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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.
مثال عملي · sum of k for k = 1 to 5
خطوة خطوة
- \sum_{k=1}^{5} k
Write the sum out.
- 1 + 2 + 3 + 4 + 5
Expand the terms.
- = 15
Closed form.
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الآن أنت اختار مشكلة، أو طبعها أو رسمها بنفسك، كل خطوة، صورة، الإجابة مخفية حتى تسأل.
الرموز المستخدمة هنا
لمس أي رمز للحصول على التعريف الكامل، والصورة، وما يعنيه كل حرف في ذلك.
كيف: Graphs: vertices, edges, degrees
- Write the sum out.
- Expand the terms.
- Closed form.
أسئلة يطرحها الناس
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.
أكثر في 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