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Lattice Graph
In graph theory, a lattice graph, mesh graph, or grid graph is a graph whose drawing, embedded in some Euclidean space , forms a regular tiling. This implies that the group of bijective transformations that send the graph to itself is a lattice in the group-theoretical sense. Typically, no clear distinction is made between such a graph in the more abstract sense of graph theory, and its drawing in space (often the plane or 3D space). This type of graph may more shortly be called just a lattice, mesh, or grid. Moreover, these terms are also commonly used for a finite section of the infinite graph, as in "an 8 × 8 square grid". The term lattice graph has also been given in the literature to various other kinds of graphs with some regular structure, such as the Cartesian product of a number of complete graphs. Square grid graph A common type of a lattice graph (known under different names, such as square grid graph) is the graph whose vertices correspond to the p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Grid Graph
In graph theory, a lattice graph, mesh graph, or grid graph is a Graph (discrete mathematics), graph whose graph drawing, drawing, Embedding, embedded in some Euclidean space , forms a regular tiling. This implies that the group (mathematics), group of Bijection, bijective transformations that send the graph to itself is a lattice (group), lattice in the group-theoretical sense. Typically, no clear distinction is made between such a graph in the more abstract sense of graph theory, and its drawing in space (often the plane or 3D space). This type of graph may more shortly be called just a lattice, mesh, or grid. Moreover, these terms are also commonly used for a finite section of the infinite graph, as in "an 8 × 8 square grid". The term lattice graph has also been given in the literature to various other kinds of graphs with some regular structure, such as the Cartesian product of graphs, Cartesian product of a number of complete graphs. Square grid graph A comm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipartite Graph
In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets U and V, that is every edge connects a vertex in U to one in V. Vertex sets U and V are usually called the ''parts'' of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles. The two sets U and V may be thought of as a coloring of the graph with two colors: if one colors all nodes in U blue, and all nodes in V red, each edge has endpoints of differing colors, as is required in the graph coloring problem.. In contrast, such a coloring is impossible in the case of a non-bipartite graph, such as a triangle: after one node is colored blue and another red, the third vertex of the triangle is connected to vertices of both colors, preventing it from being assigned either color. One often writes G=(U,V,E) to denote a bipartite graph whose partition has the parts U and V, with E denoting ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer Triangle
An integer triangle or integral triangle is a triangle all of whose sides have lengths that are integers. A rational triangle can be defined as one having all sides with rational Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ... length; any such rational triangle can be integrally rescaled (can have all sides multiplied by the same integer, namely a common multiple of their denominators) to obtain an integer triangle, so there is no substantive difference between integer triangles and rational triangles in this sense. However, other definitions of the term "rational triangle" also exist: In 1914 Carmichael used the term in the sense that we today use the term Heronian triangle; SomosSomos, M., "Rational triangles", http://grail.eecs.csuohio.edu/~somos/rattri.html uses it to refer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pick's Theorem
In geometry, Pick's theorem provides a formula for the area of a simple polygon with integer vertex coordinates, in terms of the number of integer points within it and on its boundary. The result was first described by Georg Alexander Pick in 1899. It was popularized in English by Hugo Steinhaus in the 1950 edition of his book ''Mathematical Snapshots''. It has multiple proofs, and can be generalized to formulas for certain kinds of non-simple polygons. Formula Suppose that a polygon has integer coordinates for all of its vertices. Let i be the number of integer points interior to the polygon, and let b be the number of integer points on its boundary (including both vertices and points along the sides). Then the area A of this polygon is: A = i + \frac - 1. The example shown has i=7 interior points and b=8 boundary points, so its area is A=7+\tfrac-1=10 square units. Proofs Via Euler's formula One proof of this theorem involves subdividing the polygon into triangles with three ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lattice Path
In combinatorics, a lattice path in the -dimensional integer lattice of length with steps in the set , is a sequence of vectors such that each consecutive difference v_i - v_ lies in . A lattice path may lie in any lattice in , but the integer lattice is most commonly used. An example of a lattice path in of length 5 with steps in S = \lbrace (2,0), (1,1), (0,-1) \rbrace is L = \lbrace (-1,-2), (0,-1), (2,-1), (2,-2), (2,-3), (4,-3) \rbrace . North-East lattice paths A North-East (NE) lattice path is a lattice path in \mathbb^2 with steps in S = \lbrace (0,1), (1,0) \rbrace . The (0,1) steps are called North steps and denoted by N 's; the (1,0) steps are called East steps and denoted by E 's. NE lattice paths most commonly begin at the origin. This convention allows us to encode all the information about a NE lattice path L in a single permutation word. The length of the word gives us the number of steps of the lattice path, k . The order of the N 's an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wazir (chess)
The wazir or vazir ''(word used for describing Queen in Indian chess)'' is a fairy chess piece that may move a single square vertically or horizontally. In notation, it is given the symbol ''W''. In this article, the wazir is represented by an inverted rook. Name etymology The name wazīr (vazir) (Arabic/Persian: وزير from Middle Persian vichir) means "minister" in several West and South Asian languages and is found in English as vizier. Wazīr (Vazir) is also the Arabic and Persian name of the queen. History and nomenclature The wazir is a very old piece, appearing in some very early chess variants, such as Tamerlane chess. The wazir also appears in some historical large shogi variants, such as in dai shogi under the name ''angry boar'' (嗔猪 ''shinchō''). The general in xiangqi moves like a wazir but may not leave its palace or end its turn in check. Value The wazir by itself is not much more powerful than a pawn, but as an additional power to other pieces, it is worth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fairy Chess Piece
A fairy chess piece, variant chess piece, unorthodox chess piece, or heterodox chess piece is a chess piece not used in conventional chess but incorporated into certain chess variants and some chess problems. Compared to conventional pieces, fairy pieces vary mostly in the way they move, but they may also follow special rules for capturing, promotions, etc. Because of the distributed and uncoordinated nature of unorthodox chess development, the same piece can have different names, and different pieces can have the same name in various contexts. Most are symbolised as inverted or rotated icons of the standard pieces in diagrams, and the meanings of these "wildcards" must be defined in each context separately. Pieces invented for use in chess variants rather than problems sometimes instead have special icons designed for them, but with some exceptions (the princess, empress, and occasionally amazon), many of these are not used beyond the individual games for which they were invented ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chessboard
A chessboard is a used to play chess. It consists of 64 squares, 8 rows by 8 columns, on which the chess pieces are placed. It is square in shape and uses two colours of squares, one light and one dark, in a chequered pattern. During play, the board is oriented such that each player's near-right corner square is a light square. The columns of a chessboard are known as ', the rows are known as ', and the lines of adjoining same-coloured squares (each running from one edge of the board to an adjacent edge) are known as '. Each square of the board is named using algebraic, descriptive, or numeric chess notation; algebraic notation is the FIDE standard. In algebraic notation, using White's perspective, files are labeled ''a'' through ''h'' from left to right, and ranks are labeled ''1'' through ''8'' from bottom to top; each square is identified by the file and rank which it occupies. The a- through d-files comprise the , while the e- through h-files comprise the . History and evo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chess Piece
A chess piece, or chessman, is a game piece that is placed on a chessboard to play the game of chess. It can be either White and Black in chess, white or black, and it can be one of six types: King (chess), king, Queen (chess), queen, Rook (chess), rook, Bishop (chess), bishop, Knight (chess), knight, or Pawn (chess), pawn. Chess sets generally come with sixteen pieces of each color. Additional pieces, usually an extra queen per color, may be provided for use in Promotion (chess), promotion. Number of pieces Each player begins with sixteen pieces (but see the #Usage of the term piece, subsection below for other usage of the term ''piece''). The pieces that belong to each player are distinguished by color: the lighter colored pieces are referred to as "white" and the player that owns them as "White", whereas the darker colored pieces are referred to as "black" and the player that owns them as "Black". In a standard game, each of the two players begins with the following sixteen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rook (chess)
The rook (; ♖, ♜) is a piece in the game of chess. It may move any number of squares horizontally or vertically without jumping, and it may an enemy piece on its path; additionally, it may participate in castling. Each player starts the game with two rooks, one in each corner on their own side of the board. Formerly, the rook (from Persian رخ ''rokh''/''rukh'', meaning "chariot") was alternatively called the tower, marquess, rector, and comes (count or earl). The term "castle" is considered to be informal, incorrect, or old-fashioned. Placement and movement The white rooks start on squares a1 and h1, while the black rooks start on a8 and h8. The rook moves horizontally or vertically, through any number of unoccupied squares (see diagram). The rook cannot jump over pieces. The rook may capture an enemy piece by moving to the square on which the enemy piece stands, removing it from play. The rook also participates with the king in a special move called castling, wherein i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rook's Graph
In graph theory, a rook's graph is a 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 each edge connects two squares on the same row (rank) or on the same column (file) as each other, the squares that a rook can move between. These graphs can be constructed for chessboards of any rectangular shape, and can be defined mathematically as the Cartesian product of two complete graphs, as the two-dimensional Hamming graphs, or as the line graphs of complete bipartite 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 every other pair at the same distance (they are distance-transitive). For chessboards with relatively prime dimensions, they are circulant graphs. With one exception, they can be dist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hanan Grid
In geometry, the Hanan grid of a finite set of points in the plane is obtained by constructing vertical and horizontal lines through each point in . The main motivation for studying the Hanan grid stems from the fact that it is known to contain a minimum length rectilinear Steiner tree The rectilinear Steiner tree problem, minimum rectilinear Steiner tree problem (MRST), or rectilinear Steiner minimum tree problem (RSMT) is a variant of the geometric Steiner tree problem in the plane, in which the Euclidean distance is replaced w ... for . It is named after Maurice Hanan, who was first to investigate the rectilinear Steiner minimum tree and introduced this graph.M. HananOn Steiner's problem with rectilinear distance J. SIAM Appl. Math. 14 (1966), 255 - 265. References {{reflist Graph families Geometric graphs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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