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Bidiakis Cube
In the mathematical field of graph theory, the bidiakis cube is a 3-regular graph with 12 vertices and 18 edges. Construction The bidiakis cube is a cubic Hamiltonian graph and can be defined by the LCF notation 6,4,-4sup>4. The bidiakis cube can also be constructed from a cube by adding edges across the top and bottom faces which connect the centres of opposite sides of the faces. The two additional edges need to be perpendicular to each other. With this construction, the bidiakis cube is a polyhedral graph, and can be realized as a convex polyhedron. Therefore, by Steinitz's theorem, it is a 3-vertex-connected simple planar graph.Branko Grünbaum, '' Convex Polytopes'', 2nd edition, prepared by Volker Kaibel, Victor Klee, and Günter M. Ziegler, 2003, , , 466pp. Algebraic properties The bidiakis cube is not a vertex-transitive graph and its full automorphism group is isomorphic to the dihedral group of order 8, the group of symmetries of a square, including both rotations a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K-vertex-connected Graph
In graph theory, a connected graph is said to be -vertex-connected (or -connected) if it has more than vertices and remains connected whenever fewer than vertices are removed. The vertex-connectivity, or just connectivity, of a graph is the largest for which the graph is -vertex-connected. Definitions A graph (other than a complete graph) has connectivity ''k'' if ''k'' is the size of the smallest subset of vertices such that the graph becomes disconnected if you delete them. Complete graphs are not included in this version of the definition since they cannot be disconnected by deleting vertices. The complete graph with ''n'' vertices has connectivity ''n'' − 1, as implied by the first definition. An equivalent definition is that a graph with at least two vertices is ''k''-connected if, for every pair of its vertices, it is possible to find ''k'' vertex-independent paths connecting these vertices; see Menger's theorem . This definition produces the same ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Individual Graphs
An individual is that which exists as a distinct entity. Individuality (or self-hood) is the state or quality of being an individual; particularly (in the case of humans) of being a person unique from other people and possessing one's own needs or goals, rights and responsibilities. The concept of an individual features in diverse fields, including biology, law, and philosophy. Etymology From the 15th century and earlier (and also today within the fields of statistics and metaphysics) ''individual'' meant " indivisible", typically describing any numerically singular thing, but sometimes meaning "a person". From the 17th century on, ''individual'' has indicated separateness, as in individualism. Law Although individuality and individualism are commonly considered to mature with age/time and experience/wealth, a sane adult human being is usually considered by the state as an "individual person" in law, even if the person denies individual culpability ("I followed instruct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chromatic Index
In graph theory, an edge coloring of a graph is an assignment of "colors" to the edges of the graph so that no two incident edges have the same color. For example, the figure to the right shows an edge coloring of a graph by the colors red, blue, and green. Edge colorings are one of several different types of graph coloring. The edge-coloring problem asks whether it is possible to color the edges of a given graph using at most different colors, for a given value of , or with the fewest possible colors. The minimum required number of colors for the edges of a given graph is called the chromatic index of the graph. For example, the edges of the graph in the illustration can be colored by three colors but cannot be colored by two colors, so the graph shown has chromatic index three. By Vizing's theorem, the number of colors needed to edge color a simple graph is either its maximum degree or . For some graphs, such as bipartite graphs and high-degree planar graphs, the number of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chromatic Number
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] |
Characteristic Polynomial
In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The characteristic polynomial of an endomorphism of a finite-dimensional vector space is the characteristic polynomial of the matrix of that endomorphism over any base (that is, the characteristic polynomial does not depend on the choice of a basis). The characteristic equation, also known as the determinantal equation, is the equation obtained by equating the characteristic polynomial to zero. In spectral graph theory, the characteristic polynomial of a graph is the characteristic polynomial of its adjacency matrix. Motivation In linear algebra, eigenvalues and eigenvectors play a fundamental role, since, given a linear transformation, an eigenvector is a vector whose direction is not changed by the transformation, and the corresponding eigenva ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square (geometry)
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adjacent sides. It is the only regular polygon whose internal angle, central angle, and external angle are all equal (90°), and whose diagonals are all equal in length. A square with vertices ''ABCD'' would be denoted . Characterizations A convex quadrilateral is a square if and only if it is any one of the following: * A rectangle with two adjacent equal sides * A rhombus with a right vertex angle * A rhombus with all angles equal * A parallelogram with one right vertex angle and two adjacent equal sides * A quadrilateral with four equal sides and four right angles * A quadrilateral where the diagonals are equal, and are the perpendicular bisectors of each other (i.e., a rhombus with equal diagonals) * A convex quadrilateral with successiv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex-transitive Graph
In the mathematical field of graph theory, a vertex-transitive graph is a graph in which, given any two vertices and of , there is some automorphism :f : G \to G\ such that :f(v_1) = v_2.\ In other words, a graph is vertex-transitive if its automorphism group acts transitively on its vertices.. A graph is vertex-transitive if and only if its graph complement is, since the group actions are identical. Every symmetric graph without isolated vertices is vertex-transitive, and every vertex-transitive graph is regular. However, not all vertex-transitive graphs are symmetric (for example, the edges of the truncated tetrahedron), and not all regular graphs are vertex-transitive (for example, the Frucht graph and Tietze's graph). Finite examples Finite vertex-transitive graphs include the symmetric graphs (such as the Petersen graph, the Heawood graph and the vertices and edges of the Platonic solids). The finite Cayley graphs (such as cube-connected cycles) are also ve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Günter M
Gunter or Günter may refer to: * Gunter rig, a type of rig used in sailing, especially in small boats * Gunter Annex, Alabama, a United States Air Force installation * Gunter, Texas, city in the United States People Surname * Chris Gunter (born 1989), Welsh footballer with Cardiff City, Tottenham Hotspur, Nottingham Forest and Reading * Cornell Gunter (1936–1990), American R&B singer, brother of Shirley Gunter * David Gunter (1933–2005), English footballer with Southampton, brother of Phil Gunter * Edmund Gunter (1581–1626), British mathematician and inventor, known for: ** Gunter's chain ** Gunter's rule * James Gunter (1745–1819), English confectioner, fruit grower and scientific gardener * Jen Gunter (born 1966), Canadian-American gynecologist & author * Gordon Gunter (1909–1998), American marine biologist and fisheries scientist * Matthew Alan Gunter (born 1957), United States Episcopal bishop * Phil Gunter (1932–2007), English footballer with Portsmout ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Victor Klee
Victor LaRue Klee, Jr. (September 18, 1925 – August 17, 2007) was a mathematician specialising in convex sets, functional analysis, analysis of algorithms, optimization, and combinatorics. He spent almost his entire career at the University of Washington in Seattle. Life Born in San Francisco, Vic Klee earned his B.A. degree in 1945 with high honors from Pomona College, majoring in mathematics and chemistry. He did his graduate studies, including a thesis on Convex Sets in Linear Spaces, and received his PhD in mathematics from the University of Virginia in 1949. After teaching for several years at the University of Virginia, he moved in 1953 to the University of Washington in Seattle, Washington, where he was a faculty member for 54 years. He died in Lakewood, Ohio. Research Klee wrote more than 240 research papers. He proposed Klee's measure problem and the art gallery problem. Kleetopes are also named after him, as is the Klee–Minty cube, which shows that the simpl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Volker Kaibel
{{disambiguation ...
Volker may refer to: * Volker (name), including a list of people with the given name or surname * Volker, Kansas City, a historic neighborhood in Kansas City * Volker Boulevard, Kansas City * ''Alien Nations'' (German: ''Die Völker''), a real-time strategy video game released in 1999 See also * VolkerWessels, a Dutch construction company ** VolkerRail, a railway infrastructure services company based in Doncaster, England, owned by VolkerWessels * Voelcker (other) * Voelker (other) Voelker is a surname. Notable people with the surname include: *Joe Voelker (Born 1987), and Mike Voelker (Born 1982), Famous brothers from Florida * Bobby Voelker (born 1979), American mixed martial artist * Christopher Voelker (born 1961), Americ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Polytopes
''Convex Polytopes'' is a graduate-level mathematics textbook about convex polytopes, higher-dimensional generalizations of three-dimensional convex polyhedra. It was written by Branko Grünbaum, with contributions from Victor Klee, Micha Perles, and G. C. Shephard, and published in 1967 by John Wiley & Sons. It went out of print in 1970. A second edition, prepared with the assistance of Volker Kaibel, Victor Klee, and Günter M. Ziegler, was published by Springer-Verlag in 2003, as volume 221 of their book series Graduate Texts in Mathematics. ''Convex Polytopes'' was the winner of the 2005 Leroy P. Steele Prize for mathematical exposition, given by the American Mathematical Society. The Basic Library List Committee of the Mathematical Association of America has recommended its inclusion in undergraduate mathematics libraries. Topics The book has 19 chapters. After two chapters introducing background material in linear algebra, topology, and convex geometry, two more chapters ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
