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Complete Graph
In the 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. A complete digraph is a directed graph in which every pair of distinct vertices is connected by a pair of unique edges (one in each direction). Graph theory itself is typically dated as beginning with Leonhard Euler's 1736 work on the Seven Bridges of Königsberg. However, drawings of complete graphs, with their vertices placed on the points of a regular polygon, had already appeared in the 13th century, in the work of Ramon Llull. Such a drawing is sometimes referred to as a mystic rose. Properties The complete graph on vertices is denoted by . Some sources claim that the letter in this notation stands for the German word , but the German name for a complete graph, , does not contain the letter , and other sources state that the notation honors the contributions of Kazimierz Kuratowski to graph theory. has edg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Graph K7
Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies that there are no "gaps" in the real numbers * Complete metric space, a metric space in which every Cauchy sequence converges * Complete uniform space, a uniform space where every Cauchy net in converges (or equivalently every Cauchy filter converges) * Complete measure, a measure space where every subset of every null set is measurable * Completion (algebra), at an ideal * Completeness (cryptography) * Completeness (statistics), a statistic that does not allow an unbiased estimator of zero * Complete graph, an undirected graph in which every pair of vertices has exactly one edge connecting them * Complete tree (abstract data type), a tree with every level filled, except possibly the last * Complete category, a category ''C'' where every d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Drawing
Graph drawing is an area of mathematics and computer science combining methods from geometric graph theory and information visualization to derive two-dimensional depictions of graph (discrete mathematics), graphs arising from applications such as social network analysis, cartography, linguistics, and bioinformatics. A drawing of a graph or network diagram is a pictorial representation of the vertex (graph theory), vertices and edge (graph theory), edges of a graph. This drawing should not be confused with the graph itself: very different layouts can correspond to the same graph., p. 6. In the abstract, all that matters is which pairs of vertices are connected by edges. In the concrete, however, the arrangement of these vertices and edges within a drawing affects its understandability, usability, fabrication cost, and aesthetics. The problem gets worse if the graph changes over time by adding and deleting edges (dynamic graph drawing) and the goal is to preserve the user's men ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tournament (graph Theory)
In graph theory, a tournament is a directed graph with exactly one edge between each two vertices, in one of the two possible directions. Equivalently, a tournament is an orientation of an undirected complete graph. (However, as directed graphs, tournaments are not complete: complete directed graphs have two edges, in both directions, between each two vertices.) Equivalently, a tournament is a complete asymmetric relation. The name ''tournament'' comes from interpreting the graph as the outcome of a round-robin tournament, a game where each player is paired against every other exactly once. In a tournament, the vertices represent the players, and the edges between players point from the winner to the loser. Many of the important properties of tournaments were investigated by H. G. Landau in 1953 to model dominance relations in flocks of chickens. Tournaments are also heavily studied in voting theory, where they can represent partial information about voter preferences am ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orientation (graph Theory)
In graph theory, an orientation of an undirected graph is an assignment of a direction to each edge, turning the initial graph into a directed graph. Oriented graphs A directed graph is called an oriented graph if none of its pairs of vertices is linked by two mutually symmetric edges. Among directed graphs, the oriented graphs are the ones that have no 2-cycles (that is at most one of and may be arrows of the graph). A tournament is an orientation of a complete graph. A polytree is an orientation of an undirected tree. Sumner's conjecture states that every tournament with vertices contains every polytree with vertices. The number of non-isomorphic oriented graphs with vertices (for ) is : 1, 2, 7, 42, 582, 21480, 2142288, 575016219, 415939243032, … . Tournaments are in one-to-one correspondence with complete directed graphs (graphs in which there is a directed edge in one or both directions between every pair of distinct vertices). A complete directed graph can be con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Empty Graph
In the mathematical field of graph theory, the term "null graph" may refer either to the order-zero graph, or alternatively, to any edgeless graph (the latter is sometimes called an "empty graph"). Order-zero graph The order-zero graph, , is the unique graph having no vertices (hence its order is zero). It follows that also has no edges. Thus the null graph is a regular graph of degree zero. Some authors exclude from consideration as a graph (either by definition, or more simply as a matter of convenience). Whether including as a valid graph is useful depends on context. On the positive side, follows naturally from the usual set-theoretic definitions of a graph (it is the ordered pair for which the vertex and edge sets, and , are both empty), in proofs it serves as a natural base case for mathematical induction, and similarly, in recursively defined data structures is useful for defining the base case for recursion (by treating the null tree as the child of miss ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complement Graph
In the mathematical field of graph theory, the complement or inverse of a graph is a graph on the same vertices such that two distinct vertices of are adjacent if and only if they are not adjacent in . That is, to generate the complement of a graph, one fills in all the missing edges required to form a complete graph, and removes all the edges that were previously there.. The complement is not the set complement of the graph; only the edges are complemented. Definition Let be a simple graph and let consist of all 2-element subsets of . Then is the complement of , where is the relative complement of in . For directed graphs, the complement can be defined in the same way, as a directed graph on the same vertex set, using the set of all 2-element ordered pairs of in place of the set in the formula above. In terms of the adjacency matrix ''A'' of the graph, if ''Q'' is the adjacency matrix of the complete graph of the same number of vertices (i.e. all entries ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex Cut
In graph theory, a vertex subset is a vertex separator (or vertex cut, separating set) for nonadjacent vertices and if the removal of from the graph separates and into distinct connected components. Examples Consider a grid graph with rows and columns; the total number of vertices is . For instance, in the illustration, , , and . If is odd, there is a single central row, and otherwise there are two rows equally close to the center; similarly, if is odd, there is a single central column, and otherwise there are two columns equally close to the center. Choosing to be any of these central rows or columns, and removing from the graph, partitions the graph into two smaller connected subgraphs and , each of which has at most vertices. If (as in the illustration), then choosing a central column will give a separator with r \leq \sqrt vertices, and similarly if then choosing a central row will give a separator with at most \sqrt vertices. Thus, every grid graph has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connectivity (graph Theory)
In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) that need to be removed to separate the remaining nodes into two or more Connected component (graph theory), isolated subgraphs. It is closely related to the theory of flow network, network flow problems. The connectivity of a graph is an important measure of its resilience as a network. Connected vertices and graphs In an undirected graph , two vertex (graph theory), vertices and are called connected if contains a Path (graph theory), path from to . Otherwise, they are called disconnected. If the two vertices are additionally connected by a path of length (that is, they are the endpoints of a single edge), the vertices are called adjacent. A Graph (discrete mathematics), graph is said to be connected if every pair of vertices in the graph is connected. This means that there is a Path (graph theory), path between every ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clique (graph Theory)
In graph theory, a clique ( or ) is a subset of vertices of an undirected graph such that every two distinct vertices in the clique are adjacent. That is, a clique of a graph G is an induced subgraph of G that is complete. Cliques are one of the basic concepts of graph theory and are used in many other mathematical problems and constructions on graphs. Cliques have also been studied in computer science: the task of finding whether there is a clique of a given size in a graph (the clique problem) is NP-complete, but despite this hardness result, many algorithms for finding cliques have been studied. Although the study of complete subgraphs goes back at least to the graph-theoretic reformulation of Ramsey theory by , the term ''clique'' comes from , who used complete subgraphs in social networks to model cliques of people; that is, groups of people all of whom know each other. Cliques have many other applications in the sciences and particularly in bioinformatics. Definiti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degree (graph Theory)
In graph theory, the degree (or valency) of a vertex of a graph is the number of edges that are incident to the vertex; in a multigraph, a loop contributes 2 to a vertex's degree, for the two ends of the edge. The degree of a vertex v is denoted \deg(v) or \deg v. The maximum degree of a graph G is denoted by \Delta(G), and is the maximum of G's vertices' degrees. The minimum degree of a graph is denoted by \delta(G), and is the minimum of G's vertices' degrees. In the multigraph shown on the right, the maximum degree is 5 and the minimum degree is 0. In a regular graph, every vertex has the same degree, and so we can speak of ''the'' degree of the graph. A complete graph (denoted K_n, where n is the number of vertices in the graph) is a special kind of regular graph where all vertices have the maximum possible degree, n-1. In a signed graph, the number of positive edges connected to the vertex v is called positive deg(v) and the number of connected negative edges is enti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Graph
In graph theory, a regular graph is a Graph (discrete mathematics), graph where each Vertex (graph theory), vertex has the same number of neighbors; i.e. every vertex has the same Degree (graph theory), degree or valency. A regular directed graph must also satisfy the stronger condition that the indegree and outdegree of each internal vertex are equal to each other. A regular graph with vertices of degree is called a graph or regular graph of degree . Special cases Regular graphs of degree at most 2 are easy to classify: a graph consists of disconnected vertices, a graph consists of disconnected edges, and a graph consists of a disjoint union of graphs, disjoint union of cycle (graph theory), cycles and infinite chains. A graph is known as a cubic graph. A strongly regular graph is a regular graph where every adjacent pair of vertices has the same number of neighbors in common, and every non-adjacent pair of vertices has the same number of neighbors in common. The smal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangular Number
A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in the triangular arrangement with dots on each side, and is equal to the sum of the natural numbers from 1 to . The first 100 terms sequence of triangular numbers, starting with the 0th triangular number, are Formula The triangular numbers are given by the following explicit formulas: where \textstyle is notation for a binomial coefficient. It represents the number of distinct pairs that can be selected from objects, and it is read aloud as " plus one choose two". The fact that the nth triangular number equals n(n+1)/2 can be illustrated using a visual proof. For every triangular number T_n, imagine a "half-rectangle" arrangement of objects corresponding to the triangular number, as in the figure below. Copying this arrangement ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
