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Chapter 7: Problem 22

How many (undirected) edges are there in the complete graphs \(K_{6}, K_{7}\), and \(K_{n}\), where \(n \in \mathbf{Z}^{+} ?\)

Short Answer

Expert verified
There are 15 undirected edges in the complete graph \(K_{6}\), 21 undirected edges in the complete graph \(K_{7}\), and in general, \(\frac{n(n - 1)}{2}\) undirected edges in the complete graph \(K_{n}\), where \(n\) is a positive integer.

Step by step solution

01

Calculation for \(K_{6}\)

To calculate the number of edges in the complete graph \(K_{6}\), substitute \(n = 6\) in the formula. It results in \(\frac{6(6 - 1)}{2} = 15\). So, there are 15 undirected edges in the complete graph \(K_{6}\).
02

Calculation for \(K_{7}\)

Similarly, substitute \(n = 7\) in the formula to calculate the number of edges in \(K_{7}\). The calculation is \(\frac{7(7 - 1)}{2} = 21\). So, there are 21 undirected edges in the complete graph \(K_{7}\).
03

General calculation for \(K_{n}\)

For a general \(n\)-vertex complete graph \(K_{n}\), where \(n\) is a positive integer, the number of edges can be calculated by substituting \(n\) in the formula, which results in \(\frac{n(n - 1)}{2}\) edges. This is the general solution for any \(n\).

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