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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 — 5 choose 2. Think it: graphs are relations drawn; the adjacency matrix makes them linear algebra.

ምሳሌ: sum of k for k = 1 to 5

Calculus · Series-And-Sums

Sum k for k = 1 to 5

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

እርምጃ በደረጃ

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

    Write the sum out.

  2. 1 + 2 + 3 + 4 + 5

    Expand the terms.

  3. = 15

    Closed form.

መልሱን አሳይ
15

Symbols used here

\sum_{k=1}^{n} a_k
summation
Add a_k for k = 1 up to n.
n!
factorial
n × (n−1) × … × 1; the number of orderings of n things. 0! = 1.
\binom{n}{k}
binomial coefficient, "n choose k"
Number of k-element subsets of n things: n!/(k!(n−k)!).
\prod_{k=1}^{n} a_k
product
Multiply a_k for k = 1 up to n.
\emptyset,\ |A|
empty set, cardinality
The set with no elements; the number of elements of A.

How to: Graphs: vertices, edges, degrees

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

Questions people ask

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.

የራስዎን ይሞክሩ

በ 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