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Incidence Graph
In combinatorial mathematics, a Levi graph or incidence graph is a bipartite graph associated with an incidence structure.. See in particulap. 181 From a collection of points and lines in an incidence geometry or a projective configuration, we form a graph with one vertex per point, one vertex per line, and an edge for every incidence between a point and a line. They are named for Friedrich Wilhelm Levi, who wrote about them in 1942. The Levi graph of a system of points and lines usually has girth at least six: Any 4- cycles would correspond to two lines through the same two points. Conversely any bipartite graph with girth at least six can be viewed as the Levi graph of an abstract incidence structure. Levi graphs of configurations are biregular, and every biregular graph with girth at least six can be viewed as the Levi graph of an abstract configuration.. Levi graphs may also be defined for other types of incidence structure, such as the incidences between points and planes i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Levi Graph Of Pappus Configuration
Levi (; ) was, according to the Book of Genesis, the third of the six sons of Jacob and Leah (Jacob's third son), and the founder of the Israelite Tribe of Levi (the Levites, including the Kohanim) and the great-grandfather of Aaron, Moses and Miriam. Certain religious and political functions were reserved for the Levites. Origins The Torah suggests that the name ''Levi'' refers to Leah's hope for Jacob to ''join'' with her, implying a derivation from ''yillaweh'', meaning ''he will join'', but scholars suspect that it may simply mean ''priest'', either as a loan word from the Minaean ''lawi'u'', meaning ''priest'', or by referring to those people who were ''joined'' to the Ark of the Covenant. Another possibility is that the Levites originated as migrants and that the name Levites indicates their ''joining'' with either the Israelites in general or the earlier Israelite priesthood in particular. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Petersen Graph
In graph theory, the generalized Petersen graphs are a family of cubic graphs formed by connecting the vertices of a regular polygon to the corresponding vertices of a star polygon. They include the Petersen graph and generalize one of the ways of constructing the Petersen graph. The generalized Petersen graph family was introduced in 1950 by H. S. M. Coxeter and was given its name in 1969 by Mark Watkins. Definition and notation In Watkins' notation, ''G''(''n'', ''k'') is a graph with vertex set :\ and edge set :\ where subscripts are to be read modulo ''n'' and ''k'' < ''n''/2. Some authors use the notation ''GPG''(''n'', ''k''). Coxeter's notation for the same graph would be + , a combination of the Schläfli symbols for the regular ''n''-gon and star polygon from which the graph is formed. The Petersen graph itself is ''G''(5, 2) or + . Any generalized Petersen graph can also be constructed from a voltage graph with two vertices, two self-loops, and one other ed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Families Of Sets
Family (from la, familia) is a group of people related either by consanguinity (by recognized birth) or affinity (by marriage or other relationship). The purpose of the family is to maintain the well-being of its members and of society. Ideally, families offer predictability, structure, and safety as members mature and learn to participate in the community. Historically, most human societies use family as the primary locus of attachment, nurturance, and socialization. Anthropologists classify most family organizations as matrifocal (a mother and her children), patrifocal (a father and his children), conjugal (a wife, her husband, and children, also called the nuclear family), avuncular (a man, his sister, and her children), or extended (in addition to parents and children, may include grandparents, aunts, uncles, or cousins). The field of genealogy aims to trace family lineages through history. The family is also an important economic unit studied in family economics. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ljubljana Graph
In the mathematical field of graph theory, the Ljubljana graph is an undirected bipartite graph with 112 vertices and 168 edges. It is a cubic graph with diameter 8, radius 7, chromatic number 2 and chromatic index 3. Its girth is 10 and there are exactly 168 cycles of length 10 in it. There are also 168 cycles of length 12. Construction The Ljubljana graph is Hamiltonian and can be constructed from the LCF notation : [47, -23, -31, 39, 25, -21, -31, -41, 25, 15, 29, -41, -19, 15, -49, 33, 39, -35, -21, 17, -33, 49, 41, 31, -15, -29, 41, 31, -15, -25, 21, 31, -51, -25, 23, 9, -17, 51, 35, -29, 21, -51, -39, 33, -9, -51, 51, -47, -33, 19, 51, -21, 29, 21, -31, -39]2. The Ljubljana graph is the Levi graph of the Ljubljana configuration, a quadrangle-free configuration with 56 lines and 56 points. In this configuration, each line contains exactly 3 points, each point belongs to exactly 3 lines and any two lines intersect in at most one point. Algebraic properties The automorphism ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Möbius Configuration
In geometry, the Möbius configuration or Möbius tetrads is a certain configuration in Euclidean space or projective space, consisting of two mutually inscribed tetrahedra: each vertex of one tetrahedron lies on a face plane of the other tetrahedron and vice versa. Thus, for the resulting system of eight points and eight planes, each point lies on four planes (the three planes defining it as a vertex of a tetrahedron and the fourth plane from the other tetrahedron that it lies on), and each plane contains four points (the three tetrahedron vertices of its face, and the vertex from the other tetrahedron that lies on it). Möbius's theorem The configuration is named after August Ferdinand Möbius, who in 1828 proved that, if two tetrahedra have the property that seven of their vertices lie on corresponding face planes of the other tetrahedron, then the eighth vertex also lies on the plane of its corresponding face, forming a configuration of this type. This incidence theorem is tru ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypercube Graph
In graph theory, the hypercube graph is the graph formed from the vertices and edges of an -dimensional hypercube. For instance, the cube graph is the graph formed by the 8 vertices and 12 edges of a three-dimensional cube. has vertices, edges, and is a regular graph with edges touching each vertex. The hypercube graph may also be constructed by creating a vertex for each subset of an -element set, with two vertices adjacent when their subsets differ in a single element, or by creating a vertex for each -digit binary number, with two vertices adjacent when their binary representations differ in a single digit. It is the -fold Cartesian product of the two-vertex complete graph, and may be decomposed into two copies of connected to each other by a perfect matching. Hypercube graphs should not be confused with cubic graphs, which are graphs that have exactly three edges touching each vertex. The only hypercube graph that is a cubic graph is the cubical graph . Constru ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cremona–Richmond Configuration
In mathematics, the Cremona–Richmond configuration is a configuration of 15 lines and 15 points, having 3 points on each line and 3 lines through each point, and containing no triangles. It was studied by and . It is a generalized quadrangle with parameters (2,2). Its Levi graph is the Tutte–Coxeter graph. Symmetry The points of the Cremona–Richmond configuration may be identified with the 15=\tbinom unordered pairs of elements of a six-element set; these pairs are called ''duads''. Similarly, the lines of the configuration may be identified with the 15 ways of partitioning the same six elements into three pairs; these partitions are called ''synthemes''. Identified in this way, a point of the configuration is incident to a line of the configuration if and only if the duad corresponding to the point is one of the three pairs in the syntheme corresponding to the line. The symmetric group of all permutations of the six elements underlying this system of duads and synthemes act ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tutte Eight-cage
William Thomas Tutte OC FRS FRSC (; 14 May 1917 – 2 May 2002) was an English and Canadian codebreaker and mathematician. During the Second World War, he made a brilliant and fundamental advance in cryptanalysis of the Lorenz cipher, a major Nazi German cipher system which was used for top-secret communications within the Wehrmacht High Command. The high-level, strategic nature of the intelligence obtained from Tutte's crucial breakthrough, in the bulk decrypting of Lorenz-enciphered messages specifically, contributed greatly, and perhaps even decisively, to the defeat of Nazi Germany. He also had a number of significant mathematical accomplishments, including foundation work in the fields of graph theory and matroid theory. Tutte's research in the field of graph theory proved to be of remarkable importance. At a time when graph theory was still a primitive subject, Tutte commenced the study of matroids and developed them into a theory by expanding from the work that Hassler ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gray Graph
Grey (more common in British English) or gray (more common in American English) is an intermediate color between black and white. It is a neutral or achromatic color, meaning literally that it is "without color", because it can be composed of black and white. It is the color of a cloud-covered sky, of ash and of lead. The first recorded use of ''grey'' as a color name in the English language was in 700 CE.Maerz and Paul ''A Dictionary of Color'' New York:1930 McGraw-Hill Page 196 ''Grey'' is the dominant spelling in European and Commonwealth English, while ''gray'' has been the preferred spelling in American English; both spellings are valid in both varieties of English. In Europe and North America, surveys show that grey is the color most commonly associated with neutrality, conformity, boredom, uncertainty, old age, indifference, and modesty. Only one percent of respondents chose it as their favorite color. Etymology ''Grey'' comes from the Middle English or , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Möbius–Kantor Configuration
In geometry, the Möbius–Kantor configuration is a configuration consisting of eight points and eight lines, with three points on each line and three lines through each point. It is not possible to draw points and lines having this pattern of incidences in the Euclidean plane, but it is possible in the complex projective plane. Coordinates asked whether there exists a pair of polygons with ''p'' sides each, having the property that the vertices of one polygon lie on the lines through the edges of the other polygon, and vice versa. If so, the vertices and edges of these polygons would form a projective configuration. For p = 4 there is no solution in the Euclidean plane, but found pairs of polygons of this type, for a generalization of the problem in which the points and edges belong to the complex projective plane In mathematics, the complex projective plane, usually denoted P2(C), is the two-dimensional complex projective space. It is a complex manifold of complex dimension ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Möbius–Kantor Graph
In the mathematical field of graph theory, the Möbius–Kantor graph is a symmetric bipartite cubic graph with 16 vertices and 24 edges named after August Ferdinand Möbius and Seligmann Kantor. It can be defined as the generalized Petersen graph ''G''(8,3): that is, it is formed by the vertices of an octagon, connected to the vertices of an eight-point star in which each point of the star is connected to the points three steps away from it. Möbius–Kantor configuration asked whether there exists a pair of polygons with ''p'' sides each, having the property that the vertices of one polygon lie on the lines through the edges of the other polygon, and vice versa. If so, the vertices and edges of these polygons would form a projective configuration. For ''p'' = 4 there is no solution in the Euclidean plane, but found pairs of polygons of this type, for a generalization of the problem in which the points and edges belong to the complex projective plane. That is, in Kantor's so ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cage (graph Theory)
In the mathematical area of graph theory, a cage is a regular graph that has as few vertices as possible for its girth. Formally, an is defined to be a graph in which each vertex has exactly neighbors, and in which the shortest cycle has length exactly . An is an with the smallest possible number of vertices, among all . A is often called a . It is known that an exists for any combination of and . It follows that all exist. If a Moore graph exists with degree and girth , it must be a cage. Moreover, the bounds on the sizes of Moore graphs generalize to cages: any cage with odd girth must have at least :1+r\sum_^(r-1)^i vertices, and any cage with even girth must have at least :2\sum_^(r-1)^i vertices. Any with exactly this many vertices is by definition a Moore graph and therefore automatically a cage. There may exist multiple cages for a given combination of and . For instance there are three nonisomorphic , each with 70 vertices: the Balaban 10-cage, the Ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
