Complete graph

Complete graph
K7, a complete graph with 7 vertices
Vertices	n
Edges	$\frac{n (n-1)}{2}$
Diameter	1
Girth	3 if n ≥ 3
Automorphisms	n! (Sn)
Chromatic number	n
Chromatic index	n if n is odd n − 1 if n is even
Spectral Gap	n
Properties	(n − 1)-regular Symmetric graph Vertex-transitive Edge-transitive Unit distance Strongly regular Integral
Notation	Kn
v · mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. Properties The complete graph on n vertices has n(n − 1)/2 edges (a triangular number), and is denoted by Kn (from the German komplett).[1] It is a regular graph of degree n − 1. All complete graphs are their own maximal cliques. They are maximally connected as the only vertex cut which disconnects the graph is the complete set of vertices. The complement graph of a complete graph is an empty graph. If the edges of a complete graph are each given an orientation, the resulting directed graph is called a tournament. Geometry and topology A complete graph with n nodes represents the edges of an (n − 1)-simplex. Geometrically K3 forms the edge set of a triangle, K4 a tetrahedron, etc. The Császár polyhedron, a nonconvex polyhedron with the topology of a torus, has the complete graph K7 as its skeleton. Every neighborly polytope in four or more dimensions also has a complete skeleton. K1 through K4 are all planar graphs. However, every planar drawing of a complete graph with five or more vertices must contain a crossing, and the nonplanar complete graph K5 plays a key role in the characterizations of planar graphs: by Kuratowski's theorem, a graph is planar if and only if it contains neither K5 nor the complete bipartite graph K3,3 as a subdivision, and by Wagner's theorem the same result holds for graph minors in place of subdivisions. As part of the Petersen family, K6 plays a similar role as one of the forbidden minors for linkless embedding.[2] Examples Complete graphs on n vertices, for n between 1 and 12, are shown below along with the numbers of edges: K1: 0 K2: 1 K3: 3 K4: 6 K5: 10 K6: 15 K7: 21 K8: 28 K9: 36 K10: 45 K11: 55 K12: 66 References ^ Gries, David; Schneider, Fred B. (1993), A Logical Approach to Discrete Math, Springer-Verlag, p. 436 . ^ Robertson, Neil; Seymour, P. D.; Thomas, Robin (1993), "Linkless embeddings of graphs in 3-space", Bulletin of the American Mathematical Society 28 (1): 84–89, arXiv:math/9301216, doi:10.1090/S0273-0979-1993-00335-5, MR1164063 . External links Weisstein, Eric W., "Complete Graph" from MathWorld. Wikimedia Foundation. 2010. Игры ⚽ Поможем сделать НИР Justin (historian) Nibelungenlied Look at other dictionaries: complete graph — noun A graph where every pair of vertices is connected by an edge …   Wiktionary Complete bipartite graph — A complete bipartite graph with m = 5 and n = 3 Vertices n + m Edges mn …   Wikipedia Graph pebbling — is a mathematical game and area of interest played on a graph with pebbles on the vertices. Game play is composed of a series of pebbling moves. A pebbling move on a graph consists of taking two pebbles off one vertex and placing one on an… …   Wikipedia Graph coloring — A proper vertex coloring of the Petersen graph with 3 colors, the minimum number possible. In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called colors to elements of a graph… …   Wikipedia Graph theory — In mathematics and computer science, graph theory is the study of graphs : mathematical structures used to model pairwise relations between objects from a certain collection. A graph in this context refers to a collection of vertices or nodes and …   Wikipedia Graph (mathematics) — This article is about sets of vertices connected by edges. For graphs of mathematical functions, see Graph of a function. For statistical graphs, see Chart. Further information: Graph theory A drawing of a labeled graph on 6 vertices and 7 edges …   Wikipedia graph theory — Math. the branch of mathematics dealing with linear graphs. [1965 70] * * * Mathematical theory of networks. A graph consists of nodes (also called points or vertices) and edges (lines) connecting certain pairs of nodes. An edge that connects a… …   Universalium Graph factorization — Not to be confused with Factor graph. 1 factorization of Desargues graph: each color class is a 1 factor …   Wikipedia Graph isomorphism — In graph theory, an isomorphism of graphs G and H is a bijection between the vertex sets of G and H such that any two vertices u and v of G are adjacent in G if and only if ƒ(u) and ƒ(v) are adjacent in H. This kind of bijection is commonly… …   Wikipedia Graph homomorphism — Not to be confused with graph homeomorphism. In the mathematical field of graph theory a graph homomorphism is a mapping between two graphs that respects their structure. More concretely it maps adjacent vertices to adjacent vertices. Contents 1… …   Wikipedia 18+ © Academic, 2000-2024 Contact us: Technical Support, Advertising Dictionaries export, created on PHP, Joomla, Drupal, WordPress, MODx. Mark and share Search through all dictionaries Translate… Search Internet Share the article and excerpts Direct link … Do a right-click on the link above and select “Copy Link”