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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.

हल किया गया उदाहरण · sum of k for k = 1 to 5

Calculus · Series-And-Sums

Sum k for k = 1 to 5

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

चरण दर चरण

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

    Write the sum out.

  2. 1 + 2 + 3 + 4 + 5

    Expand the terms.

  3. = 15

    Closed form.

जवाब दिखाएँ
15

अब आप कोई प्रश्न चुनें, या अपना प्रश्न टाइप करें या बनाएं। हर चरण, एक चित्र, और उत्तर तब तक छिपा रहता है जब तक आप न पूछें।

अपना काम सहेजकर रखें

एक स्वतंत्र वृत्तान्त प्रत्येक पाठ पर आगे कहता है, जो आपने पूरा किया है, आपके हल किये गये समस्याओं का रिकॉर्ड, और एक शिक्षक जिसके बारे में आप पूछ सकते हैं ।

साइन अप करें लॉगिन

यहाँ उपयोग में संकेत

पूरी परिभाषा, चित्र, और इसमें हर अक्षर का अर्थ क्या है?

कैसे: Graphs: vertices, edges, degrees

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

लोग जो सवाल पूछते हैं

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.

More 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