Exercise 2.5.18. Consider a directed graph of 
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 nodes has a connection to the 
other nodes. This would normally produce a total of 
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 
to node 
counts as a connection for both nodes.) The total number of edges in the complete graph is therefore 
.
For example, a 2-node complete graph has 
edge, a 3-node complete graph has 
edges, a 4-node complete graph has 
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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