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.
Arbejdstxrd eksempel · 5 choose 2
Trin for trin
- \binom{5}{2} = \frac{5!}{2!\,(5-2)!}
Unordered selections: n! / (k! (n−k)!).
- = \frac{120}{2 \times 6} = 10
Afslør svaret
Nu er du Vælg et problem, eller skriv eller tegne din egen. Hvert skridt, et billede, svaret skjult, indtil du spørger.
En gratis konto tilføjer noter om hver lektion, en rekord over, hvad du har afsluttet, dine løste problemer på ét sted, og en vejleder, du kan spørge om denne side. Matematik selv er åben for alle, logget på eller ej.
Tilmeld dig LoginSymboler brugt her
Tryk på ethvert symbol for den fulde definition, et billede, og hvad hvert bogstav i det betyder.
Hvordan: Paths, cycles, trees, Euler and Hamilton
- Unordered selections: n! / (k! (n−k)!).
Spørgsmål folk spørger
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.
Mere i Combinatorics & Graph Theory
The counting principlesPigeonhole principle and inclusion, exclusionBinomial coefficients and Pascal's triangleRecurrences and generating functionsGraphs: vertices, edges, degreesColouring and planar graphs