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Graphs: vertices, edges, degrees

The basic objects of graph theory and the handshake lemma.

A graph is a set of vertices with edges between some pairs. The handshake lemma: the degrees sum to twice the number of edges, so the number of odd-degree vertices is even. Picture it: dots and lines; a complete graph on 5 vertices has 10 edges, which is 5 choose 2. Think it: graphs are relations drawn; the adjacency matrix makes them linear algebra.

Gewerkte voorbeeld · sum of k for k = 1 to 5

Calculus · Series-And-Sums

Sum k for k = 1 to 5

\sum_{k=1}^{5} k

Stap met stap

  1. \sum_{k=1}^{5} k

    Write the sum out.

  2. 1 + 2 + 3 + 4 + 5

    Expand the terms.

  3. = 15

    Closed form.

Openbaar die antwoord
15

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Hou jou eie werk

'n Vry rekening voeg notas by oor elke les, 'n verslag van wat jy klaar gemaak het, jou opgelos probleme op een plek, en' n tugmeester wat jy kan vra oor hierdie bladsy. Die wiskunde self is oop vir almal, geteken in of nie.

Meld op Meld aan

Simbools wat hier gebruik word

Maak enige simbool van die volledige definisie,'n prent en wat elke letter daarin beteken.

Hoe om: Graphs: vertices, edges, degrees

  1. Write the sum out.
  2. Expand the terms.
  3. Closed form.

Vrae wat mense vra

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.

Meer in Combinatorics & Graph Theory

The counting principlesPigeonhole principle and inclusion, exclusionBinomial coefficients and Pascal's triangleRecurrences and generating functionsPaths, cycles, trees, Euler and HamiltonColouring and planar graphs