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Turán's Theorem
In graph theory, Turán's theorem bounds the number of edges that can be included in an undirected graph that does not have a complete subgraph of a given size. It is one of the central results of extremal graph theory, an area studying the largest or smallest graphs with given properties, and is a special case of the forbidden subgraph problem on the maximum number of edges in a graph that does not have a given subgraph. An example of an n-vertex graph that does not contain any (r+1)-vertex clique K_ may be formed by partitioning the set of n vertices into r parts of equal or nearly equal size, and connecting two vertices by an edge whenever they belong to two different parts. The resulting graph is the Turán graph T(n,r). Turán's theorem states that the Turán graph has the largest number of edges among all -free -vertex graphs. Turán's theorem, and the Turán graphs giving its extreme case, were first described and studied by Hungarian mathematician Pál Turán in 1941. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Turán's Method
In mathematics, Turán's method provides lower bounds for exponential sums and complex power sums. The method has been applied to problems in equidistribution. The method applies to sums of the form : s_\nu = \sum_^N b_n z_n^\nu \ where the ''b'' and ''z'' are complex numbers and ''ν'' runs over a range of integers. There are two main results, depending on the size of the complex numbers ''z''. Turán's first theorem The first result applies to sums ''s''ν where , z_n, \ge 1 for all ''n''. For any range of ''ν'' of length ''N'', say ''ν'' = ''M'' + 1, ..., ''M'' + ''N'', there is some ''ν'' with , ''s''''ν'', at least ''c''(''M'', ''N''), ''s''0, where : c(M,N) = \left(\right)^ \ . The sum here may be replaced by the weaker but simpler \left(\right)^. We may deduce the Fabry gap theorem from this result. Turán's second theorem The second result applies to sums ''s''ν where , z_n, \le 1 for all ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
