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.

Ohatra · 5 choose 2

Discrete-Math · Counting

5 choose 2

5,\ 2

Dingana amin'ny dingana

  1. \binom{5}{2} = \frac{5!}{2!\,(5-2)!}

    Unordered selections: n! / (k! (n−k)!).

  2. = \frac{120}{2 \times 6} = 10

Asehoy ny valinteny
\binom{5}{2} = 10

Ankehitriny ianao Misafidiana olana iray, na soraty na esory ny anao manokana. Ny dingana tsirairay, sary iray, ny valiny miafina mandra-pahafantaranao.

Aoka ho miasa ny anao

Ny kaonty maimaimpoana dia mampiditra ny fanamarihana amin'ny lesona tsirairay, ny tahirin'ny zavatra vitanao, ny olana voavahao ao amin'ny toerana iray, ary ny mpampianatra iray izay azonao anontanio momba ity pejy ity. misokatra ho an'ny rehetra ny matematika, na efa niditra na tsy niditra.

Misoratra anarana Fidirana

Marika ampiasaina eto

Tsindrio ny marika iray mba hahitana ny famaritana feno, sary, ary ny dikan'ny litera tsirairay ao aminy.

Ahoana ny fomba: Paths, cycles, trees, Euler and Hamilton

  1. Unordered selections: n! / (k! (n−k)!).

Fanontaniana napetrak'ireo olona

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.

Mbola maro ao Combinatorics & Graph Theory

The counting principlesPigeonhole principle and inclusion, exclusionBinomial coefficients and Pascal's triangleRecurrences and generating functionsGraphs: vertices, edges, degreesColouring and planar graphs