Linear Algebra and Its Applications, Exercise 2.5.18

Exercise 2.5.18. Consider a directed graph of n nodes, where every node has a edge connecting it to every other node. How many edges are in this complete graph?

Answer: Each of the n nodes has a connection to the n-1 other nodes. This would normally produce a total of n(n-1) connections; however in order to avoid double counting a given edge we have to divide this value by 2. (In other words an edge from node i to node j counts as a connection for both nodes.) The total number of edges in the complete graph is therefore n(n-1)/2.

For example, a 2-node complete graph has 2 \cdot (2-1)/2 = 2/2 = 1 edge, a 3-node complete graph has  3 \cdot (3-1)/2 = 6/2 = 3 edges, a 4-node complete graph has 4 \cdot (4-1)/2 = 12/2 = 6 edges, and so on.

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.

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