Connectivity (graph theory)
In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements that need to be removed to separate the remaining nodes into isolated subgraphs. It is related to the theory of network flow problems; the connectivity of a graph is an important measure of its resilience as a network. An undirected graph is connected. In a connected graph, there are no unreachable vertices. A graph, not connected is disconnected. An undirected graph G is said to be disconnected if there exist two nodes in G such that no path in G has those nodes as endpoints. A graph with just one vertex is connected. An edgeless graph with two or more vertices is disconnected. In an undirected graph G, two vertices u and v are called connected if G contains a path from u to v. Otherwise, they are called disconnected. If the two vertices are additionally connected by a path of length 1, i.e. by a single edge, the vertices are called adjacent. A graph is said to be connected.
A connected component is a maximal connected subgraph of G. Each vertex belongs to one connected component, as does each edge. A directed graph is called weakly connected if replacing all of its directed edges with undirected edges produces a connected graph, it is connected if it contains a directed path from u to v or a directed path from v to u for every pair of vertices u, v. It is connected, diconnected, or strong if it contains a directed path from u to v and a directed path from v to u for every pair of vertices u, v; the strong components are the maximal connected subgraphs. A cut, vertex cut, or separating set of a connected graph G is a set of vertices whose removal renders G disconnected; the connectivity or vertex connectivity κ is the size of a minimal vertex cut. A graph is called k-vertex-connected if its vertex connectivity is k or greater. More any graph G is said to be k-connected if it contains at least k+1 vertices, but does not contain a set of k − 1 vertices whose removal disconnects the graph.
In particular, a complete graph with n vertices, denoted Kn, has no vertex cuts at all, but κ = n − 1. A vertex cut for two vertices u and v is a set of vertices whose removal from the graph disconnects u and v; the local connectivity κ is the size of a smallest vertex cut separating u and v. Local connectivity is symmetric for undirected graphs. Moreover, except for complete graphs, κ equals the minimum of κ over all nonadjacent pairs of vertices u, v. 2-connectivity is called biconnectivity and 3-connectivity is called triconnectivity. A graph G, connected but not 2-connected is sometimes called separable. Analogous concepts can be defined for edges. In the simple case in which cutting a single, specific edge would disconnect the graph, that edge is called a bridge. More an edge cut of G is a set of edges whose removal renders the graph disconnected; the edge-connectivity λ is the size of a smallest edge cut, the local edge-connectivity λ of two vertices u, v is the size of a smallest edge cut disconnecting u from v. Again, local edge-connectivity is symmetric.
A graph is called k-edge-connected. A graph is said to be maximally connected. A graph is said to be maximally edge-connected. A graph is said to be super-κ if every minimum vertex cut isolates a vertex. A graph is said to be hyper-connected or hyper-κ if the deletion of each minimum vertex cut creates two components, one of, an isolated vertex. A graph is semi-hyper-connected or semi-hyper-κ if any minimum vertex cut separates the graph into two components. More precisely: a G connected graph is said to be super-connected or super-κ if all minimum vertex-cuts consist of the vertices adjacent with one vertex. A G connected graph is said to be super-edge-connected or super-λ if all minimum edge-cuts consist of the edges incident on some vertex. A cutset X of G is called a non-trivial cutset if X does not contain the neighborhood N of any vertex u ∉ X; the superconnectivity κ1 of G is: κ1 = min. A non-trivial edge-cut and the edge-superconnectivity λ1 are defined analogously. One of the most important facts about connectivity in graphs is Menger's theorem, which characterizes the connectivity and edge-connectivity of a graph in terms of the number of independent paths between vertices.
If u and v are vertices of a graph G a collection of paths between u and v is called independent if no two of them share a vertex. The collection is edge-independent if no two paths in it share an edge; the number of mutually independent paths between u and v is written as κ′, the number of mutually edge-independent paths between u and v is written as λ′. Menger's theorem asserts that for distinct vertices u,v, λ equals λ′, if u is not adjacent to v κ equals κ′; this fact is a special case of the max-flow min-cut theorem. The problem of determining whether two vertices in a graph are connected can be solved efficiently using a search algorithm, such as breadth-first search. More it is easy to determine computationally whether a graph is connected, or to count the number of connected compone
László "Laci" Babai is a Hungarian professor of computer science and mathematics at the University of Chicago. His research focuses on computational complexity theory, algorithms and finite groups, with an emphasis on the interactions between these fields. In 1968, Babai won a gold medal at the International Mathematical Olympiad. Babai studied mathematics at Eötvös Loránd University from 1968 to 1973, received a Ph. D. from the Hungarian Academy of Sciences in 1975, received a D. Sc. from the Hungarian Academy of Sciences in 1984. He held a teaching position at Eötvös Loránd University since 1971. In 1995, he began a joint appointment in the mathematics department at Chicago and gave up his position at Eötvös Loránd, he is the author of over 180 academic papers. His notable accomplishments include the introduction of interactive proof systems, the introduction of the term Las Vegas algorithm, the introduction of group theoretic methods in graph isomorphism testing. In November 2015, he announced a quasipolynomial time algorithm for the graph isomorphism problem.
He is editor-in-chief of the refereed online journal Theory of Computing. Babai was involved in the creation of the Budapest Semesters in Mathematics program and first coined the name. From 10 November to 1 December 2015, Babai gave three lectures on «Graph Isomorphism in Quasipolynomial Time» in the «Combinatorics and Theoretical Computer Science» Seminar at the University of Chicago, he outlined a proof to show that the Graph isomorphism problem can be solved in quasi-polynomial time. A video of the first talk was published on 10 December 2015, a preprint was uploaded to arXiv.org on 11 December 2015, a fix was published on 14 January 2017. The paper is undergoing review. In 1988, Babai won the Hungarian State Prize, in 1990 he was elected as a corresponding member of the Hungarian Academy of Sciences, in 1994 he became a full member. In 1999 the Budapest University of Technology and Economics awarded him an honorary doctorate. In 1993, Babai was awarded the Gödel Prize together with Shafi Goldwasser, Silvio Micali, Shlomo Moran, Charles Rackoff, for their papers on interactive proof systems.
In 2015, he was elected a fellow of the American Academy of Arts and Sciences, won the Knuth Prize. Professor László Babai’s algorithm is next big step in conquering isomorphism in graphs // Published on Nov 20, 2015 Division of the Physical Sciences / The University of Chicago Mathematician claims breakthrough in complexity theory, by Adrian Cho 10 November 2015 17:45 // Posted in Math, Science AAAS News A Quasipolynomial Time Algorithm for Graph Isomorphism: The Details + Background on Graph Isomorphism + The Main Result // Math ∩ Programming. Posted on November 12, 2015 by j2kun Landmark Algorithm Breaks 30-Year Impasse, Algorithm Solves Graph Isomorphism in Record Time // Quanta Magazine. By: Erica Klarreich, December 14, 2015 A Little More on the Graph Isomorphism Algorithm // November 21, 2015, by RJLipton+KWRegan Бабай приблизился к решению «проблемы тысячелетия» // Наука Lenta.ru, 14:48, 20 ноября 2015copy from Lenta.ru // texnomaniya.ru, 20 ноября 2015Опубликован быстрый алгоритм для задачи изоморфизма графов // Анатолий Ализар, Хабрахабр, 16 декабря в 02:12Опубліковано швидкий алгоритм для задачі ізоморфізму графів // Джерело: Хабрахабр, перекладено 16 грудня 2015, 06:30 Media related to László Babai at Wikimedia Commons Personal website.
MathSciNet: "Items authored by Babai, László." DBLP: László Babai
In mathematics, a group is a set equipped with a binary operation which combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied, namely closure, associativity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but groups are encountered in numerous areas within and outside mathematics, help focusing on essential structural aspects, by detaching them from the concrete nature of the subject of the study. Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other. Lie groups are the symmetry groups used in the Standard Model of particle physics; the concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s.
After contributions from other fields such as number theory and geometry, the group notion was generalized and established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. In addition to their abstract properties, group theorists study the different ways in which a group can be expressed concretely, both from a point of view of representation theory and of computational group theory. A theory has been developed for finite groups, which culminated with the classification of finite simple groups, completed in 2004. Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become an active area in group theory; the modern concept of an abstract group developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4.
The 19th-century French mathematician Évariste Galois, extending prior work of Paolo Ruffini and Joseph-Louis Lagrange, gave a criterion for the solvability of a particular polynomial equation in terms of the symmetry group of its roots. The elements of such a Galois group correspond to certain permutations of the roots. At first, Galois' ideas were rejected by his contemporaries, published only posthumously. More general permutation groups were investigated in particular by Augustin Louis Cauchy. Arthur Cayley's On the theory of groups, as depending on the symbolic equation θn = 1 gives the first abstract definition of a finite group. Geometry was a second field in which groups were used systematically symmetry groups as part of Felix Klein's 1872 Erlangen program. After novel geometries such as hyperbolic and projective geometry had emerged, Klein used group theory to organize them in a more coherent way. Further advancing these ideas, Sophus Lie founded the study of Lie groups in 1884; the third field contributing to group theory was number theory.
Certain abelian group structures had been used implicitly in Carl Friedrich Gauss' number-theoretical work Disquisitiones Arithmeticae, more explicitly by Leopold Kronecker. In 1847, Ernst Kummer made early attempts to prove Fermat's Last Theorem by developing groups describing factorization into prime numbers; the convergence of these various sources into a uniform theory of groups started with Camille Jordan's Traité des substitutions et des équations algébriques. Walther von Dyck introduced the idea of specifying a group by means of generators and relations, was the first to give an axiomatic definition of an "abstract group", in the terminology of the time; as of the 20th century, groups gained wide recognition by the pioneering work of Ferdinand Georg Frobenius and William Burnside, who worked on representation theory of finite groups, Richard Brauer's modular representation theory and Issai Schur's papers. The theory of Lie groups, more locally compact groups was studied by Hermann Weyl, Élie Cartan and many others.
Its algebraic counterpart, the theory of algebraic groups, was first shaped by Claude Chevalley and by the work of Armand Borel and Jacques Tits. The University of Chicago's 1960–61 Group Theory Year brought together group theorists such as Daniel Gorenstein, John G. Thompson and Walter Feit, laying the foundation of a collaboration that, with input from numerous other mathematicians, led to the classification of finite simple groups, with the final step taken by Aschbacher and Smith in 2004; this project exceeded previous mathematical endeavours by its sheer size, in both length of proof and number of researchers. Research is ongoing to simplify the proof of this classification; these days, group theory is still a active mathematical branch, impacting many other fields. One of the most familiar groups is the set of integers Z which consists of the numbers... − 4, − 3, − − 1, 0, 1, 2, 3, 4... together with addition. The following properties of integer addition serve as a model for the group axioms given in the definition below.
For any two integers a and b, the sum a + b is an integer. That is, addition of integers always yields an integer; this property is known as closure under addition. For all integers a, b and c, + c = a +. Expressed in words
In group theory, a branch of abstract algebra, a cyclic group or monogenous group is a group, generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, it contains an element g such that every other element of the group may be obtained by applying the group operation to g or its inverse; each element can be written as a power of g in multiplicative notation, or as a multiple of g in additive notation. This element g is called a generator of the group; every infinite cyclic group is isomorphic to the additive group of the integers. Every finite cyclic group of order n is isomorphic to the additive group of Z/nZ, the integers modulo n; every cyclic group is an abelian group, every finitely generated abelian group is a direct product of cyclic groups. Every cyclic group of prime order is a simple group. In the classification of finite simple groups, one of the three infinite classes consists of the cyclic groups of prime order; the cyclic groups of prime order are thus among the building blocks from which all groups can be built.
For any element g in any group G, one can form the subgroup of all integer powers ⟨g⟩ =, called the cyclic subgroup of g. The order of g is the number of elements in ⟨g⟩. A cyclic group is a group, equal to one of its cyclic subgroups: G = ⟨g⟩ for some element g, called a generator. For a finite cyclic group with order |G| = n, this means G =, where e is the identity element and gj = gk whenever j ≡ k modulo n. An abstract group defined by this multiplication is denoted Cn, we say that G is isomorphic to the standard cyclic group Cn; such a group is isomorphic to Z/nZ, the group of integers modulo n with the addition operation, the standard cyclic group in additive notation. Under the isomorphism χ defined by χ = i the identity element e corresponds to 0, products correspond to sums, powers correspond to multiples. For example, the set of complex 6th roots of unity G = forms a group under multiplication, it is cyclic, since it is generated by the primitive root z = 1 2 + 3 2 i = e 2 π i / 6: that is, G = ⟨z⟩ = with z6 = 1.
Under a change of letters, this is isomorphic to the standard cyclic group of order 6, defined as C6 = ⟨g⟩ = with multiplication gj · gk = gj+k, so that g6 = g0 = e. These groups are isomorphic to Z/6Z = with the operation of addition modulo 6, with zk and gk corresponding to k. For example, 1 + 2 ≡ 3 corresponds to z1 · z2 = z3, 2 + 5 ≡ 1 corresponds to z2 · z5 = z7 = z1, so on. Any element generates its own cyclic subgroup, such as ⟨z2⟩ = of order 3, isomorphic to C3 and Z/3Z. Instead of the quotient notations Z/nZ, Z/, or Z/n, some authors denote a finite cyclic group as Zn, but this conflicts with the notation of number theory, where Zp denotes a p-adic number ring, or localization at a prime ideal. On the other hand, in an infinite cyclic group G = ⟨g⟩, the powers gk give distinct elements for all integers k, so that G =, G is isomorphic to the standard group C = C∞ and to Z, the additive group of the integers. An example is the first frieze group. Here there are no finite cycles, the name "cyclic" may be misleading.
To avoid this confusion, Bourbaki introduced the term monogenous group for a group with a single generator and restricted "cyclic group" to mean a finite monogenous group, avoiding the term "infinite cyclic group". The set of integers Z,with the operation of addition, forms a group, it is an infinite cyclic group, because all integers can be written by adding or subtracting the single number 1. In this group, 1 and −1 are the only generators; every infinite cyclic group is isomorphic to Z. For every positive integer n, the set of integers modulo n, again with the operation of addition, forms a finite cyclic group, denoted Z/nZ. A modular integer i is a generator of this group if i is prime to n, because these elements can generate all other elements of the group through integer addition; every finite cyclic group G is isomorphic to Z/nZ. The addition operations on integers and modular integers, used to define the cyclic groups, are the addition operations of commutative rings denoted Z and Z/nZ or Z/.
If p is a prime Z/pZ is a finite field, is denoted Fp or GF. For every positive integer n, the set of the integers modulo n that are prime to n is written as ×; this group is not always cyclic, bu
In the mathematical field of graph theory, a vertex-transitive graph is a graph G such that, given any two vertices v1 and v2 of G, there is some automorphism f: V → V such that f = v 2. In other words, a graph is vertex-transitive if its automorphism group acts transitively upon 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, every vertex-transitive graph is regular. However, not all vertex-transitive graphs are symmetric, not all regular graphs are vertex-transitive. Finite vertex-transitive graphs include the symmetric graphs; the finite Cayley graphs are vertex-transitive, as are the vertices and edges of the Archimedean solids. Potočnik and Verret have constructed a census of all connected cubic vertex-transitive graphs on at most 1280 vertices. Although every Cayley graph is vertex-transitive, there exist other vertex-transitive graphs that are not Cayley graphs.
The most famous example is the Petersen graph, but others can be constructed including the line graphs of edge-transitive non-bipartite graphs with odd vertex degrees. The edge-connectivity of a vertex-transitive graph is equal to the degree d, while the vertex-connectivity will be at least 2/3. If the degree is 4 or less, or the graph is edge-transitive, or the graph is a minimal Cayley graph the vertex-connectivity will be equal to d. Infinite vertex-transitive graphs include: infinite paths infinite regular trees, e.g. the Cayley graph of the free group graphs of uniform tessellations, including all tilings by regular polygons infinite Cayley graphs the Rado graphTwo countable vertex-transitive graphs are called quasi-isometric if the ratio of their distance functions is bounded from below and from above. A well known conjecture stated that every infinite vertex-transitive graph is quasi-isometric to a Cayley graph. A counterexample was proposed by Diestel and Leader in 2001. In 2005, Eskin and Whyte confirmed the counterexample.
Edge-transitive graph Lovász conjecture Semi-symmetric graph Zero-symmetric graph Weisstein, Eric W. "Vertex-transitive graph". MathWorld. A census of small connected cubic vertex-transitive graphs. Primož Potočnik, Pablo Spiga, Gabriel Verret, 2012
Marie Ennemond Camille Jordan was a French mathematician, known both for his foundational work in group theory and for his influential Cours d'analyse. Jordan was educated at the École polytechnique, he was an engineer by profession. He is remembered now by name in a number of results: The Jordan curve theorem, a topological result required in complex analysis The Jordan normal form and the Jordan matrix in linear algebra In mathematical analysis, Jordan measure is an area measure that predates measure theory In group theory, the Jordan–Hölder theorem on composition series is a basic result. Jordan's theorem on finite linear groupsJordan's work did much to bring Galois theory into the mainstream, he investigated the Mathieu groups, the first examples of sporadic groups. His Traité des substitutions, on permutation groups, was published in 1870, he was an Invited Speaker of the ICM in 1920 in Strasbourg. The asteroid 25593 Camillejordan and Institut Camille Jordan are named in his honour. Camille Jordan is not to be confused with the geodesist Wilhelm Jordan or the physicist Pascual Jordan.
Cours d'analyse de l'Ecole Polytechnique. Jordan–Chevalley decomposition Jordan totient function Jordan–Schur theorem Jordan–Schönflies theorem Jordan's lemma Jordan's theorem O'Connor, John J..
Leipzig is the most populous city in the federal state of Saxony, Germany. With a population of 581,980 inhabitants as of 2017, it is Germany's tenth most populous city. Leipzig is located about 160 kilometres southwest of Berlin at the confluence of the White Elster, Pleiße and Parthe rivers at the southern end of the North German Plain. Leipzig has been a trade city since at least the time of the Holy Roman Empire; the city sits at the intersection of the Via Regia and the Via Imperii, two important medieval trade routes. Leipzig was once one of the major European centers of learning and culture in fields such as music and publishing. Leipzig became a major urban center within the German Democratic Republic after the Second World War, but its cultural and economic importance declined. Events in Leipzig in 1989 played a significant role in precipitating the fall of communism in Central and Eastern Europe through demonstrations starting from St. Nicholas Church. Since the reunification of Germany, Leipzig has undergone significant change with the restoration of some historical buildings, the demolition of others, the development of a modern transport infrastructure.
Leipzig today is an economic centre, the most livable city in Germany, according to the GfK marketing research institution and has the second-best future prospects of all cities in Germany, according to HWWI and Berenberg Bank. Leipzig Zoo is one of the most modern zoos in Europe and ranks first in Germany and second in Europe according to Anthony Sheridan. Since the opening of the Leipzig City Tunnel in 2013, Leipzig forms the centrepiece of the S-Bahn Mitteldeutschland public transit system. Leipzig is listed as a Gamma World City, Germany's "Boomtown" and as the European City of the Year 2019. Leipzig has long been a major center for music, both classical as well as modern "dark alternative music" or darkwave genres; the Oper Leipzig is one of the most prominent opera houses in Germany. It was founded in 1693, making it the third oldest opera venue in Europe after La Fenice and the Hamburg State Opera. Leipzig is home to the University of Music and Theatre "Felix Mendelssohn Bartholdy", it was during a stay in this city that Friedrich Schiller wrote his poem "Ode to Joy".
The Leipzig Gewandhaus Orchestra, established in 1743, is one of the oldest symphony orchestras in the world. Johann Sebastian Bach is one among many major composers who lived in Leipzig; the name Leipzig is derived from the Slavic word Lipsk, which means "settlement where the linden trees stand". An older spelling of the name in English is Leipsic; the Latin name Lipsia was used. The name is cognate with Lipetsk in Liepāja in Latvia. In 1937 the Nazi government renamed the city Reichsmessestadt Leipzig. Since 1989 Leipzig has been informally dubbed "Hero City", in recognition of the role that the Monday demonstrations there played in the fall of the East German regime – the name alludes to the honorary title awarded in the former Soviet Union to certain cities that played a key role in the victory of the Allies during the Second World War; the common usage of this nickname for Leipzig up until the present is reflected, for example, in the name of a popular blog for local arts and culture, Heldenstadt.de.
More the city has sometimes been nicknamed the "Boomtown of eastern Germany", "Hypezig" or "The better Berlin" for being celebrated by the media as a hip urban centre for the vital lifestyle and creative scene with many startups. Leipzig was first documented in 1015 in the chronicles of Bishop Thietmar of Merseburg as urbs Libzi and endowed with city and market privileges in 1165 by Otto the Rich. Leipzig Trade Fair, started in the Middle Ages, has become an event of international importance and is the oldest surviving trade fair in the world. There are records of commercial fishing operations on the river Pleiße in Leipzig dating back to 1305, when the Margrave Dietrich the Younger granted the fishing rights to the church and convent of St Thomas. There were a number of monasteries in and around the city, including a Franciscan monastery after which the Barfußgäßchen is named and a monastery of Irish monks near the present day Ranstädter Steinweg; the foundation of the University of Leipzig in 1409 initiated the city's development into a centre of German law and the publishing industry, towards being the location of the Reichsgericht and the German National Library.
During the Thirty Years' War, two battles took place in Breitenfeld, about 8 kilometres outside Leipzig city walls. The first Battle of Breitenfeld took place in 1631 and the second in 1642. Both battles resulted in victories for the Swedish-led side. On 24 December 1701, an oil-fueled street lighting system was introduced; the city employed light guards who had to follow a specific schedule to ensure the punctual lighting of the 700 lanterns. The Leipzig region was the arena of the 1813 Battle of Leipzig between Napoleonic France and an allied coalition of Prussia, Russia and Sweden, it was the largest battle in Europe before the First World War and the coalition victory ended Napoleon's presence in Germany and would lead to his first exile on Elba. The Monument to the Battle of the Nations celebrating the centenary of this event was completed in 1913. In addition to stimulating German nationalism, the war had a major impact in mobilizing a civic spirit in numerous volunteer activities. Many volunteer militi