|
Petersen Graph
In the mathematical field of graph theory, the Petersen graph is an undirected graph with 10 vertices and 15 edges. It is a small graph that serves as a useful example and counterexample for many problems in graph theory. The Petersen graph is named after Julius Petersen, who in 1898 constructed it to be the smallest bridgeless cubic graph with no three- edge-coloring. Although the graph is generally credited to Petersen, it had in fact first appeared 12 years earlier, in a paper by . Kempe observed that its vertices can represent the ten lines of the Desargues configuration, and its edges represent pairs of lines that do not meet at one of the ten points of the configuration. Donald Knuth states that the Petersen graph is "a remarkable configuration that serves as a counterexample to many optimistic predictions about what might be true for graphs in general." The Petersen graph also makes an appearance in tropical geometry. The cone over the Petersen graph is naturally ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Julius Petersen
Julius Peter Christian Petersen (16 June 1839 in Sorø, West Zealand – 5 August 1910 in Copenhagen) was a Denmark, Danish mathematician. His contributions to the field of mathematics led to the birth of graph theory. Biography Petersen's interests in mathematics were manifold, including: geometry, complex analysis, number theory, mathematical physics, mathematical economics, cryptography and graph theory. His famous paper ''Die Theorie der regulären graphs'' was a fundamental contribution to modern graph theory as we know it today. In 1898, he presented a counterexample to Peter Guthrie Tait, Tait's claimed theorem about 1-factorability of 3-regular graphs, which is nowadays known as the "Petersen graph". In cryptography and mathematical economics he made contributions which today are seen as pioneering. He published a systematic treatment of geometry, geometrical constructions (with straightedge and compass) in 1880. A French language, French translation was reprinted in 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tropical Geometry
In mathematics, tropical geometry is the study of polynomials and their geometric properties when addition is replaced with minimization and multiplication is replaced with ordinary addition: : x\oplus y=\min\, : x\otimes y=x+y. So for example, the classical polynomial x^3+xy+y^4 would become \min\. Such polynomials and their solutions have important applications in optimization problems, for example the problem of optimizing departure times for a network of trains. Tropical geometry is a variant of algebraic geometry in which polynomial graphs resemble piecewise linear meshes, and in which numbers belong to the tropical semiring instead of a field. Because classical and tropical geometry are closely related, results and methods can be converted between them. Algebraic varieties can be mapped to a tropical counterpart and, since this process still retains some geometric information about the original variety, it can be used to help prove and generalize classical results from ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Matching
In graph theory, a perfect matching in a graph is a matching that covers every vertex of the graph. More formally, given a graph with edges and vertices , a perfect matching in is a subset of , such that every vertex in is adjacent to exactly one edge in . The adjacency matrix of a perfect matching is a symmetric permutation matrix. A perfect matching is also called a 1-factor; see Graph factorization for an explanation of this term. In some literature, the term complete matching is used. Every perfect matching is a maximum-cardinality matching, but the opposite is not true. For example, consider the following graphs:Alan Gibbons, Algorithmic Graph Theory, Cambridge University Press, 1985, Chapter 5. : In graph (b) there is a perfect matching (of size 3) since all 6 vertices are matched; in graphs (a) and (c) there is a maximum-cardinality matching (of size 2) which is not perfect, since some vertices are unmatched. A perfect matching is also a minimum-size edge cov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Bipartite Graph
In the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set..Electronic edition page 17. Graph theory itself is typically dated as beginning with Leonhard Euler's 1736 work on the Seven Bridges of Königsberg. However, drawings of complete bipartite graphs were already printed as early as 1669, in connection with an edition of the works of Ramon Llull edited by Athanasius Kircher. Llull himself had made similar drawings of complete graphs three centuries earlier.. Definition A complete bipartite graph is a graph whose vertices can be partitioned into two subsets and such that no edge has both endpoints in the same subset, and every possible edge that could connect vertices in different subsets is part of the graph. That is, it is a bipartite graph such that for every two vertices and, is an edge in . A complete bipartite graph w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Graph
In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is connected by a pair of unique edges (one in each direction). Graph theory itself is typically dated as beginning with Leonhard Euler's 1736 work on the Seven Bridges of Königsberg. However, drawings of complete graphs, with their vertices placed on the points of a regular polygon, had already appeared in the 13th century, in the work of Ramon Llull. Such a drawing is sometimes referred to as a mystic rose. Properties The complete graph on vertices is denoted by . Some sources claim that the letter in this notation stands for the German word , but the German name for a complete graph, , does not contain the letter , and other sources state that the notation honors the contributions of Kazimierz Kuratowski to graph theory. has edg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minor (graph Theory)
Minor may refer to: Common meanings * Minor (law), a person not under the age of certain legal activities. * Academic minor, a secondary field of study in undergraduate education Mathematics * Minor (graph theory), a relation of one graph to another * Minor (matroid theory), a relation of one matroid to another * Minor (linear algebra), the determinant of a square submatrix Music * Minor chord * Minor interval * Minor key * Minor scale People * Minor (given name), a masculine given name * Minor (surname), a surname Places in the United States * Minor, Alabama, a census-designated place * Minor, Virginia, an unincorporated community * Minor Creek (California) * Minor Creek (Missouri) * Minor Glacier, Wyoming Sports * Minor, a grade in Gaelic games; also, a person who qualifies to play in that grade * Minor league, a sports league not regarded as a premier league ** Minor League Baseball Minor League Baseball (MiLB) is a professional baseball organization ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Planar Graph
In graph theory, a planar graph is a graph (discrete mathematics), graph that can be graph embedding, embedded in the plane (geometry), plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a plane graph, or a planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points. Every graph that can be drawn on a plane can be drawn on the sphere as well, and vice versa, by means of stereographic projection. Plane graphs can be encoded by combinatorial maps or rotation systems. An equivalence class of topologically equivalent drawings on the sphere, usually with addit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dodecahedron
In geometry, a dodecahedron (; ) or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three Kepler–Poinsot polyhedron, regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120. Some dodecahedra have the same combinatorial structure as the regular dodecahedron (in terms of the graph formed by its vertices and edges), but their pentagonal faces are not regular: The #Pyritohedron, pyritohedron, a common crystal form in pyrite, has pyritohedral symmetry, while the #Tetartoid, tetartoid has tetrahedral symmetry. The rhombic dodecahedron can be seen as a limiting case of the pyritohedron, and it has octahedral symmetry. The elongated dodecahedron and trapezo-rhombic dodecahedron variations, along with the rhombic dodecahedra, are space-filling polyhedra, space-filling. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hemi-dodecahedron
In geometry, a hemi-dodecahedron is an abstract polytope, abstract, regular polyhedron, containing half the Face (geometry), faces of a regular dodecahedron. It can be realized as a projective polyhedron (a tessellation of the real projective plane by 6 pentagons), which can be visualized by constructing the projective plane as a wikt:hemisphere, hemisphere where opposite points along the boundary are connected and dividing the hemisphere into three equal parts. It has 6 pentagonal faces, 15 edges, and 10 vertices. Projections It can be projected symmetrically inside of a 10-sided or 12-sided perimeter: : Petersen graph From the point of view of graph theory this is an embedding of the Petersen graph on a real projective plane. With this embedding, the dual graph is ''K''6 (the complete graph with 6 vertices) --- see hemi-icosahedron. See also *57-cell – an abstract regular 4-polytope constructed from 57 hemi-dodecahedra. *hemi-icosahedron *hemi-cube (geometry), hemi-cube ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Odd Graph
In the mathematics, mathematical field of graph theory, the odd graphs are a family of symmetric graphs defined from certain set systems. They include and generalize the Petersen graph. The odd graphs have high odd girth, meaning that they contain long parity (mathematics), odd-length cycle (graph theory), cycles but no short ones. However their name comes not from this property, but from the fact that each edge (graph theory), edge in the graph has an "odd man out", an element that does not participate in the two sets connected by the edge. Definition and examples The odd graph O_n has one vertex (graph theory), vertex for each of the (n-1)-element subsets of a (2n-1)-element set. Two vertices are connected by an edge if and only if the corresponding subsets are disjoint sets, disjoint. That is, O_n is the Kneser graph KG(2n-1,n-1). O_2 is a triangle, while O_3 is the familiar Petersen graph. The generalized odd graphs are defined as distance-regular graphs with diameter (grap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subset
In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ''B''. The relationship of one set being a subset of another is called inclusion (or sometimes containment). ''A'' is a subset of ''B'' may also be expressed as ''B'' includes (or contains) ''A'' or ''A'' is included (or contained) in ''B''. A ''k''-subset is a subset with ''k'' elements. When quantified, A \subseteq B is represented as \forall x \left(x \in A \Rightarrow x \in B\right). One can prove the statement A \subseteq B by applying a proof technique known as the element argument:Let sets ''A'' and ''B'' be given. To prove that A \subseteq B, # suppose that ''a'' is a particular but arbitrarily chosen element of A # show that ''a'' is an element of ''B''. The validity of this technique ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kneser Graph
Kneser is a surname. Notable people with the surname include: * Adolf Kneser (1862–1930), mathematician * Hellmuth Kneser (1898–1973), mathematician, son of Adolf Kneser * Martin Kneser (1928–2004), mathematician, son of Hellmuth Kneser {{surname ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
