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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] |
4x4 Sudoku Graph
Four-wheel drive, also called 4×4 ("four by four") or 4WD, refers to a two-axled vehicle drivetrain capable of providing torque to all of its wheels simultaneously. It may be full-time or on-demand, and is typically linked via a transfer case providing an additional output drive shaft and, in many instances, additional gear ranges. A four-wheel drive vehicle with torque supplied to both axles is described as "all-wheel drive" (AWD). However, "four-wheel drive" typically refers to a set of specific components and functions, and intended off-road application, which generally complies with modern use of the terminology. Definitions Four-wheel-drive systems were developed in many different markets and used in many different vehicle platforms. There is no universally accepted set of terminology that describes the various architectures and functions. The terms used by various manufacturers often reflect marketing rather than engineering considerations or significant technical diff ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Graph Theory
In mathematics, spectral graph theory is the study of the properties of a graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors of matrices associated with the graph, such as its adjacency matrix or Laplacian matrix. The adjacency matrix of a simple undirected graph is a real symmetric matrix and is therefore orthogonally diagonalizable; its eigenvalues are real algebraic integers. While the adjacency matrix depends on the vertex labeling, its spectrum is a graph invariant, although not a complete one. Spectral graph theory is also concerned with graph parameters that are defined via multiplicities of eigenvalues of matrices associated to the graph, such as the Colin de Verdière number. Cospectral graphs Two graphs are called cospectral or isospectral if the adjacency matrices of the graphs are isospectral, that is, if the adjacency matrices have equal multisets of eigenvalues. Cospectral graphs need not be isomorphic, but isomorphic gr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parametric Families Of Graphs
Parametric may refer to: Mathematics *Parametric equation, a representation of a curve through equations, as functions of a variable *Parametric statistics, a branch of statistics that assumes data has come from a type of probability distribution * Parametric derivative, a type of derivative in calculus *Parametric model, a family of distributions that can be described using a finite number of parameters *Parametric oscillator, a harmonic oscillator whose parameters oscillate in time *Parametric surface, a particular type of surface in the Euclidean space R3 *Parametric family, a family of objects whose definitions depend on a set of parameters Science * Parametric process, in optical physics, any process in which an interaction between light and matter does not change the state of the material *Spontaneous parametric down-conversion, in quantum optics, a source of entangled photon pairs and of single photons *Optical parametric amplifier, a type of laser light source that em ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Notices Of The American Mathematical Society
''Notices of the American Mathematical Society'' is the membership journal of the American Mathematical Society (AMS), published monthly except for the combined June/July issue. The first volume appeared in 1953. Each issue of the magazine since January 1995 is available in its entirety on the journal web site. Articles are peer-reviewed by an editorial board of mathematical experts. Since 2019, the editor-in-chief is Erica Flapan. The cover regularly features mathematical visualizations. The ''Notices'' is self-described to be the world's most widely read mathematical journal. As the membership journal of the American Mathematical Society, the ''Notices'' is sent to the approximately 30,000 AMS members worldwide, one-third of whom reside outside the United States. By publishing high-level exposition, the ''Notices'' provides opportunities for mathematicians to find out what is going on in the field. Each issue contains one or two such expository articles that describe current d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brouwer–Haemers Graph
In the mathematical field of graph theory, the Brouwer–Haemers graph is a 20- regular undirected graph with 81 vertices and 810 edges. It is a strongly regular graph, a distance-transitive graph, and a Ramanujan graph. Although its construction is folklore, it was named after Andries Brouwer and Willem H. Haemers, who proved its uniqueness as a strongly regular graph. Construction The Brouwer–Haemers graph has several related algebraic constructions. One of the simplest is as a degree-4 generalized Paley graph: it can be defined by making a vertex for each element in the finite field GF(81) and an edge for every two elements that differ by a fourth power. Properties The Brouwer–Haemers graph is the unique strongly regular graph with parameters (81, 20, 1, 6). This means that it has 81 vertices, 20 edges per vertex, 1 triangle per edge, and 6 length-two paths connecting each non-adjacent pair of vertices. As a strongly regular graph with the ... [...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] |
Abelian Group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel. The concept of an abelian group underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood and fully classified. Definition An abelian group is a set A, together with an operation \cdot that combines any two elements a and b of A to form another element of A, denoted a \cdot b. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by \lambda, is the factor by which the eigenvector is scaled. Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. Loosely speaking, in a multidimensional vector space, the eigenvector is not rotated. Formal definition If is a linear transformation from a vector space over a field into itself and is a nonzero vector in , then is an eigenvector of if is a scalar multiple of . This can be written as T(\mathbf) = \lambda \mathbf, where is a scalar in , known as the eigenvalue, characteristic value, or characteristic roo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjacency Matrix
In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph. In the special case of a finite simple graph, the adjacency matrix is a (0,1)-matrix with zeros on its diagonal. If the graph is undirected (i.e. all of its edges are bidirectional), the adjacency matrix is symmetric. The relationship between a graph and the eigenvalues and eigenvectors of its adjacency matrix is studied in spectral graph theory. The adjacency matrix of a graph should be distinguished from its incidence matrix, a different matrix representation whose elements indicate whether vertex–edge pairs are incident or not, and its degree matrix, which contains information about the degree of each vertex. Definition For a simple graph with vertex set , the adjacency matrix is a square matrix such that its element is one when there is an edge from vertex to vert ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Coloring
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 subject to certain constraints. In its simplest form, it is a way of coloring the 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 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 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 a vertex coloring of its line graph, and a face coloring of a plane graph is just a vertex coloring of its dual. However, non-vertex coloring problems are often stated and studied as-is. This is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics Of Sudoku
The mathematics of Sudoku refers to the use of mathematics to study Sudoku puzzles to answer questions such as ''"How many filled Sudoku grids are there?"'', "''What is the minimal number of clues in a valid puzzle?''" and ''"In what ways can Sudoku grids be symmetric?"'' through the use of combinatorics and group theory. The analysis of Sudoku falls is generally divided between analyzing the properties of unsolved puzzles (such as the minimum possible number of given clues) and analyzing the properties of solved puzzles. Initial analysis was largely focused on enumerating solutions, with results first appearing in 2004. For classical Sudoku, the number of filled grids is 6,670,903,752,021,072,936,960 (), which reduces to 5,472,730,538 essentially different solutions under the validity preserving transformations. There are 26 types of symmetry, but they can only be found in about 0.005% of all filled grids. A puzzle with a unique solution must have at least 17 clues, and the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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