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Erdős–Stone Theorem
In extremal graph theory, the Erdős–Stone theorem is an asymptotic result generalising Turán's theorem to bound the number of edges in an ''H''-free graph for a non-complete graph ''H''. It is named after Paul Erdős and Arthur Stone (mathematician), Arthur Stone, who proved it in 1946, and it has been described as the “fundamental theorem of extremal graph theory”. Statement for Turán graphs The ''extremal number'' ex(''n''; ''H'') is defined to be the maximum number of edges in a graph with ''n'' vertices not containing a subgraph isomorphic to ''H''; see the Forbidden subgraph problem for more examples of problems involving the extremal number. Turán's theorem says that ex(''n''; ''K''''r'') = ''t''''r'' − 1(''n''), the number of edges of the Turán graph ''T''(''n'', r − 1), and that the Turán graph is the unique such extremal graph. The Erdős–Stone theorem extends this result to ''H'' = ''K''''r''('' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extremal Graph Theory
Extremal graph theory is a branch of combinatorics, itself an area of mathematics, that lies at the intersection of extremal combinatorics and graph theory. In essence, extremal graph theory studies how global properties of a graph influence local substructure. Results in extremal graph theory deal with quantitative connections between various Graph property, graph properties, both global (such as the number of vertices and edges) and local (such as the existence of specific subgraphs), and problems in extremal graph theory can often be formulated as optimization problems: how big or small a parameter of a graph can be, given some constraints that the graph has to satisfy? A graph that is an optimal solution to such an optimization problem is called an extremal graph, and extremal graphs are important objects of study in extremal graph theory. Extremal graph theory is closely related to fields such as Ramsey theory, spectral graph theory, computational complexity theory, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zarankiewicz Problem
The Zarankiewicz problem, an unsolved problem in mathematics, asks for the largest possible number of edges in a bipartite graph that has a given number of vertices and has no complete bipartite subgraphs of a given size.. Reprint of 1978 Academic Press edition, . It belongs to the field of extremal graph theory, a branch of combinatorics, and is named after the Polish mathematician Kazimierz Zarankiewicz, who proposed several special cases of the problem in 1951. Problem statement A bipartite graph G=(U\cup V,E) consists of two disjoint sets of vertices U and V, and a set of edges each of which connects a vertex in U to a vertex in V. No two edges can both connect the same pair of vertices. A complete bipartite graph is a bipartite graph in which every pair of a vertex from U and a vertex from V is connected to each other. A complete bipartite graph in which U has s vertices and V has t vertices is denoted K_. If G=(U\cup V,E) is a bipartite graph, and there exists a set of s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extremal Graph Theory
Extremal graph theory is a branch of combinatorics, itself an area of mathematics, that lies at the intersection of extremal combinatorics and graph theory. In essence, extremal graph theory studies how global properties of a graph influence local substructure. Results in extremal graph theory deal with quantitative connections between various Graph property, graph properties, both global (such as the number of vertices and edges) and local (such as the existence of specific subgraphs), and problems in extremal graph theory can often be formulated as optimization problems: how big or small a parameter of a graph can be, given some constraints that the graph has to satisfy? A graph that is an optimal solution to such an optimization problem is called an extremal graph, and extremal graphs are important objects of study in extremal graph theory. Extremal graph theory is closely related to fields such as Ramsey theory, spectral graph theory, computational complexity theory, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of The London Mathematical Society
The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical Society and the Operational Research Society (ORS). History The Society was established on 16 January 1865, the first president being Augustus De Morgan. The earliest meetings were held in University College, but the Society soon moved into Burlington House, Piccadilly. The initial activities of the Society included talks and publication of a journal. The LMS was used as a model for the establishment of the American Mathematical Society in 1888. Mary Cartwright was the first woman to be President of the LMS (in 1961–62). The Society was granted a royal charter in 1965, a century after its foundation. In 1998 the Society moved from rooms in Burlington House into De Morgan House (named after the society's first president), at 57–5 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bulletin Of The London Mathematical Society
The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical Society and the Operational Research Society (ORS). History The Society was established on 16 January 1865, the first president being Augustus De Morgan. The earliest meetings were held in University College, but the Society soon moved into Burlington House, Piccadilly. The initial activities of the Society included talks and publication of a journal. The LMS was used as a model for the establishment of the American Mathematical Society in 1888. Mary Cartwright was the first woman to be President of the LMS (in 1961–62). The Society was granted a royal charter in 1965, a century after its foundation. In 1998 the Society moved from rooms in Burlington House into De Morgan House (named after the society's first president), at 57–58 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elsevier
Elsevier ( ) is a Dutch academic publishing company specializing in scientific, technical, and medical content. Its products include journals such as ''The Lancet'', ''Cell (journal), Cell'', the ScienceDirect collection of electronic journals, ''Trends (journals), Trends'', the ''Current Opinion (Elsevier), Current Opinion'' series, the online citation database Scopus, the SciVal tool for measuring research performance, the ClinicalKey search engine for clinicians, and the ClinicalPath evidence-based cancer care service. Elsevier's products and services include digital tools for Data management platform, data management, instruction, research analytics, and assessment. Elsevier is part of the RELX Group, known until 2015 as Reed Elsevier, a publicly traded company. According to RELX reports, in 2022 Elsevier published more than 600,000 articles annually in over 2,800 journals. As of 2018, its archives contained over 17 million documents and 40,000 Ebook, e-books, with over one b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Little-o Notation
Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by German mathematicians Paul Bachmann, Edmund Landau, and others, collectively called Bachmann–Landau notation or asymptotic notation. The letter O was chosen by Bachmann to stand for '' Ordnung'', meaning the order of approximation. In computer science, big O notation is used to classify algorithms according to how their run time or space requirements grow as the input size grows. In analytic number theory, big O notation is often used to express a bound on the difference between an arithmetical function and a better understood approximation; one well-known example is the remainder term in the prime number theorem. Big O notation is also used in many other fields to provide similar estimates. Big O notation characterizes functions according to their growth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clique (graph Theory)
In graph theory, a clique ( or ) is a subset of vertices of an undirected graph such that every two distinct vertices in the clique are adjacent. That is, a clique of a graph G is an induced subgraph of G that is complete. Cliques are one of the basic concepts of graph theory and are used in many other mathematical problems and constructions on graphs. Cliques have also been studied in computer science: the task of finding whether there is a clique of a given size in a graph (the clique problem) is NP-complete, but despite this hardness result, many algorithms for finding cliques have been studied. Although the study of complete subgraphs goes back at least to the graph-theoretic reformulation of Ramsey theory by , the term ''clique'' comes from , who used complete subgraphs in social networks to model cliques of people; that is, groups of people all of whom know each other. Cliques have many other applications in the sciences and particularly in bioinformatics. Definiti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypergraph
In mathematics, a hypergraph is a generalization of a Graph (discrete mathematics), graph in which an graph theory, edge can join any number of vertex (graph theory), vertices. In contrast, in an ordinary graph, an edge connects exactly two vertices. Formally, a directed hypergraph is a pair (X,E), where X is a set of elements called ''nodes'', ''vertices'', ''points'', or ''elements'' and E is a set of pairs of subsets of X. Each of these pairs (D,C)\in E is called an ''edge'' or ''hyperedge''; the vertex subset D is known as its ''tail'' or ''domain'', and C as its ''head'' or ''codomain''. The order of a hypergraph (X,E) is the number of vertices in X. The size of the hypergraph is the number of edges in E. The order of an edge e=(D,C) in a directed hypergraph is , e, = (, D, ,, C, ): that is, the number of vertices in its tail followed by the number of vertices in its head. The definition above generalizes from a directed graph to a directed hypergraph by defining the h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kövari-Sós-Turán Theorem
The Zarankiewicz problem, an unsolved problem in mathematics, asks for the largest possible number of edges in a bipartite graph that has a given number of vertices and has no complete bipartite subgraphs of a given size.. Reprint of 1978 Academic Press edition, . It belongs to the field of extremal graph theory, a branch of combinatorics, and is named after the Polish mathematician Kazimierz Zarankiewicz, who proposed several special cases of the problem in 1951. Problem statement A bipartite graph G=(U\cup V,E) consists of two disjoint sets of vertices U and V, and a set of edges each of which connects a vertex in U to a vertex in V. No two edges can both connect the same pair of vertices. A complete bipartite graph is a bipartite graph in which every pair of a vertex from U and a vertex from V is connected to each other. A complete bipartite graph in which U has s vertices and V has t vertices is denoted K_. If G=(U\cup V,E) is a bipartite graph, and there exists a set of s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipartite Graph
In the mathematics, mathematical field of graph theory, a bipartite graph (or bigraph) is a Graph (discrete mathematics), graph whose vertex (graph theory), vertices can be divided into two disjoint sets, disjoint and Independent set (graph theory), independent sets U and V, that is, every edge (graph theory), edge connects a Vertex (graph theory), vertex in U to one in V. Vertex sets U and V are usually called the ''parts'' of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycle (graph theory), cycles. The two sets U and V may be thought of as a graph coloring, coloring of the graph with two colors: if one colors all nodes in U blue, and all nodes in V red, each edge has endpoints of differing colors, as is required in the graph coloring problem.. In contrast, such a coloring is impossible in the case of a non-bipartite graph, such as a Gallery of named graphs, triangle: after one node is colored blue and another red, the third vertex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Asymptotic
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates Limit of a function#Limits at infinity, tends to infinity. In projective geometry and related contexts, an asymptote of a curve is a line which is tangent to the curve at a point at infinity. The word asymptote is derived from the Greek language, Greek ἀσύμπτωτος (''asumptōtos'') which means "not falling together", from ἀ Privative alpha, priv. + σύν "together" + πτωτ-ός "fallen". The term was introduced by Apollonius of Perga in his work on conic sections, but in contrast to its modern meaning, he used it to mean any line that does not intersect the given curve. There are three kinds of asymptotes: ''horizontal'', ''vertical'' and ''oblique''. For curves given by the graph of a function, graph of a function (mathematics), function , horizontal asymptotes are horizontal lines tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
