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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.
ਕੰਮ ਉਦਾਹਰਨ · sum of k for k = 1 to 5
ਕਦਮ ਦਰ ਕਦਮ
- \sum_{k=1}^{5} k
Write the sum out.
- 1 + 2 + 3 + 4 + 5
Expand the terms.
- = 15
Closed form.
ਜਵਾਬ ਦਿਓ
ਹੁਣ ਤੁਸੀਂ ਇੱਕ ਸਮੱਸਿਆ ਚੁਣੋ, ਜਾਂ ਆਪਣਾ ਆਪ ਲਿਖੋ ਜਾਂ ਖਿੱਚੋ। ਹਰ ਕਦਮ, ਇੱਕ ਤਸਵੀਰ, ਜਵਾਬ ਜਦੋਂ ਤੱਕ ਤੁਸੀਂ ਪੁੱਛਦੇ ਹੋ, ਲੁਕਿਆ ਰਹਿੰਦਾ ਹੈ।
ਇੱਥੇ ਵਰਤੇ ਨਿਸ਼ਾਨ
ਕਿਸੇ ਵੀ ਨਿਸ਼ਾਨ ਨੂੰ ਦਬਾਉ, ਜਿਸ ਵਿੱਚ ਪੂਰੀ ਪਰਿਭਾਸ਼ਾ, ਇੱਕ ਤਸਵੀਰ ਅਤੇ ਹਰ ਅੱਖਰ ਦਾ ਅਰਥ ਹੈ ।
ਕਿਵੇਂ: Graphs: vertices, edges, degrees
- Write the sum out.
- Expand the terms.
- Closed form.
ਲੋਕ ਪੁੱਛਦੇ ਹਨ
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.
ਹੋਰ ਵਿੱਚ 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