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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
Compact Riemann surface of genus 3
In hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact Riemann surface of genus 3 with the highest possible order automorphism
Klein_quartic
Planar graph with 4 nodes and 5 edges
Hamiltonian graph. A graph is diamond-free if it has no diamond as an induced subgraph. The triangle-free graphs are diamond-free graphs, since every
Diamond_graph
of graphs contains definitions of graphs and graph families. For collected definitions of graph theory terms that do not refer to individual graph types
List_of_graphs
Mathematical abelian group
component, also results in the Klein four-group. Among the simple connected graphs, the one with the fewest entities admits the Klein four-group as its automorphism
Klein_four-group
various individual (finite) graphs. The columns 'vertices', 'edges', 'radius', 'diameter', 'girth', 'P' (whether the graph is planar), χ (chromatic number)
List of graphs by edges and vertices
List_of_graphs_by_edges_and_vertices
Graph often embedded in the Klein bottle
suffice. (The Klein bottle is the only surface for which the Heawood conjecture fails.) The Franklin graph can be embedded in the Klein bottle so that
Franklin_graph
Undirected graph with 14 vertices
Trivalent Symmetric Graphs of Small Order" (PDF). Australasian Journal of Combinatorics. 11: 146. Royle, G. "Cubic Symmetric Graphs (The Foster Census)
Heawood_graph
Cubic graph with 28 vertices and 42 edges
graph K2, the Petersen graph, the Coxeter graph and two graphs derived from the Petersen and Coxeter graphs by replacing each vertex with a triangle.
Coxeter_graph
Graph property
complete graphs and disconnected graphs from this definition. Every distance-transitive graph is distance regular. Indeed, distance-regular graphs were introduced
Distance-regular_graph
Embedding a graph in a topological space, often Euclidean
and only if it does not have one of the seven graphs of the Petersen family as a minor. The Petersen graph and associated map embedded in the projective
Graph_embedding
Edge that connects a node to itself
Where graphs are defined so as to allow loops and multiple edges, a graph without loops or multiple edges is often distinguished from other graphs by calling
Loop_(graph_theory)
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
Computational problem of graph theory
path problem can be defined for graphs whether undirected, directed, or mixed. The definition for undirected graphs states that every edge can be traversed
Shortest_path_problem
Klein four-group Klein geometry Klein graphs Klein's inequality Klein model Klein polyhedron Klein surface Klein quadric Klein quartic Kleinian group Kleinian
List of things named after Felix Klein
List_of_things_named_after_Felix_Klein
Planar maps require at most four colors
color theorem for triangulated graphs to prove it for all planar graphs, and without loss of generality we assume the graph is triangulated. Suppose v, e
Four_color_theorem
Theorem on graph coloring on surfaces
colors, rather than 7 as predicted by the formula. The Franklin graph can be drawn on the Klein bottle in a way that forms six mutually-adjacent regions, showing
Heawood_conjecture
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
Set of edges without common vertices
for special classes of graphs such as bipartite planar graphs, as described in the main article. In a weighted bipartite graph, the optimization problem
Matching_(graph_theory)
Branch of geometry that studies combinatorial properties and constructive methods
polytope, unit disk graphs, and visibility graphs. Topics in this area include: Graph drawing Polyhedral graphs Random geometric graphs Voronoi diagrams
Discrete_geometry
28 lines which touch a general quartic plane curve in two places
Manivel (2006). Dejter, Italo J. (2011), "From the Coxeter graph to the Klein graph", Journal of Graph Theory, 70: 1–9, arXiv:1002.1960, doi:10.1002/jgt.20597
Bitangents_of_a_quartic
Natural number
Matrix Theory to Laplacian Matrices of Graphs. CRC Press. p. 197. ISBN 978-1-4398-6339-8. ... The complete graph on the largest number of vertices that
4
Method for finding minimum spanning trees
vertices in G (assuming E ≥ V). In planar graphs, and more generally in families of graphs closed under graph minor operations, it can be made to run in
Borůvka's_algorithm
Mathematical classification
of small integers – 1 through 6, depending on the graph. The ordinary ADE graphs are the only graphs that admit a positive labeling with the following
ADE_classification
Conjecture about coloring graphs
mathematics Conjecture: If k complete graphs, each having exactly k vertices, have the property that every pair of complete graphs has at most one shared vertex
Erdős–Faber–Lovász_conjecture
Five coplanar points have a subset forming a convex quadrilateral
because it led to the marriage of mathematicians George Szekeres and Esther Klein) is the following statement: Theorem—any set of five points in the plane
Happy_ending_problem
Maximal subgraph whose vertices can reach each other
component is a maximal clique. These graphs may be produced as the transitive closures of arbitrary undirected graphs, for which finding the transitive closure
Component_(graph_theory)
Faria, L.; Klein, S.; Sritharan, R. (2007), "On the complexity of the sandwich problems for strongly chordal graphs and chordal bipartite graphs", Theoretical
Graph_sandwich_problem
postman problem) for mixed graphs (having both directed and undirected edges). The program is solvable in polynomial time if the graph has all undirected or
List_of_NP-complete_problems
Any planar graph can be subdivided by removing a few vertices
In graph theory, the planar separator theorem is a form of isoperimetric inequality for planar graphs, that states that any planar graph can be split
Planar_separator_theorem
American computer scientist
Klein is an American computer scientist and professor at Brown University. His research focuses on algorithms for optimization problems in graphs. Klein
Philip_N._Klein
Measure of similarity between two graphs
between two graphs is related to the string edit distance between strings. With the interpretation of strings as connected, directed acyclic graphs of maximum
Graph_edit_distance
Curve for failure rates over time
curve is a failure rate graph that curves up at both ends, similar in shape to a bathtub. The term can also apply to any graph with this shape. Many but
Bathtub_curve
Unsolved problem on graph coloring
needed to color biplanar graphs? More unsolved problems in mathematics The Earth–Moon problem is an unsolved problem on graph coloring in mathematics.
Earth–Moon_problem
Upper bound for number of colors that suffice to color any graph
number in 1976. Franklin proved that the chromatic number of a graph embedded in the Klein bottle can be as large as 6 {\displaystyle 6} , but never exceeds
Heawood_number
Process of generalization
they are not abstract in the sense of the objects in graph 1 below. We might look at other graphs, in a progression from cat to mammal to animal, and see
Abstraction
Least-weight tree connecting graph vertices
problem for planar graphs in linear time. By the Euler characteristic of planar graphs, m ≤ 3n - 6 ∈ O(n), so this is in time O(n). Given graph G where the nodes
Minimum_spanning_tree
Number of "holes" of a surface
ISBN 978-0-8218-3678-1 Ellis-Monaghan, Joanna A.; Moffatt, Iain (2013). Graphs on Surfaces: Dualities, Polynomials, and Knots. New York, NY: Springer New
Genus_(mathematics)
"On Two-Factors of Bipartite Regular Graphs." Discrete Math. 41, 35-41, 1982. Owens, P. J. "Bipartite Cubic Graphs and a Shortness Exponent." Discrete
Horton_graph
Graph theory problem: find a matching containing the most edges
efficient algorithms exist for special kinds of bipartite graphs: For sparse bipartite graphs, the maximum matching problem can be solved in O ~ ( E 10
Maximum-cardinality_matching
Graph drawing used to study Riemann surfaces
Hamiltonian paths on the icosahedral graph. Recognizably modern dessins d'enfants and Belyi functions were used by Felix Klein. Klein called these diagrams Linienzüge
Dessin_d'enfant
Type of plane partition
Algorithm for Voronoi Diagram in log*(n) (PDF) (Report). Aurenhammer, Franz; Klein, Rolf; Lee, Der-Tsai (2013). Voronoi Diagrams and Delaunay Triangulations
Voronoi_diagram
Algorithm used for pathfinding and graph traversal
meaning that it is formulated in terms of weighted graphs: starting from a specific starting node of a graph, it aims to find a path to the given goal node
A*_search_algorithm
Knowledge base that represents semantic relations between concepts in a network
semantic networks such as the existential graphs of Charles Sanders Peirce or the related conceptual graphs of John F. Sowa. These have expressive power
Semantic_network
NP-hard problem in combinatorial optimization
performance that ranges from 1% less efficient, for graphs with 10–20 nodes, to 11% less efficient for graphs with 120 nodes. The apparent ease with which humans
Travelling_salesman_problem
Solid with twenty equal triangular faces
Chudnovsky, Maria; Seymour, Paul (2005). "The structure of claw-free graphs" (PDF). In Bridget S. Webb (ed.). Surveys in combinatorics 2005. London
Regular_icosahedron
Topological invariant in mathematics
plane graphs by the same V − E + F {\displaystyle \ V-E+F\ } formula as for polyhedral surfaces, where F is the number of faces in the graph, including
Euler_characteristic
(2006) to prove the strong perfect graph theorem that the Berge graphs are indeed the same as the perfect graphs. Chudnovsky et al. were unable to prove
Skew_partition
Pictorial representation of the behavior of subatomic particles
device of covariant perturbation theory, the graphs were called Feynman–Dyson diagrams or Dyson graphs, because the path integral was unfamiliar when
Feynman_diagram
Solid with six equal square faces
Cartesian product of graphs: two graphs connecting the pair of vertices with an edge to form a new graph. In the case of the cubical graph, it is the product
Cube
Operation in graph theory
distance-hereditary graphs is equal to the class of the circle graphs of the Euler tours of some 4-regular graph. Locally equivalent graphs have the same rank-width
Local_complementation
Mathematical group that can be generated as the set of powers of a single element
generators as well. The Cayley graphs of cyclic groups with arbitrary generator sets are called circulant graphs. These graphs may be represented geometrically
Cyclic_group
Probability of survival beyond any specified time
The graphs below show examples of hypothetical survival functions. The x-axis is time. The y-axis is the proportion of subjects surviving. The graphs show
Survival_function
dienanthate, and undecylate, as well as polyestradiol phosphate—for more graphs. Estradiol esters like estradiol valerate and estradiol cypionate can be
Pharmacokinetics_of_estradiol
Agrawal, Ajit; Klein, Philip; Ravi, R. (1993), "Cutting down on Fill Using Nested Dissection: Provably Good Elimination Orderings", Graph Theory and Sparse
Nested_dissection
Graph metric of electrical resistance between nodes
edge being replaced by a resistance of one ohm. It is a metric on graphs. On a graph G, the resistance distance Ωi,j between two vertices vi and vj is
Resistance_distance
Type of chart to show a project schedule
Institute 2021, Glossary §3 Definitions. Gantt 1910. Morris 1997, p. 7. Klein 1999, p. 49. Richman 2002, pp. 97, 117, 276. Kumar, Pankaja Pradeep (2005)
Gantt_chart
Construction in graph theory
of the weights of the two corresponding arrows in the McKay graphs of A and B. Felix Klein proved that the finite subgroups of SL ( 2 , C ) {\displaystyle
McKay_graph
Shape in the geometry of numbers
of a Klein polyhedron is called a sail. Associated with the sail V {\displaystyle \textstyle V} of an irrational cone are two graphs: the graph Γ e (
Klein_polyhedron
Concept in graph theory
define NZ flows for arbitrary graphs (not just planar), we can use this to extend face-colorings to non-planar graphs. G is 2-face-colorable if and only
Nowhere-zero_flow
Directed graph describing citations in documents
of citation graphs, which is the fact that there is no standardized format or way of citing. This makes the construction of these graphs very difficult
Citation_graph
Graph modeling collaboration in a social network
well-studied collaboration graphs include: Collaboration graph of mathematicians also known as the Erdős collaboration graph, where two mathematicians
Collaboration_graph
Four-dimensional analogue of the cube
Criterion (PDF). 19th Japan Conference on Discrete and Computational Geometry, Graphs, and Games (JCDCG3 2016). Tokyo. Kemp, Martin (1 January 1998). "Dali's
Tesseract
Instrument used to measure distances
without any rules on it). Furthermore, it is also used to draw accurate graphs and tables. A ruler and compass construction is a construction that uses
Ruler
Two 3-regular graphs with 18 vertices and 27 edges
In the mathematical field of graph theory, the Blanuša snarks are two 3-regular graphs with 18 vertices and 27 edges. They were discovered by Yugoslavian
Blanuša_snarks
Kepler–Poinsot polyhedron
(2nd ed.). Springer. p. 46. Grünbaum, Branko (2007). "Graphs of polyhedra; polyhedra as graphs". Discrete Mathematics. 307 (3–5): 445–463. doi:10.1016/j
Small_stellated_dodecahedron
Non-orientable surface with one edge
graph but cannot be drawn without crossings on a plane. Another family of graphs that can be embedded on the Möbius strip, but not on the plane, are the
Möbius_strip
President of the United States (2017–2021; since 2025)
Falsehood in the Age of Trump. Crown/Archetype. ISBN 978-0-525-57484-2. Klein, Naomi (2017). No Is Not Enough. Penguin Books. ISBN 978-0-14-198679-1.
Donald_Trump
Free online crowdsourced encyclopedia
Harvard's Berkman Klein Center for Internet & Society said that he suspected much of the page-view decline was due to Knowledge Graphs, stating, "If you
Wikipedia
Theory of subatomic structure
belongs to a special class of functions called modular functions, whose graphs form a certain kind of repeating pattern. Although this function appears
String_theory
Relationship between two lines that meet at a right angle
and y 2 ( x ) = m 2 x + b 2 {\displaystyle y_{2}(x)=m_{2}x+b_{2}} , the graphs of the functions will be perpendicular if m 1 m 2 = − 1. {\displaystyle
Perpendicular
Awareness of facts, or competency
Condition, § 6. Doing Without Justification? Klein 1998, Lead Section, § 3. Warrant Zagzebski 1999, p. 100 Klein 1998, § 2. Propositional Knowledge Is Not
Knowledge
Two-dimensional manifold
the sphere is the boundary of the solid ball. Other surfaces arise as graphs of functions of two variables; see the figure at right. However, surfaces
Surface_(topology)
Mexican mathematician
iterated clique graphs" Discrete Mathematics 282(1–3): 263–265 (2004) Xueliang Li, Víctor Neumann-Lara, Eduardo Rivera-Campo "On a tree graph defined by a
Víctor_Neumann-Lara
Field of mathematics which studies incidence structures
Martin; Ziegler, Günter M. (2010), "Lines in the plane and decompositions of graphs", Proofs from The Book, Springer, pp. 63–67, doi:10.1007/978-3-642-00856-6_10
Incidence_geometry
Pharmaceutical compound
14 January 2017. Wilson CB, Nizet V, Maldonado Y, Klein JO, Remington JS (2015). Remington and Klein's Infectious Diseases of the Fetus and Newborn Infant
Pharmacology_of_bicalutamide
Theorem relating graph minors and topological embeddings
graphs describe the rough structure H-free graphs when H is planar. When H is nonplanar, we also need to consider k-clique-sums of a list of graphs,
Graph_structure_theorem
Cayley–Hamilton theorem in linear algebra Cayley-Klein parameters Cayley–Klein metric Cayley–Klein model of hyperbolic geometry Cayley–Menger determinant
List of things named after Arthur Cayley
List_of_things_named_after_Arthur_Cayley
Classification of algorithm
(see, for example, Reingold's algorithm for connectivity in undirected graphs). As Lipton states: This alone could be important and often is a great reason
Galactic_algorithm
Hypothetical physical concept
Hermann Weyl, Arthur Eddington, David Hilbert, Theodor Kaluza, Oskar Klein (see Kaluza–Klein theory), and most notably, Albert Einstein and his collaborators
Theory_of_everything
1920 film directed by Fritz Lang
film directed by Fritz Lang and starring Mia May, Hans Marr and Rudolf Klein-Rogge. It is also known by the alternative titles of The Wandering Picture
The_Wandering_Image
Country in Central Europe
Czech Hydrometeorological Institute, Prague, 2007. ISBN 80-244-1626-3, graphs 1.5 and 1.6 "Czech absolute record temperature registered near Prague".
Czech_Republic
Country in Southeastern Europe
original on 15 January 2023. Retrieved 3 March 2024. "Eurostat – Tables, Graphs and Maps Interface (TGM) table". Eurostat. 17 October 2013. Archived from
Bulgaria
Number, approximately 3.14
eigenvalue. For example, an idealized vibrating string can be modelled as the graph of a function f on the unit interval [0, 1], with fixed ends f(0) = f(1)
Pi
Sequential model-based optimization of expensive black-box functions
Information Processing Systems. Vol. 25. pp. 2951–2959. arXiv:1206.2944. Klein, Aaron; Falkner, Stefan; Bartels, Simon; Hennig, Philipp; Hutter, Frank
Bayesian_optimization
Partial differential equation
applied to particle position and momentum distributions, it is known as the Klein–Kramers equation. The case with zero diffusion is the continuity equation
Fokker–Planck_equation
Global system of connected computer networks
International Telecommunication Union (ITU). Retrieved 25 May 2015. Internet users graphs Archived 9 May 2020 at the Wayback Machine, Market Information and Statistics
Internet
Swedish activist (born 2003)
becoming vegan, upcycling, and giving up flying. She has said she showed them graphs and data, but when that did not work, she warned her family that they were
Greta_Thunberg
U.S. presidential administration since 2025
Archived from the original on March 17, 2025. Retrieved December 10, 2025. Klein, Betsy; Polantz, Katelyn; Cohen, Zachary (February 17, 2025). "Trump appears
Second presidency of Donald Trump
Second_presidency_of_Donald_Trump
dominant in U.S. as it doubles weekly". CNBC. Retrieved January 6, 2023. Klein B, Luhby T (January 11, 2023). "Biden administration renews COVID-19 public
COVID-19 pandemic in the United States
COVID-19_pandemic_in_the_United_States
1923 film
and starring Rudolf Klein-Rogge, Lucie Mannheim and Gustav von Wangenheim. It was shot at Babelsberg Studios in Berlin. Rudolf Klein-Rogge as Der Herr vom
The_Stone_Rider
Hungarian mathematician (1913–1996)
1933, Erdős and several other students, including George Szekeres, Esther Klein (later Szekeres), her lifelong friend Márta Wachsberger (later Svéd), and
Paul_Erdős
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
Concept in mathematical knot theory
Adv. Soviet Math. 16: 137. Watanabe, Tadayuki (2007). "Knotted trivalent graphs and construction of the LMO invariant from triangulations". Osaka J. Math
Quantum_invariant
American social news and discussion site
Retrieved May 24, 2015. Vanhemert, Kyle (January 9, 2014). "Fascinating graphs show how Reddit got huge by going mainstream". Wired. Archived from the
23 mathematical problems stated in 1900
OCLC 2331329. Chung, Fan R. K.; Graham, Ronald L. (1999-06-01). Erdös on Graphs: his legacy of unsolved problems. Natick, Mass: A K Peters/CRC Press.
Hilbert's_problems
Worldwide economic depression (1929–1939)
Franklin D. Roosevelt: Launching the New Deal, ch. 19, Little, Brown & Co. Klein, Lawrence R. (1947), The Keynesian Revolution, New York: Macmillan, pp. 56–58
Great_Depression
Political party in the United States
while highly informative, is by no means definitive. As David Shor and Ezra Klein have both said, the metric really serves as a proxy for class polarization
Democratic Party (United States)
Democratic_Party_(United_States)
American multinational conglomerate (1892–2024)
board, as did Higginson, Coolidge and Edison... Carlson 1991, pp. 294–296. Klein, Maury (2008). The Power Makers: Steam, Electricity, and the Men Who Invented
General_Electric
American comedian and social media personality (born 1986)
Sings: YouTube Progress Graphs" Archived December 21, 2014, at the Wayback Machine and "Colleen Ballinger: YouTube Progress Graphs" Archived December 23
Colleen_Ballinger
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