Math 2.0

Combinatorics Proposition

Number of edges of a complete graph

Builds on Binomial coefficientComplete graph

Statement

Let nn be a natural number. Then the complete graph KnK_n has exactly (n2)=n(n−1)2\binom{n}{2} = \frac{n (n - 1)}{2} edges.

Remarks

The edges of KnK_n are exactly the 22-element subsets of the vertex set. The number of k-element subsets gives the count. The handshaking lemma gives the same count: n(n−1)=2∣E∣n (n - 1) = 2 |E|. Each simple graph on nn vertices has at most (n2)\binom{n}{2} edges.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

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Binomial coefficient, definition, no evidence levelBinomial coefficientdefinitionComplete graph, example, no evidence levelComplete graphexampleNumber of edges of a complete graph, statement, no evidence levelNumber of edges of a …statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Binomial coefficient definition
  2. Complete graph example
  3. Number of edges of a complete graph statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Complete_graph, Properties
About this page
Tag05RN90E14Y
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Kindstatement, proposition
WritingW0 stub
MSC05C30
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UsesBinomial coefficient, Complete graph

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