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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.

Urugero · sum of k for k = 1 to 5

Calculus · Series-And-Sums

Sum k for k = 1 to 5

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

ku

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

    Write the sum out.

  2. 1 + 2 + 3 + 4 + 5

    Expand the terms.

  3. = 15

    Closed form.

i
15

Guhitamo A, Cyangwa Ubwoko: Cyangwa Gushushanya. Intera, A Ishusho, i Gihishwe & Kugeza & Kubaza.

Kuri Gushyiraho

A Kigenga Konti: & OYA & Ongera Ibisobanuro: ku, A Ibijyana Bya, in Rimwe, na A Kubaza Bigyanye iyi Ipaji:. Imibare ni Gufungura Kuri, in Cyangwa OYA.

Kwiyandikisha Ifashayinjira

Byakoreshejwe

Icyo ari cyo cyose Ibimenyetso ya: i Byuzuye Insobanuro:, A Ishusho, na buri Ibaruwa: in.

Kuri: Graphs: vertices, edges, degrees

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

Abaturage

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.

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