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Crown Graph
In graph theory, a branch of mathematics, a crown graph on vertices is an undirected graph with two sets of vertices and and with an edge from to whenever . The crown graph can be viewed as a complete bipartite graph from which the edges of a perfect matching have been removed, as the bipartite double cover of a complete graph, as the tensor product , as the complement of the Cartesian direct product of and , or as a bipartite Kneser graph representing the 1-item and -item subsets of an -item set, with an edge between two subsets whenever one is contained in the other. Examples The 6-vertex crown graph forms a cycle, and the 8-vertex crown graph is isomorphic to the graph of a cube. In the Schläfli double six, a configuration of 12 lines and 30 points in three-dimensional space, the twelve lines intersect each other in the pattern of a 12-vertex crown graph. Properties The number of edges in a crown graph is the pronic number . Its achromatic number is : one ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Crown Graphs
A crown is a traditional form of head adornment, or hat, worn by monarchs as a symbol of their power and dignity. A crown is often, by extension, a symbol of the monarch's government or items endorsed by it. The word itself is used, particularly in Commonwealth countries, as an abstract name for the monarchy itself (and, by extension, the state of which said monarch is head) as distinct from the individual who inhabits it (that is, ''The Crown''). A specific type of crown (or coronet for lower ranks of peerage) is employed in heraldry under strict rules. Indeed, some monarchies never had a physical crown, just a heraldic representation, as in the constitutional kingdom of Belgium. Variations * Costume headgear imitating a monarch's crown is also called a crown hat. Such costume crowns may be worn by actors portraying a monarch, people at costume parties, or ritual "monarchs" such as the king of a Carnival krewe, or the person who found the trinket in a king cake. * The nup ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Achromatic Number
In graph theory, a complete coloring is a (proper) vertex coloring In graph theory, graph coloring is a methodic assignment of labels traditionally called "colors" to elements of a graph. The assignment is subject to certain constraints, such as that no two adjacent elements have the same color. Graph coloring i ... in which every pair of colors appears on ''at least'' one pair of adjacent Vertex (graph theory), vertices. Equivalently, a complete coloring is minimal in the sense that it cannot be transformed into a proper coloring with fewer colors by merging pairs of color classes. The achromatic number of a graph is the maximum number of colors possible in any complete coloring of . A complete coloring is the opposite of a harmonious coloring, which requires every pair of colors to appear on ''at most'' one pair of adjacent vertices. Complexity theory Finding is an optimization problem. The decision problem for complete coloring can be phrased as: :INSTANCE: a graph and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Metric Space
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), function called a metric or distance function. Metric spaces are a general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a Conceptual metaphor , metaphorical, rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of characters that need to be changed to get from one string to another. Since they are very general, metric spaces are a tool used in many different bra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Greedy Coloring
In the study of graph coloring problems in mathematics and computer science, a greedy coloring or sequential coloring is a coloring of the vertices of a graph formed by a greedy algorithm that considers the vertices of the graph in sequence and assigns each vertex its first available color. Greedy colorings can be found in linear time, but they do not, in general, use the minimum number of colors possible. Different choices of the sequence of vertices will typically produce different colorings of the given graph, so much of the study of greedy colorings has concerned how to find a good ordering. There always exists an ordering that produces an optimal coloring, but although such orderings can be found for many special classes of graphs, they are hard to find in general. Commonly used strategies for vertex ordering involve placing higher-degree vertices earlier than lower-degree vertices, or choosing vertices with fewer available colors in preference to vertices that are less ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Ménage Problem
In combinatorics, combinatorial mathematics, the ménage problem or problème des ménages asks for the number of different ways in which it is possible to seat a set of male-female couples at a round dining table so that men and women alternate and nobody sits next to his or her partner. (''Ménage'' is the French language, French word for "household", referring here to a male-female couple.) This problem was formulated in 1891 by Édouard Lucas and independently, a few years earlier, by Peter Guthrie Tait in connection with knot theory. For a number of couples equal to 3, 4, 5, ... the number of seating arrangements is :12, 96, 3120, 115200, 5836320, 382072320, 31488549120, ... . Mathematicians have developed formulas and recurrence equations for computing these numbers and related sequences of numbers. Along with their applications to etiquette and knot theory, these numbers also have a graph theory, graph theoretic interpretation: they count the numbers of matching (graph theor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Hamiltonian Cycle
In the mathematics, mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path (graph theory), path in an undirected or directed graph that visits each vertex (graph theory), vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a cycle (graph theory), cycle that visits each vertex exactly once. A Hamiltonian path that starts and ends at adjacent vertices can be completed by adding one more edge to form a Hamiltonian cycle, and removing any edge from a Hamiltonian cycle produces a Hamiltonian path. The computational problems of determining whether such paths and cycles exist in graphs are NP-complete; see Hamiltonian path problem for details. Hamiltonian paths and cycles are named after William Rowan Hamilton, who invented the icosian game, now also known as ''Hamilton's puzzle'', which involves finding a Hamiltonian cycle in the edge graph of the dodecahedron. Hamilton solved this problem using the icosian calculus, an algebraic structur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Etiquette
Etiquette ( /ˈɛtikɛt, -kɪt/) can be defined as a set of norms of personal behavior in polite society, usually occurring in the form of an ethical code of the expected and accepted social behaviors that accord with the conventions and norms observed and practiced by a society, a social class, or a social group. In modern English usage, the French word ''étiquette'' (label and tag) dates from the year 1750 and also originates from the French word for "ticket," possibly symbolizing a person’s entry into society through proper behavior. There are many important historical figures that have helped to shape the meaning of the term as well as provide varying perspectives. History In , the Ancient Egyptian vizier Ptahhotep wrote '' The Maxims of Ptahhotep'' (), a didactic book of precepts extolling civil virtues such as truthfulness, self-control, and kindness towards other people. Recurrent thematic motifs in the maxims include learning by listening to other people, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Rook's Graph
In graph theory, a rook's graph is an undirected graph that represents all legal moves of the rook chess piece on a chessboard. Each vertex of a rook's graph represents a square on a chessboard, and there is an edge between any two squares sharing a row (rank) or column (file), the squares that a rook can move between. These graphs can be constructed for chessboards of any rectangular shape. Although rook's graphs have only minor significance in chess lore, they are more important in the abstract mathematics of graphs through their alternative constructions: rook's graphs are the Cartesian product of two complete graphs, and are the line graphs of complete bipartite graphs. The square rook's graphs constitute the two-dimensional Hamming graphs. Rook's graphs are highly symmetric, having symmetries taking every vertex to every other vertex. In rook's graphs defined from square chessboards, more strongly, every two edges are symmetric, and every pair of vertices is symmetric to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Cartesian Product Of Graphs
In graph theory, the Cartesian product of graphs and is a graph such that: * the vertex set of is the Cartesian product ; and * two vertices and are adjacent in if and only if either ** and is adjacent to in , or ** and is adjacent to in . The Cartesian product of graphs is sometimes called the box product of graphs arary 1969 The operation is associative, as the graphs and are naturally isomorphic. The operation is commutative as an operation on isomorphism classes of graphs, and more strongly the graphs and are naturally isomorphic, but it is not commutative as an operation on labeled graphs. The notation has often been used for Cartesian products of graphs, but is now more commonly used for another construction known as the tensor product of graphs. The square symbol is intended to be an intuitive and unambiguous notation for the Cartesian product, since it shows visually the four edges resulting from the Cartesian product of two edges. Examples * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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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] [Amazon] |
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Central Binomial Coefficient
In mathematics the ''n''th central binomial coefficient is the particular binomial coefficient : = \frac \textn \geq 0. They are called central since they show up exactly in the middle of the even-numbered rows in Pascal's triangle. The first few central binomial coefficients starting at ''n'' = 0 are: :, , , , , , 924, 3432, 12870, 48620, ...; Combinatorial interpretations and other properties The central binomial coefficient \binom is the number of arrangements where there are an equal number of two types of objects. For example, when n=2, the binomial coefficient \binom is equal to 6, and there are six arrangements of two copies of ''A'' and two copies of ''B'': ''AABB'', ''ABAB'', ''ABBA'', ''BAAB'', ''BABA'', ''BBAA''. The same central binomial coefficient \binom is also the number of words of length 2''n'' made up of ''A'' and ''B'' within which, as one reads from left to right, there are never more ''B'' than ''A'' at any point. For example, when n=2, there are six ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Bipartite Dimension
In the mathematical fields of graph theory and combinatorial optimization, the bipartite dimension or biclique cover number of a graph ''G'' = (''V'', ''E'') is the minimum number of bicliques (that is complete bipartite subgraphs), needed to cover all edges in ''E''. A collection of bicliques covering all edges in ''G'' is called a biclique edge cover, or sometimes biclique cover. The bipartite dimension of ''G'' is often denoted by the symbol ''d''(''G''). Example An example for a biclique edge cover is given in the following diagrams: Image:Bipartite-dimension-bipartite-graph.svg, A bipartite graph... Image:Bipartite-dimension-biclique-cover.svg, ...and a covering with four bicliques Image:Bipartite-dimension-red-biclique.svg, the red biclique from the cover Image:Bipartite-dimension-blue-biclique.svg, the blue biclique from the cover Image:Bipartite-dimension-green-biclique.svg, the green biclique from the cover Image:Bipartite-dimension-black-biclique.svg, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |