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The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite. Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
Kruskal's algorithm [1] finds a minimum spanning forest of an undirected edge-weighted graph. If the graph is connected , it finds a minimum spanning tree . It is a greedy algorithm that in each step adds to the forest the lowest-weight edge that will not form a cycle . [ 2 ]
In combinatorics, he is known for Kruskal's tree theorem (1960), which is also interesting from a mathematical logic perspective since it can only be proved nonconstructively. Kruskal also applied his work in linguistics , in an experimental lexicostatistical study of Indo-European languages , together with the linguists Isidore Dyen and Paul ...
Knaster–Tarski theorem (order theory) Kruskal's tree theorem (order theory) Shannon's expansion theorem (Boolean algebra) Stone's representation theorem for Boolean algebras (mathematical logic) Szpilrajn extension theorem (axiom of choice)
A notorious attempt to legislate the value of pi as 3.2. Infinite monkey theorem: An infinite number of monkeys typing on an infinite number of typewriters will (almost surely) produce all possible written texts. Interesting number paradox: Either all natural numbers are interesting or else none of them are. Kruskal's tree theorem
Kruskal's tree theorem, which has applications in computer science, is also undecidable from the Peano axioms but provable in set theory. In fact Kruskal's tree theorem (or its finite form) is undecidable in a much stronger system codifying the principles acceptable on basis of a philosophy of mathematics called predicativism.
Numberphile is an educational YouTube channel featuring videos that explore topics from a variety of fields of mathematics. [2] [3] In the early days of the channel, each video focused on a specific number, but the channel has since expanded its scope, [4] featuring videos on more advanced mathematical concepts such as Fermat's Last Theorem, the Riemann hypothesis [5] and Kruskal's tree ...
Let tree Y 2 be the graph obtained by removing edge f from and adding edge e to tree Y 1. It is easy to show that tree Y 2 is connected, has the same number of edges as tree Y 1 , and the total weights of its edges is not larger than that of tree Y 1 , therefore it is also a minimum spanning tree of graph P and it contains edge e and all the ...