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Graph representing connectivity between cliques of another graph
In graph theory, a branch of mathematics, the simplex graph κ(G) of an undirected graph G is itself a graph, with one node for each clique (a set of mutually
Simplex_graph
Topics referred to by the same term
problems Simplex graph, derived from the cliques of another graph Simplex noise, a method for constructing an n-dimensional noise function Simplex plot,
Simplex_(disambiguation)
Procedures for constructing new graphs in graph theory
dual graph; medial graph; quotient graph; double graph; simplex graph; YΔ- and ΔY-transformation; Mycielskian. Binary operations create a new graph from
Graph_operations
Topics referred to by the same term
graph, the intersection graph of maximal cliques Simplex graph, a graph with a vertex for each clique in the original graph, with an edge between vertices
Clique_graph_(disambiguation)
Adjacent subset of an undirected graph
complex of a graph G is an abstract simplicial complex X(G) with a simplex for every clique in G A simplex graph is an undirected graph κ(G) with a vertex
Clique_(graph_theory)
Multi-dimensional generalization of triangle
0-dimensional simplex is a point, a 1-dimensional simplex is a line segment, a 2-dimensional simplex is a triangle, a 3-dimensional simplex is a tetrahedron
Simplex
Algorithm for linear programming
Dantzig's simplex algorithm (or simplex method) is an algorithm for linear programming. The name of the algorithm is derived from the concept of a simplex and
Simplex_algorithm
Graph in which every two vertices are adjacent
The simplex, which is identical to a complete graph of n + 1 {\displaystyle n+1} vertices, where n {\displaystyle n} is the dimension of the simplex. Bang-Jensen
Complete_graph
Planar graph with quadrilateral faces
(the simplex graph of K3), the Cartesian product of an edge and a claw K1,3 (the simplex graph of a claw), and the graphs formed from a gear graph by adding
Squaregraph
Graph with a median for each three vertices
subclass of the median graphs. A polyomino is a special case of a squaregraph and therefore also forms a median graph. The simplex graph κ ( G ) {\displaystyle
Median_graph
edge graph of a convex polytope is a finite simple graph. It is connected, since a path between any two vertices can be obtained from the simplex algorithm
Graph_of_a_polytope
Algorithm in graph theory
In mathematical optimization, the network simplex algorithm is a graph theoretic specialization of the simplex algorithm. The algorithm is usually formulated
Network_simplex_algorithm
Family of graphs based on the Fibonacci sequence
representations. The Fibonacci cube of order n is the simplex graph of the complement graph of an n-vertex path graph. That is, each vertex in the Fibonacci cube
Fibonacci_cube
Abstract simplicial complex describing a graph's cliques
represented by a simplex of dimension k – 1. The 1-skeleton of X(G) (also known as the underlying graph of the complex) is an undirected graph with a vertex
Clique_complex
Numerical optimization algorithm
The Nelder–Mead method (also downhill simplex method, amoeba method, or polytope method) is a numerical method used to find a local minimum or maximum
Nelder–Mead_method
Method to solve optimization problems
Are there pivot rules which lead to polynomial-time simplex variants? Do all polytopal graphs have polynomially bounded diameter? These questions concern
Linear_programming
Barycentric plot on three variables
A ternary plot, ternary graph, triangle plot, simplex plot, or Gibbs triangle is a barycentric plot on three variables which sum to a constant. It graphically
Ternary_plot
Mathematical problem
question can be phrased in graph theoretic terms as follows. Let G be the unit distance graph of the plane: an infinite graph with all points of the plane
Hadwiger–Nelson_problem
Tiling of n-dimensional space
it is called the 5-simplex honeycomb, with Coxeter graph , filling space by 5-simplex, rectified 5-simplex, and birectified 5-simplex facets. In 6 dimensions
Simplicial_honeycomb
Computer compiler optimization technique
register allocation), or across function boundaries traversed via call-graph (interprocedural register allocation). When done per function/procedure
Register_allocation
Coxeter-Dynkin diagram as a cyclic graph of n+1 nodes with two adjacent nodes ringed. It is composed of n-simplex facets, along with all truncated n-simplices
Cyclotruncated simplicial honeycomb
Cyclotruncated_simplicial_honeycomb
Solid with eight equal triangular faces
octahedron give rise to a graph, a discrete structure drawn in a plane. The name is octahedral graph. The octahedral graph is an example of a four-connected
Regular_octahedron
Abstract regular polyhedron with 10 triangular faces
5-dimensional 5-simplex which has a complete graph of edges, but only contains half of the (20) faces. From the point of view of graph theory this is an
Hemi-icosahedron
Four-dimensional analogue of the cube
The dissection of the tesseract into instances of its characteristic simplex (a particular orthoscheme with Coxeter diagram ) is the most basic direct
Tesseract
Overview of and topical guide to algorithms
algorithm Numerical integration Monte Carlo method Linear programming Simplex algorithm Interior-point method Integer programming Dynamic programming
Outline_of_algorithms
Linear programming algorithm
optimization, the revised simplex method is a variant of George Dantzig's simplex method for linear programming. The revised simplex method is mathematically
Revised_simplex_method
Balanced complete multipartite graph
formed by embedding a Turán graph onto the vertices of a regular simplex. An n-vertex graph G is a subgraph of a Turán graph T(n,r) if and only if G admits
Turán_graph
Sequence of locally optimal choices
distributed hash table. Graph theory is a rich source of greedy algorithms. Computing scientists frequently use greedy algorithms to compute graph invariants. Dijkstra's
Greedy_algorithm
Generalization of graph theory
hypergraph is a generalization of a graph in which an edge can join any number of vertices. In contrast, in an ordinary graph, an edge connects exactly two
Hypergraph
Archimedean solid with 8 faces
World Cup. In the mathematical field of graph theory, a truncated tetrahedral graph is an Archimedean graph, the graph of vertices and edges of the truncated
Truncated_tetrahedron
Solid with twenty equal triangular faces
is an example of a Platonic solid and of a deltahedron. The icosahedral graph represents the skeleton of a regular icosahedron. Many polyhedra and other
Regular_icosahedron
Topological space formed from distances
its 1-skeleton is the unit disk graph of its points. It contains a simplex for every clique in the unit disk graph, so it is the clique complex or flag
Vietoris–Rips_complex
Seven-dimensional geometric object
There are exactly three such convex regular 7-polytopes: {3,3,3,3,3,3} - 7-simplex {4,3,3,3,3,3} - 7-cube {3,3,3,3,3,4} - 7-orthoplex There are no nonconvex
Uniform_7-polytope
Algorithms for solving convex optimization problems
polynomial—in contrast to the simplex method, which has exponential run-time in the worst case. Practically, they run as fast as the simplex method—in contrast to
Interior-point_method
polytope is a polytope formed from a simplex by repeatedly gluing another simplex onto one of its facets. Every simplex is itself a stacked polytope. In three
Stacked_polytope
Polyhedron with four faces
three-dimensional case of the more general concept of a Euclidean simplex, and may thus also be called a 3-simplex. The tetrahedron is one kind of pyramid, which is a
Tetrahedron
Method of solving linear programming problems
solving linear programming problems using the simplex algorithm. The Big M method extends the simplex algorithm to problems that contain "greater-than"
Big_M_method
Natural number
vertices a polyhedron can have. The regular tetrahedron, also called a 3-simplex, is the simplest Platonic solid. It has four regular triangles as faces
4
Regular polytope dual to the hypercube in any number of dimensions
hypercube. The vertex-edge graph of an n-dimensional cross-polytope is the Turán graph T(2n, n) (also known as a cocktail party graph ). In 1 dimension the
Cross-polytope
Independent set in a graph
In graph theory, a rainbow-independent set (ISR) is an independent set in a graph, in which each vertex has a different color. Formally, let G = (V, E)
Rainbow-independent_set
geometry, a stericated 5-simplex is a convex uniform 5-polytope with fourth-order truncations (sterication) of the regular 5-simplex. There are six unique
Stericated_5-simplexes
Tessellation of convex uniform polyhedron cells
represent a mirror removal operation. If an end-node is removed, another simplex (tetrahedral) family is generated. If a hole has two branches, a Vinberg
Paracompact uniform honeycombs
Paracompact_uniform_honeycombs
Shape with nine sides
represents an orthographic projection of the 9 vertices and 36 edges of the 8-simplex. Temples of the Baháʼí Faith, called Baháʼí Houses of Worship, are required
Nonagon
Study of mathematical algorithms for optimization problems
discrete optimization, in which an object such as an integer, permutation or graph must be found from a countable set. A problem with continuous variables
Mathematical_optimization
Mathematical optimization problem restricted to integers
nearest integers, it is not feasible for the ILP. See projection into simplex The following is a reduction from minimum vertex cover to integer programming
Integer_programming
Convex polytope, the n-dimensional analogue of a square and a cube
of tesseract and 16-cell. The graph of the n-hypercube's edges is isomorphic to the Hasse diagram of the (n−1)-simplex's face lattice. This can be seen
Hypercube
Mathematical object
vertices, but not any graph can be plotted in R 2 {\displaystyle \mathbb {R} ^{2}} in this way. If K is the standard combinatorial n-simplex, then | K | {\displaystyle
Abstract_simplicial_complex
graphically by a tetrahedral graph, which is in a dual configuration of the fundamental domain tetrahedron. In the graph, each node represents a face
Goursat_tetrahedron
{\displaystyle \Delta _{d,k}} is a convex polytope that generalizes the simplex. It is determined by two integers d {\displaystyle d} and k {\displaystyle
Hypersimplex
Graphs of d-dimensional polytopes are d-connected
combinatorics, a branch of mathematics, Balinski's theorem is a statement about the graph-theoretic structure of three-dimensional convex polyhedra and higher-dimensional
Balinski's_theorem
Polyhedron with 8 triangles and 6 squares
positions. The graph of a cuboctahedron may be constructed as the line graph of the cubical graph, showing that it is a locally linear graph. The 24 edges
Cuboctahedron
Polytope in 8-dimensional geometry
rings in this Coxeter-Dynkin diagram: . The 421 polytope has 17,280 7-simplex and 2,160 7-orthoplex facets, and 240 vertices. Its vertex figure is the
4_21_polytope
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
Problem optimization method
substructures are usually described by means of recursion. For example, given a graph G=(V,E), the shortest path p from a vertex u to a vertex v exhibits optimal
Dynamic_programming
Fixed-point theorem for set-valued functions
set-valued function on S with the following properties: φ has a closed graph; φ(x) is non-empty and convex for all x ∈ S. Then φ has a fixed point. Set-valued
Kakutani_fixed-point_theorem
Graph with at most one cycle per component
In graph theory, a pseudoforest is an undirected graph in which every connected component has at most one cycle. That is, it is a system of vertices and
Pseudoforest
Theorem on triangulation graph colorings
particular, there must be at least one rainbow simplex. We shall first address the two-dimensional case. Consider a graph G built from the triangulation T as follows:
Sperner's_lemma
Δk correspond to non-equivalent marked metric graph structures on Fn. The set j(Δk) is called open simplex in Xn corresponding to f and is denoted S(f)
Outer_space_(mathematics)
Optimization algorithm
of algorithms that solve convex problems by hill-climbing include the simplex algorithm for linear programming and binary search. To attempt to avoid
Hill_climbing
Solving an optimization problem with a quadratic objective function
Lagrangian, conjugate gradient, gradient projection, extensions of the simplex algorithm. In the case in which Q is positive definite, the problem is
Quadratic_programming
Integer associated with a graph
the dimension of the complete graph is the same as that of the simplex having the same number of vertices. All star graphs K m , 1 {\displaystyle K_{m,1}}
Dimension_(graph_theory)
Graphical technique for data sets
Star plot Surface plot Ternary plot : A ternary plot, ternary graph, triangle plot, simplex plot, or de Finetti diagram is a barycentric plot on three variables
Plot_(graphics)
Subfield of mathematical optimization
particular measure m 0 {\displaystyle m_{0}} . For example, if there is a graph G {\displaystyle G} which contains vertices u {\displaystyle u} and v {\displaystyle
Combinatorial_optimization
Sequential model-based optimization of expensive black-box functions
Khachiyan Projective algorithm of Karmarkar Basis-exchange Simplex algorithm of Dantzig Revised simplex algorithm Criss-cross algorithm Principal pivoting algorithm
Bayesian_optimization
Abstraction of ordered linear algebra
matroid is a mathematical structure that abstracts the properties of directed graphs, vector arrangements over ordered fields, and hyperplane arrangements over
Oriented_matroid
Polytope contained by 7-polytope facets
parentheses for cross-referencing. See also a list of 8-simplex polytopes for symmetric Coxeter plane graphs of these polytopes. The B8 family has symmetry of
Uniform_8-polytope
Branch of discrete mathematics
right. One of the oldest and most accessible parts of combinatorics is graph theory, which by itself has numerous natural connections to other areas
Combinatorics
Discrete (i.e., incremental) version of infinitesimal calculus
and σi is an oriented k-simplex. In this definition, we declare that each oriented simplex is equal to the negative of the simplex with the opposite orientation
Discrete_calculus
Measure of similarity and diversity between sets
{\displaystyle k} -simplex corresponds to a probability distribution on k + 1 {\displaystyle k+1} elements, because the unit k {\displaystyle k} -simplex is the set
Jaccard_index
Optimization algorithm
computational problems that can be reduced to finding good paths through graphs. Artificial ants represent multi-agent methods inspired by the behavior
Ant colony optimization algorithms
Ant_colony_optimization_algorithms
Algorithm for finding zeros of functions
than x0. Geometrically, (x1, 0) is the x-intercept of the tangent to the graph of f at (x0, f(x0)): that is, the improved guess, x1, is the unique root
Newton's_method
Graph-theoretic description of polyhedra
planar graph, and every 3-connected planar graph can be represented as the graph of a convex polyhedron. For this reason, the 3-connected planar graphs are
Steinitz's_theorem
Type of mathematical set
a maximal simplex, i.e., any simplex in a complex that is not a face of any larger simplex. (Note the difference from a "face" of a simplex). A pure simplicial
Simplicial_complex
Optimization algorithm
Khachiyan Projective algorithm of Karmarkar Basis-exchange Simplex algorithm of Dantzig Revised simplex algorithm Criss-cross algorithm Principal pivoting algorithm
Limited-memory_BFGS
Four-dimensional analogue of the tetrahedron
hypertetrahedron, pentachoron, pentatope, pentahedroid, tetrahedral pyramid, or 4-simplex (Coxeter's α4 polytope), the simplest possible convex 4-polytope, and is
5-cell
Optimization method
Khachiyan Projective algorithm of Karmarkar Basis-exchange Simplex algorithm of Dantzig Revised simplex algorithm Criss-cross algorithm Principal pivoting algorithm
Broyden–Fletcher–Goldfarb–Shanno algorithm
Broyden–Fletcher–Goldfarb–Shanno_algorithm
polytopes with A6 symmetry. There is one self-dual regular form, the 6-simplex with 7 vertices. Each can be visualized as symmetric orthographic projections
A6_polytope
algorithm for constructing maximum-cardinality matching on graphs. Coloring algorithm: algorithms for graph (vertex or edge) coloring (subject to constraints,
List_of_algorithms
Graph theory model
partial k-trees. The graphs formed by the edges and vertices of k-dimensional stacked polytopes, polytopes formed by starting from a simplex and then repeatedly
K-tree
Pictorial representation of the behavior of subatomic particles
_{\mathrm {simplex} }u^{n}e^{-u\left(v_{0}D_{0}+v_{1}D_{1}+v_{2}D_{2}\cdots +v_{n}D_{n}\right)}\,dv_{1}\cdots dv_{n}\,du\,,} where the simplex is the region
Feynman_diagram
Algorithm used to solve non-linear least squares problems
b=102} used in the initial curve. Only when the parameters in the last graph are chosen closest to the original, are the curves fitting exactly. This
Levenberg–Marquardt_algorithm
On lengths of shortest paths in convex polytopes
combinatorics, the Hirsch conjecture is the statement that the edge-vertex graph of an n-facet polytope in d-dimensional Euclidean space has diameter no
Hirsch_conjecture
Numerical approximation algorithm
Khachiyan Projective algorithm of Karmarkar Basis-exchange Simplex algorithm of Dantzig Revised simplex algorithm Criss-cross algorithm Principal pivoting algorithm
Iterative_method
the 5-simplex itself. Vertices of the rectified 5-simplex are located at the edge-centers of the 5-simplex. Vertices of the birectified 5-simplex are located
Rectified_5-simplexes
Combinitorics of Polyhedra
not true. For simplicial polytopes (polytopes in which every facet is a simplex), it is often convenient to transform these vectors, producing a different
Polyhedral_combinatorics
Polytope constructed from alternation of a hypercube
formed. The 2n facets become 2n (n − 1)-demicubes, and 2n - 1 (n − 1)-simplex facets are formed in place of the deleted vertices. They have been named
Demihypercube
Algorithm for computing the maximal flow of a network
shortest augmenting paths. The introduction of the concepts of the level graph and blocking flow enable Dinic's algorithm to achieve its performance. Dinitz
Dinic's_algorithm
Polytope whose facets are all simplices
graph. For a simplicial tiling, examples are triangular tiling and Laves tiling. Simplicial 4-polytopes include: convex regular 4-polytope 4-simplex,
Simplicial_polytope
Isometric subgraph of a hypercube
median graphs are partial cubes. The trees and hypercube graphs are examples of median graphs. Since the median graphs include the squaregraphs, simplex graphs
Partial_cube
Algorithm to compute the maximum flow in a flow network
algorithm EdmondsKarp is input: graph (graph[v] should be the list of edges coming out of vertex v in the original graph and their corresponding constructed
Edmonds–Karp_algorithm
Geometric configuration of ten points and lines
Graphs associated with the Desargues configuration include the Desargues graph (its graph of point-line incidences) and the Petersen graph (its graph
Desargues_configuration
Collective behavior of decentralized, self-organized systems
technique useful in problems that deal with finding better paths through graphs. Artificial 'ants'—simulation agents—locate optimal solutions by moving
Swarm_intelligence
Optimization technique
1965: Matyas proposes random optimization. 1965: Nelder and Mead propose a simplex heuristic, which was shown by Powell to converge to non-stationary points
Metaheuristic
Optimization algorithm
adjacent picture. Here, f is assumed to be defined on the plane, and that its graph has a bowl shape. The blue curves are the contour lines, that is, the regions
Gradient_descent
Class of algorithms for solving constrained optimization problems
Khachiyan Projective algorithm of Karmarkar Basis-exchange Simplex algorithm of Dantzig Revised simplex algorithm Criss-cross algorithm Principal pivoting algorithm
Augmented_Lagrangian_method
Uniform 6-polytope
leaves the 5-simplex, . Removing the node on the end of the 2-length branch leaves the 5-orthoplex in its alternated form: (211), . Every simplex facet touches
2_21_polytope
Numerical optimization algorithm
Khachiyan Projective algorithm of Karmarkar Basis-exchange Simplex algorithm of Dantzig Revised simplex algorithm Criss-cross algorithm Principal pivoting algorithm
Berndt–Hall–Hall–Hausman algorithm
Berndt–Hall–Hall–Hausman_algorithm
Shape with six sides
5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221 Uniform 7-polytope 7-simplex 7-orthoplex
Hexagon
Combinatorial optimization method
solves the linear program without the integer constraint using the regular simplex algorithm. When an optimal solution is obtained, and this solution has
Branch_and_cut
Graph without four-vertex star subgraphs
polytopes are claw-free, including the graph of the tetrahedron and more generally of any simplex (a complete graph), the graph of the octahedron and more generally
Claw-free_graph
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