)
List_of_theorems
Branch of mathematics
this period was the systematic study of projective geometry by Girard Desargues (1591–1661). Projective geometry studies properties of shapes which are
Geometry
Field of knowledge
include: Projective geometry, introduced in the 16th century by Girard Desargues, extends Euclidean geometry by adding points at infinity at which parallel
Mathematics
Curve from a cone intersecting a plane
concept of limits. Kepler first used the term 'foci' in 1604. Girard Desargues and Blaise Pascal developed a theory of conics using an early form of
Conic_section
Unsolved problem in combinatorial geometry
(2010), "Complexity of some geometric and topological problems" (PDF), Graph Drawing, 17th International Symposium, GS 2009, Chicago, IL, USA, September
Kobon_triangle_problem
most significant for projective planes due to the universal validity of Desargues' theorem in higher dimensions. In contrast, the analytic approach is to
Incidence_(geometry)
Branch of mathematics
Gérard Desargues approached geometry from a different perspective, developing the synthetic notions of projective geometry. Pascal and Desargues also studied
Algebraic_geometry
Uniform polychoron
vertices, and 10 of the triangle faces represent a self dual symmetric Desargues configuration, (103), seen here perspective projected into 3-dimensions
Rectified_5-cell
Press/Harvard University Press. O'Connor, John J; Edmund F. Robertson "Gérard Desargues". MacTutor History of Mathematics archive. Rao, C. Radhakrishna (1992)
List of people considered father or mother of a scientific field
List_of_people_considered_father_or_mother_of_a_scientific_field
French polymath (1596–1650)
everything he saw during the siege. He also met French mathematician Girard Desargues. In the autumn of that year, in the residence of the papal nuncio Guidi
René_Descartes
Mathematical model of the physical space
(a line), or x2 + y2 = 7 (a circle). Also in the 17th century, Girard Desargues, motivated by the theory of perspective, introduced the concept of idealized
Euclidean_geometry
Dutch mathematician and physicist (1629–1695)
in 1655 and encountered the work of Fermat, Blaise Pascal and Girard Desargues years earlier. He eventually published what was, at the time, the most
Christiaan_Huygens
Geometric system with a finite number of points
Desarguesian planes (those that are isomorphic with a PG(2, q)) satisfy Desargues's theorem and are projective planes over finite fields, but there are many
Finite_geometry
The latter is known to imply the former (Hessenberg). The theorem of Desargues expresses a kind of homogeneity of the plane. In general, it holds in
Topological_geometry
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