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Kneser Graph
In graph theory, the Kneser graph (alternatively ) is the graph whose vertices correspond to the -element subsets of a set of elements, and where two vertices are adjacent if and only if the two corresponding sets are disjoint. Kneser graphs are named after Martin Kneser, who first investigated them in 1956. Examples The Kneser graph is the complete graph on vertices. The Kneser graph is the complement of the line graph of the complete graph on vertices. The Kneser graph is the odd graph ; in particular is the Petersen graph (see top right figure). The Kneser graph , visualized on the right. Properties Basic properties The Kneser graph K(n,k) has \tbinom vertices. Each vertex has exactly \tbinom neighbors. The Kneser graph is vertex transitive and arc transitive. When k=2, the Kneser graph is a strongly regular graph, with parameters ( \tbinom, \tbinom, \tbinom, \tbinom ). However, it is not strongly regular when k>2, as different pairs of nonadjacent vert ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kneser Graph KG(5,2)
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, Kneser ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edge-transitive Graph
In the mathematical field of graph theory, an edge-transitive graph is a graph such that, given any two edges and of , there is an automorphism of that maps to . In other words, a graph is edge-transitive if its automorphism group acts transitively on its edges. Examples and properties The number of connected simple edge-transitive graphs on n vertices is 1, 1, 2, 3, 4, 6, 5, 8, 9, 13, 7, 19, 10, 16, 25, 26, 12, 28 ... Edge-transitive graphs include all symmetric graph, such as the vertices and edges of the cube. Symmetric graphs are also vertex-transitive (if they are connected), but in general edge-transitive graphs need not be vertex-transitive. Every connected edge-transitive graph that is not vertex-transitive must be bipartite, (and hence can be colored with only two colors), and either semi-symmetric or biregular.. Examples of edge but not vertex transitive graphs include the complete bipartite graphs K_ where m ≠ n, which includes the star graphs K_. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian Cycle
In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a cycle that visits each vertex exactly once. A Hamiltonian path that starts and ends at adjacent vertices can be completed by adding one more edge to form a Hamiltonian cycle, and removing any edge from a Hamiltonian cycle produces a Hamiltonian path. Determining whether such paths and cycles exist in graphs (the Hamiltonian path problem and Hamiltonian cycle problem) are NP-complete. Hamiltonian paths and cycles are named after William Rowan Hamilton who invented the icosian game, now also known as ''Hamilton's puzzle'', which involves finding a Hamiltonian cycle in the edge graph of the dodecahedron. Hamilton solved this problem using the icosian calculus, an algebraic structure based on roots of unity with many similarities to the quaternions (also invented by H ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractional Chromatic Number
Fractional coloring is a topic in a young branch of graph theory known as fractional graph theory. It is a generalization of ordinary graph coloring. In a traditional graph coloring, each vertex in a graph is assigned some color, and adjacent vertices — those connected by edges — must be assigned different colors. In a fractional coloring however, a ''set'' of colors is assigned to each vertex of a graph. The requirement about adjacent vertices still holds, so if two vertices are joined by an edge, they must have no colors in common. Fractional graph coloring can be viewed as the linear programming relaxation of traditional graph coloring. Indeed, fractional coloring problems are much more amenable to a linear programming approach than traditional coloring problems. Definitions A ''b''-fold coloring of a graph ''G'' is an assignment of sets of size ''b'' to vertices of a graph such that adjacent vertices receive disjoint sets. An ''a'':''b''-coloring is a ''b''-fold color ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorial Proof
In mathematics, the term ''combinatorial proof'' is often used to mean either of two types of mathematical proof: * A proof by double counting. A combinatorial identity is proven by counting the number of elements of some carefully chosen set in two different ways to obtain the different expressions in the identity. Since those expressions count the same objects, they must be equal to each other and thus the identity is established. * A bijective proof. Two sets are shown to have the same number of members by exhibiting a bijection, i.e. a one-to-one correspondence, between them. The term "combinatorial proof" may also be used more broadly to refer to any kind of elementary proof in combinatorics. However, as writes in his review of (a book about combinatorial proofs), these two simple techniques are enough to prove many theorems in combinatorics and number theory. Example An archetypal double counting proof is for the well known formula for the number \tbinom nk of ''k''-combi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jiří Matoušek (mathematician)
Jiří (Jirka) Matoušek (10 March 1963 – 9 March 2015) was a Czech mathematician working in computational geometry and algebraic topology. He was a professor at Charles University in Prague and the author of several textbooks and research monographs. Biography Matoušek was born in Prague. In 1986, he received his Master's degree at Charles University under Miroslav Katětov. From 1986 until his death he was employed at the Department of Applied Mathematics of Charles University, holding a professor position since 2000. He was also a visiting and later full professor at ETH Zurich. In 1996, he won the European Mathematical Society prize and in 2000 he won the Scientist award of the Learned Society of the Czech Republic. In 1998 he was an Invited Speaker of the International Congress of Mathematicians in Berlin. He became a fellow of the Learned Society of the Czech Republic in 2005. Matoušek's paper on computational aspects of algebraic topology won the Best Paper awa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morgan Prize
:''Distinguish from the De Morgan Medal awarded by the London Mathematical Society.'' The Morgan Prize (full name Frank and Brennie Morgan Prize for Outstanding Research in Mathematics by an Undergraduate Student) is an annual award given to an undergraduate student in the US, Canada, or Mexico who demonstrates superior mathematics research. The $1,200 award, endowed by Mrs. Frank Morgan of Allentown, Pennsylvania, was founded in 1995. The award is made jointly by the American Mathematical Society, the Mathematical Association of America, and the Society for Industrial and Applied Mathematics. The Morgan Prize has been described as the highest honor given to an undergraduate in mathematics. Previous winners ;1995 :Winner: Kannan Soundararajan (Analytic Number Theory, University of Michigan) :Honorable mention: Kiran Kedlaya (Harvard University) ;1996 :Winner: Manjul Bhargava (Algebra, Harvard University) :Honorable mention: Lenhard Ng (Harvard University) ;1997 :Winner: Jade Vi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Gale
David (; , "beloved one") (traditional spelling), , ''Dāwūd''; grc-koi, Δαυΐδ, Dauíd; la, Davidus, David; gez , ዳዊት, ''Dawit''; xcl, Դաւիթ, ''Dawitʿ''; cu, Давíдъ, ''Davidŭ''; possibly meaning "beloved one". was, according to the Hebrew Bible, the third king of the United Kingdom of Israel. In the Books of Samuel, he is described as a young shepherd and harpist who gains fame by slaying Goliath, a champion of the Philistines, in southern Canaan. David becomes a favourite of Saul, the first king of Israel; he also forges a notably close friendship with Jonathan, a son of Saul. However, under the paranoia that David is seeking to usurp the throne, Saul attempts to kill David, forcing the latter to go into hiding and effectively operate as a fugitive for several years. After Saul and Jonathan are both killed in battle against the Philistines, a 30-year-old David is anointed king over all of Israel and Judah. Following his rise to power, David co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borsuk–Ulam Theorem
In mathematics, the Borsuk–Ulam theorem states that every continuous function from an ''n''-sphere into Euclidean ''n''-space maps some pair of antipodal points to the same point. Here, two points on a sphere are called antipodal if they are in exactly opposite directions from the sphere's center. Formally: if f: S^n \to \R^n is continuous then there exists an x\in S^n such that: f(-x)=f(x). The case n=1 can be illustrated by saying that there always exist a pair of opposite points on the Earth's equator with the same temperature. The same is true for any circle. This assumes the temperature varies continuously in space. The case n=2 is often illustrated by saying that at any moment, there is always a pair of antipodal points on the Earth's surface with equal temperatures and equal barometric pressures, assuming that both parameters vary continuously in space. The Borsuk–Ulam theorem has several equivalent statements in terms of odd functions. Recall that S^n is the ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Imre Bárány
Imre Bárány (Mátyásföld, Budapest, 7 December 1947) is a Hungarian mathematician, working in combinatorics and discrete geometry. He works at the Rényi Mathematical Institute of the Hungarian Academy of Sciences, and has a part-time appointment at University College London. Notable results * He gave a surprisingly simple alternative proof of László Lovász's theorem on Kneser graphs. * He gave a new proof of the Borsuk–Ulam theorem. * Bárány gave a colored version of Carathéodory's theorem. * He solved an old problem of James Joseph Sylvester on the probability of random point sets in convex position. * With Van H. Vu proved a central limit theorem on random points in convex bodies. * With Zoltán Füredi he gave an algorithm for mental poker. * With Füredi he proved that no deterministic polynomial time algorithm determines the volume of convex bodies in dimension ''d'' within a multiplicative error ''d''''d''. * With Füredi and János Pach he proved th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Combinatorics
The mathematical discipline of topological combinatorics is the application of topological and algebro-topological methods to solving problems in combinatorics. History The discipline of combinatorial topology used combinatorial concepts in topology and in the early 20th century this turned into the field of algebraic topology. In 1978 the situation was reversed—methods from algebraic topology were used to solve a problem in combinatorics—when László Lovász proved the Kneser conjecture, thus beginning the new field of topological combinatorics. Lovász's proof used the Borsuk–Ulam theorem and this theorem retains a prominent role in this new field. This theorem has many equivalent versions and analogs and has been used in the study of fair division problems. In another application of homological methods to graph theory, Lovász proved both the undirected and directed versions of a conjecture of András Frank: Given a ''k''-connected graph ''G'', ''k'' points v_1,\ldo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
László Lovász
László Lovász (; born March 9, 1948) is a Hungarian mathematician and professor emeritus at Eötvös Loránd University, best known for his work in combinatorics, for which he was awarded the 2021 Abel Prize jointly with Avi Wigderson. He was the president of the International Mathematical Union from 2007 to 2010 and the president of the Hungarian Academy of Sciences from 2014 to 2020. In graph theory, Lovász's notable contributions include the proofs of Kneser's conjecture and the Lovász local lemma, as well as the formulation of the Erdős–Faber–Lovász conjecture. He is also one of the eponymous authors of the LLL lattice reduction algorithm. Early life and education Lovász was born on March 9, 1948, in Budapest, Hungary. Lovász attended the Fazekas Mihály Gimnázium in Budapest. He won three gold medals (1964–1966) and one silver medal (1963) at the International Mathematical Olympiad. He also participated in a Hungarian game show about math prodigies. Pau ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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