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Babai's Problem
Babai's problem is a problem in algebraic graph theory first proposed in 1979 by László Babai. Babai's problem Let G be a finite group, let \operatorname(G) be the set of all irreducible characters of G, let \Gamma=\operatorname(G,S) be the Cayley graph (or directed Cayley graph) corresponding to a generating subset S of G\setminus \, and let \nu be a positive integer. Is the set : M_\nu^S=\left\ an ''invariant'' of the graph \Gamma? In other words, does \operatorname(G,S)\cong \operatorname(G,S') imply that M_\nu^S=M_\nu^? BI-group A finite group G is called a BI-group (Babai Invariant group) if \operatorname(G,S)\cong \operatorname(G,T) for some inverse closed subsets S and T of G\setminus \ implies that M_\nu^S=M_\nu^T for all positive integers \nu. Open problem Which finite groups are BI-groups? See also * List of unsolved problems in mathematics Many mathematical problems have been stated but not yet solved. These problems come from many areas of m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Graph Theory
Algebraic graph theory is a branch of mathematics in which algebraic methods are applied to problems about graphs. This is in contrast to geometric, combinatoric, or algorithmic approaches. There are three main branches of algebraic graph theory, involving the use of linear algebra, the use of group theory, and the study of graph invariants. Branches of algebraic graph theory Using linear algebra The first branch of algebraic graph theory involves the study of graphs in connection with linear algebra. Especially, it studies the spectrum of the adjacency matrix, or the Laplacian matrix of a graph (this part of algebraic graph theory is also called spectral graph theory). For the Petersen graph, for example, the spectrum of the adjacency matrix is (−2, −2, −2, −2, 1, 1, 1, 1, 1, 3). Several theorems relate properties of the spectrum to other graph properties. As a simple example, a connected graph with diameter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
László Babai
László "Laci" Babai (born July 20, 1950, in Budapest) a fellow of the American Academy of Arts and Sciences, and won the Knuth Prize. Babai was an invited speaker at the International Congresses of Mathematicians in Kyoto (1990), Zürich (1994, plenary talk), and Rio de Janeiro (2018). Sources 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 iMath Science AAAS AAAS may refer to: * American Academy of Arts and Sciences, a learned society and center for policy research; the publisher of the journal ''Dædalus'' * American Association for the Advancement of Science, an organization that supports scientifi ... News A Quasipolynomial Time Algorithm for Graph Isomorphism: The Details+ Background on Graph Isomorphism + The Main Result // Math ∩ Pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Combinatorial Theory, Series B
The ''Journal of Combinatorial Theory'', Series A and Series B, are mathematical journals specializing in combinatorics and related areas. They are published by Elsevier. ''Series A'' is concerned primarily with structures, designs, and applications of combinatorics. ''Series B'' is concerned primarily with graph and matroid theory. The two series are two of the leading journals in the field and are widely known as ''JCTA'' and ''JCTB''. The journal was founded in 1966 by Frank Harary and Gian-Carlo Rota.They are acknowledged on the journals' title pages and Web sites. SeEditorial board of JCTA Originally there was only one journal, which was split into two parts in 1971 as the field grew rapidly. An electronic, |
Irreducible Character
In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix. The character carries the essential information about the representation in a more condensed form. Georg Frobenius initially developed representation theory of finite groups entirely based on the characters, and without any explicit matrix realization of representations themselves. This is possible because a complex representation of a finite group is determined (up to isomorphism) by its character. The situation with representations over a field of positive characteristic, so-called "modular representations", is more delicate, but Richard Brauer developed a powerful theory of characters in this case as well. Many deep theorems on the structure of finite groups use characters of modular representations. Applications Characters of irreducible representations encode many import ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cayley Graph
In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem (named after Arthur Cayley), and uses a specified set of generators for the group. It is a central tool in combinatorial and geometric group theory. The structure and symmetry of Cayley graphs makes them particularly good candidates for constructing families of expander graphs. Definition Let G be a group and S be a generating set of G. The Cayley graph \Gamma = \Gamma(G,S) is an edge-colored directed graph constructed as follows: In his Collected Mathematical Papers 10: 403–405. * Each element g of G is assigned a vertex: the vertex set of \Gamma is identified with G. * Each element s of S is assigned a color c_s. * For every g \in G and s \in S, there is a directed edge of color c_s from the vertex corresponding to g to the one corresponding to gs. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lovász Conjecture
In graph theory, the Lovász conjecture (1969) is a classical problem on Hamiltonian paths in graphs. It says: : Every finite connected vertex-transitive graph contains a Hamiltonian path. Originally László Lovász stated the problem in the opposite way, but this version became standard. In 1996, László Babai published a conjecture sharply contradicting this conjecture, but both conjectures remain widely open. It is not even known if a single counterexample would necessarily lead to a series of counterexamples. Historical remarks The problem of finding Hamiltonian paths in highly symmetric graphs is quite old. As Donald Knuth describes it in volume 4 of ''The Art of Computer Programming'', the problem originated in British campanology (bell-ringing). Such Hamiltonian paths and cycles are also closely connected to Gray codes. In each case the constructions are explicit. Variants of the Lovász conjecture Hamiltonian cycle Another version of Lovász conjecture states that : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generating Set Of A Group
In abstract algebra, a generating set of a group is a subset of the group set such that every element of the group can be expressed as a combination (under the group operation) of finitely many elements of the subset and their inverses. In other words, if ''S'' is a subset of a group ''G'', then , the ''subgroup generated by S'', is the smallest subgroup of ''G'' containing every element of ''S'', which is equal to the intersection over all subgroups containing the elements of ''S''; equivalently, is the subgroup of all elements of ''G'' that can be expressed as the finite product of elements in ''S'' and their inverses. (Note that inverses are only needed if the group is infinite; in a finite group, the inverse of an element can be expressed as a power of that element.) If ''G'' = , then we say that ''S'' ''generates'' ''G'', and the elements in ''S'' are called ''generators'' or ''group generators''. If ''S'' is the empty set, then is the trivial group , since we consider ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invariant (mathematics)
In mathematics, an invariant is a property of a mathematical object (or a class of mathematical objects) which remains unchanged after operations or transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used. For example, the area of a triangle is an invariant with respect to isometries of the Euclidean plane. The phrases "invariant under" and "invariant to" a transformation are both used. More generally, an invariant with respect to an equivalence relation is a property that is constant on each equivalence class. Invariants are used in diverse areas of mathematics such as geometry, topology, algebra and discrete mathematics. Some important classes of transformations are defined by an invariant they leave unchanged. For example, conformal maps are defined as transformations of the plane that preserve angles. The discovery of invariants is an importa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Algebra And Its Applications
The ''Journal of Algebra and Its Applications'' covers both theoretical and applied algebra, with a focus on practical applications. It is published by World Scientific.{{cite journal , title=Journal of Algebra and its Applications , journal=www.scimagojr.com , url=https://www.scimagojr.com/journalsearch.php?q=11800154589&tip=sid&clean=0 According to the ''Journal Citation Reports'', the journal has a 2020 impact factor of 0.736. Abstracting and indexing The journal is abstracted and indexed in: * Mathematical Reviews * Zentralblatt MATH * Science Citation Index Expanded * Current Contents/Physical Chemical and Earth Sciences * Journal Citation Reports ''Journal Citation Reports'' (''JCR'') is an annual publicationby Clarivate Analytics (previously the intellectual property of Thomson Reuters). It has been integrated with the Web of Science and is accessed from the Web of Science-Core Collect .../Science Edition * INSPEC References Academic journals established in 2008 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Communications In Algebra
''Communications in Algebra'' is a monthly peer-reviewed scientific journal covering algebra, including commutative algebra, ring theory, module theory, non-associative algebra (including Lie algebras and Jordan algebras), group theory, and algebraic geometry. It was established in 1974 and is published by Taylor & Francis. The editor-in-chief is Scott Chapman ( Sam Houston State University). Earl J. Taft (Rutgers University) was the founding editor. Abstracting and indexing The journal is abstracted and indexed in CompuMath Citation Index, Current Contents/Chemical, Earth, and Physical Sciences, Mathematical Reviews, MathSciNet, Science Citation Index Expanded (SCIE), and Zentralblatt MATH. According to the ''Journal Citation Reports'', the journal has a 2018 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Unsolved Problems In Mathematics
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations. Some problems belong to more than one discipline and are studied using techniques from different areas. Prizes are often awarded for the solution to a long-standing problem, and some lists of unsolved problems, such as the Millennium Prize Problems, receive considerable attention. This list is a composite of notable unsolved problems mentioned in previously published lists, including but not limited to lists considered authoritative. Although this list may never be comprehensive, the problems listed here vary widely in both difficulty and importance. Lists of unsolved problems in mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Graph Theory
Algebraic graph theory is a branch of mathematics in which algebraic methods are applied to problems about graphs. This is in contrast to geometric, combinatoric, or algorithmic approaches. There are three main branches of algebraic graph theory, involving the use of linear algebra, the use of group theory, and the study of graph invariants. Branches of algebraic graph theory Using linear algebra The first branch of algebraic graph theory involves the study of graphs in connection with linear algebra. Especially, it studies the spectrum of the adjacency matrix, or the Laplacian matrix of a graph (this part of algebraic graph theory is also called spectral graph theory). For the Petersen graph, for example, the spectrum of the adjacency matrix is (−2, −2, −2, −2, 1, 1, 1, 1, 1, 3). Several theorems relate properties of the spectrum to other graph properties. As a simple example, a connected graph with diameter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
