FACTOID # 14: North Carolina has a larger Native American population than North Dakota, South Dakota and Montana combined.

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(* = Graphable)

Encyclopedia > Complete graph
 Complete graph K7, a complete graph with 7 vertices Vertices n Edges n(n − 1) / 2

A complete graph with n-nodes represents the edges of an n-simplex. Geometrically K3 relates to a triangle, K4 a tetrahedron, K5 a pentachoron, etc. A 3-simplex or tetrahedron In geometry, a simplex (plural simplexes or simplices) or n-simplex is an n-dimensional analogue of a triangle. ... A triangle. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... The pentachoron, also called a pentatope or 4-simplex, is the simplest convex regular polychoron (a type of four-dimensional geometric figure). ...

Kuratowski's theorem says that a planar graph cannot contain K5 (or the complete bipartite graph K3,3) as a minor. Since Kn includes Kn − 1, no complete graph Kn with n greater than or equal to 5 is planar. In graph theory, a planar graph is a graph that can be embedded in the plane so that no edges intersect. ... In graph theory, a planar graph is a graph that can be drawn so that no edges intersect (or that can be embedded) in the plane. ... In the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set. ... In graph theory, a graph H is called a minor of the graph G if H is isomorphic to a graph that results from a subgraph of G by zero or more edge contractions. ...

An important property of the complete graph is the quadratic number of connections. That is, the number of edges is a quadratic function of the number of nodes. As such, a complete graph can be the worst case for large connected systems like social and computer networks. f(x) = x2 - x - 2 In mathematics, a quadratic function is a polynomial function of the form , where a is nonzero. ... f(x) = x2 - x - 2 A quadratic function, in mathematics, is a polynomial function of the form , where . ... A social network is a map of the relationships between individuals, indicating the ways in which they are connected through various social familiarities ranging from casual acquaintance to close familial bonds. ... A computer network is a system for communication among two or more computers. ...

Complete graphs on n vertices, for n between 1 and 8, are shown below along with the numbers of connections:

K1:0 K2:1 K3:3 K4:6
K5:10 K6:15 K7:21 K8:28

Image File history File links Complete_graph_K1. ... Image File history File links Complete_graph_K2. ... Image File history File links This is a lossless scalable vector image. ... Image File history File links Complete_graph_K4. ... Image File history File links Complete_graph_K5. ... Image File history File links Complete_graph_K6. ... Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ...

Results from FactBites:

 Complete graph - Wikipedia, the free encyclopedia (101 words) In the mathematical field of graph theory a complete graph is a simple graph where an edge connects every pair of vertices. It is a regular graph of degree n − 1. They are maximally connected as the only vertex cut which disconnects the graph is the complete set of vertices.
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