maths.free › Combinatorics & Graph Theory › Paths, cycles, trees, Euler and Hamilton
Paths, cycles, trees, Euler and Hamilton
Connectivity, spanning trees, Euler circuits and the bridges of Königsberg.
A tree is a connected graph with no cycles: n vertices, n − 1 edges. Euler proved a graph has a circuit using every edge once exactly when every degree is even. The Königsberg bridges failed. Picture it: the seven bridges; four land masses with odd degree. Think it: Euler's condition is local (degrees) yet decides a global question, the first theorem of topology.
ຕົວຢ່າງທີ່ໄດ້ເຮັດ · 5 choose 2
ຂັ້ນຕອນຕໍ່ຂັ້ນຕອນ
- \binom{5}{2} = \frac{5!}{2!\,(5-2)!}
Unordered selections: n! / (k! (n−k)!).
- = \frac{120}{2 \times 6} = 10
ເປີດເຜີຍຄຳຕອບ
ປັດຈຸບັນທ່ານ ເລືອກບັນຫາ ຫຼື ພິມ ຫຼື ແຕ້ມເອງ. ທຸກບາດກ້າວ, ຮູບ, ຄໍາຕອບຖືກລົບຈົນກວ່າເຈົ້າຈະຖາມມັນ.
ຕົວສະແດງທີ່ໃຊ້ຢູ່ທີ່ນີ້
ກົດຕົວອັກສອນໃດໜຶ່ງເພື່ອເບິ່ງຄວາມໝາຍເຕັມ, ຮູບ ແລະ ຕົວອັກສອນແຕ່ລະຕົວໃນມັນໝາຍຄວາມວ່າແນວໃດ.
ວິທີການ: Paths, cycles, trees, Euler and Hamilton
- Unordered selections: n! / (k! (n−k)!).
ຄໍາຖາມທີ່ຄົນຖາມ
Permutation or combination?
Ask whether order matters. A lock code is a permutation (order matters); a hand of cards is a combination (it does not).
What is a graph in this sense?
Dots (vertices) joined by lines (edges), not a plot. Road maps, social networks and molecules are graphs; questions like "is there a route" and "how few colours" are graph theory.
ເພີ່ມເຕີມໃນ Combinatorics & Graph Theory
The counting principlesPigeonhole principle and inclusion, exclusionBinomial coefficients and Pascal's triangleRecurrences and generating functionsGraphs: vertices, edges, degreesColouring and planar graphs