Complete Graph: Definition & Example Video

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  • 0:04 Complete Graphs
  • 1:52 Complete Graph Properties
  • 3:48 Example
  • 5:20 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

Graphs play an important part in the world around us. This lesson will discuss the definition of a graph in mathematics, and will explore a specific type of graph called a complete graph.

Complete Graphs

Have you ever considered social media to be mathematical? Believe it or not, it is! Look at it like this: we'll represent each person in a social network as a point, and if they're friends on social media with other people in the network, then we will draw a line between those two people's points. Two examples of this are shown in the images below:


In mathematics, we call a visual representation of a network a graph. A graph is a collection of points, called vertices, and line segments connecting those points, called edges. The number of edges that belong to a vertex is called the degree of the vertex.

In these graphs, the people in the network are the vertices, the edges represent a social media friendship between two people, and the degree of each vertex represents how many friends on social media the person represented by that vertex has. Thus, the degree of Nate's vertex is 6, while the degree of Andrea's vertex is 2.

See? Social media is mathematical, and it gets even more interesting! There are two different types of graphs, and one that's particularly fascinating is the complete graph. Consider the two graph examples again. They have one very prominent difference. That is, the first graph has an edge between every single vertex in the graph, but the second graph does not (notice, for example, there is no edge between Dave and Andrea). The first example is an example of a complete graph. A complete graph is a graph that has an edge between every single vertex in the graph; we represent a complete graph with n vertices using the symbol Kn.

Therefore, the first example is the complete graph K7, and the second example isn't a complete graph at all. In the context of these examples, this tells us that in the first example, everyone in the network is friends on social media, but in the second example, some of the people in the network are not friends on social media (such as Dave and Andrea).

Complete Graph Properties

As said earlier, complete graphs are really quite fascinating. But once we know the number of vertices of a complete graph, how do we determine the total numbers of degrees and edges without manually counting each one? Well, because of how complete graphs are defined, they satisfy certain properties. Some of those properties can be calculated as follows:

  • If a complete graph has n vertices, then each vertex has degree n - 1.
  • The sum of all the degrees in a complete graph, Kn, is n(n-1).
  • The number of edges in a complete graph, Kn, is (n(n - 1)) / 2.

Putting these into the context of the social media example, our network represented by graph K7 has the following properties:

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