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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] |
Integral Graph
In the mathematical field of graph theory, an integral graph is a graph whose adjacency matrix's spectrum consists entirely of integers. In other words, a graph is an integral graph if all of the roots of the characteristic polynomial of its adjacency matrix are integers. The notion was introduced in 1974 by Frank Harary and Allen Schwenk. Examples *The complete graph ''Kn'' is integral for all ''n''. *The only cycle graphs that are integral are C_3, C_4, and C_6. *If a graph is integral, then so is its complement graph; for instance, the complements of complete graphs, edgeless graphs, are integral. If two graphs are integral, then so is their Cartesian product and strong product; for instance, the Cartesian products of two complete graphs, the rook's graphs, are integral. Similarly, the hypercube graphs, as Cartesian products of any number of complete graphs K_2, are integral. *The line graph of a regular integral graph is again integral. For instance, as the line graph of K ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distance-transitive Graph
In the mathematical field of graph theory, a distance-transitive graph is a graph such that, given any two vertices and at any distance , and any other two vertices and at the same distance, there is an automorphism of the graph that carries to and to . Distance-transitive graphs were first defined in 1971 by Norman L. Biggs and D. H. Smith. A distance-transitive graph is interesting partly because it has a large automorphism group. Some interesting finite groups are the automorphism groups of distance-transitive graphs, especially of those whose diameter is 2. Examples Some first examples of families of distance-transitive graphs include: * The Johnson graphs. * The Grassmann graphs. * The Hamming Graphs (including Hypercube graphs). * The folded cube graphs. * The square rook's graphs. * The Livingstone graph. Classification of cubic distance-transitive graphs After introducing them in 1971, Biggs Biggs may refer to: Arts and entertainment * B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangular Duoprism YW And ZW Rotations
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimensional line segments. A triangle has three internal angles, each one bounded by a pair of adjacent edges; the sum of angles of a triangle always equals a straight angle (180 degrees or π radians). The triangle is a plane figure and its interior is a planar region. Sometimes an arbitrary edge is chosen to be the ''base'', in which case the opposite vertex is called the ''apex''; the shortest segment between the base and apex is the ''height''. The area of a triangle equals one-half the product of height and base length. In Euclidean geometry, any two points determine a unique line segment situated within a unique straight line, and any three points that do not all lie on the same straight line determine a unique triangle situated within a u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sudoku Graph
In the mathematics of Sudoku, the Sudoku graph is an undirected graph whose vertices represent the cells of a (blank) Sudoku puzzle and whose edges represent pairs of cells that belong to the same row, column, or block of the puzzle. The problem of solving a Sudoku puzzle can be represented as precoloring extension on this graph. It is an integral Cayley graph. Basic properties and examples On a Sudoku board of size n^2\times n^2, the Sudoku graph has n^4 vertices, each with exactly 3n^2-2n-1 neighbors. Therefore, it is a regular graph. The total number of edges is n^4(3n^2-2n-1)/2. For instance, the graph shown in the figure above, for a 4\times 4 board, has 16 vertices and 56 edges, and is 7-regular. For the most common form of Sudoku, on a 9\times 9 board, the Sudoku graph is a 20-regular graph with 81 vertices and 810 edges. The second figure shows how to count the neighbors of each cell in a 9\times 9 board. Puzzle solutions and graph coloring Each row, column, or block ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Latin Square
Latin ( or ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken by the Latins in Latium (now known as Lazio), the lower Tiber area around Rome, Italy. Through the expansion of the Roman Republic, it became the dominant language in the Italian Peninsula and subsequently throughout the Roman Empire. It has greatly influenced many languages, including English, having contributed many words to the English lexicon, particularly after the Christianization of the Anglo-Saxons and the Norman Conquest. Latin roots appear frequently in the technical vocabulary used by fields such as theology, the sciences, medicine, and law. By the late Roman Republic, Old Latin had evolved into standardized Classical Latin. Vulgar Latin refers to the less prestigious colloquial registers, attested in inscriptions and some literary works such as those of the comic playwrights Plautus and Terence and the author Petronius. Whil ... [...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] |
Domination Number
Domination or dominant may refer to: Society * World domination, structure where one dominant power governs the planet * Colonialism in which one group (usually a nation) invades another region for material gain or to eliminate competition * Chauvinism in which a person or group consider themselves to be superior, and thus entitled to use force to dominate others * Sexual dominance involving individuals in a subset of BDSM behaviour * Hierarchy Music * Dominant (music), a diatonic scale step and diatonic function in tonal music theory Albums * ''Domination'' (Cannonball Adderley album) or the title track, 1965 * ''Domination'' (Morbid Angel album), 1995 * ''Domination'', by Domino, 2004 * ''Domination'', by Morifade, 2004 Songs * "Domination" (song), by Pantera, 1990 * "Domination", by Band-Maid from ''World Domination'', 2018 * "Domination", by Symphony X from '' Paradise Lost'', 2007 * "Domination", by Way Out West from '' Way Out West'', 1996 * "Domination", by Within ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Independence Number
Independence is a condition of a nation, country, or state, in which residents and population, or some portion thereof, exercise self-government, and usually sovereignty, over its territory. The opposite of independence is the status of a dependent territory or colony. The commemoration of the independence day of a country or nation celebrates when a country is free from all forms of colonialism; free to build a country or nation without any interference from other nations. Definition Whether the attainment of independence is different from revolution has long been contested, and has often been debated over the question of violence as legitimate means to achieving sovereignty. In general, revolutions aim only to redistribute power with or without an element of emancipation, such as in democratization ''within'' a state, which as such may remain unaltered. For example, the Mexican Revolution (1910) chiefly refers to a multi-factional conflict that eventually led to a new ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strong Perfect Graph Theorem
In graph theory, the strong perfect graph theorem is a forbidden graph characterization of the perfect graphs as being exactly the graphs that have neither odd holes (odd-length induced cycles of length at least 5) nor odd antiholes (complements of odd holes). It was conjectured by Claude Berge in 1961. A proof by Maria Chudnovsky, Neil Robertson, Paul Seymour, and Robin Thomas was announced in 2002 and published by them in 2006. The proof of the strong perfect graph theorem won for its authors a $10,000 prize offered by Gérard Cornuéjols of Carnegie Mellon University and the 2009 Fulkerson Prize. Statement A perfect graph is a graph in which, for every induced subgraph, the size of the maximum clique equals the minimum number of colors in a coloring of the graph; perfect graphs include many well-known graph classes including the bipartite graphs, chordal graphs, and comparability graphs. In his 1961 and 1963 works defining for the first time this class of graphs, Claude ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Induced Subgraph
In graph theory, an induced subgraph of a graph is another graph, formed from a subset of the vertices of the graph and ''all'' of the edges, from the original graph, connecting pairs of vertices in that subset. Definition Formally, let G=(V,E) be any graph, and let S\subseteq V be any subset of vertices of . Then the induced subgraph G is the graph whose vertex set is S and whose edge set consists of all of the edges in E that have both endpoints in S . That is, for any two vertices u,v\in S , u and v are adjacent in G if and only if they are adjacent in G . The same definition works for undirected graphs, directed graphs, and even multigraphs. The induced subgraph G may also be called the subgraph induced in G by S , or (if context makes the choice of G unambiguous) the induced subgraph of S . Examples Important types of induced subgraphs include the following. * Induced paths are induced subgraphs that are paths. The shortest path between any two vertices in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clique Number
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. Definitions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Coloring
In graph theory, graph coloring is a methodic assignment of labels traditionally called "colors" to elements of a Graph (discrete mathematics), graph. The assignment is subject to certain constraints, such as that no two adjacent elements have the same color. Graph coloring is a special case of graph labeling. In its simplest form, it is a way of coloring the Vertex (graph theory), vertices of a graph such that no two adjacent vertices are of the same color; this is called a vertex coloring. Similarly, an ''edge coloring'' assigns a color to each Edge (graph theory), edges so that no two adjacent edges are of the same color, and a face coloring of a planar graph assigns a color to each Face (graph theory), face (or region) so that no two faces that share a boundary have the same color. Vertex coloring is often used to introduce graph coloring problems, since other coloring problems can be transformed into a vertex coloring instance. For example, an edge coloring of a graph is just ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
