Graphs and NetworksEuler

The simplest graph consists of a single vertex. We can easily check that Euler’s equation works.

Let us add a new vertex to our graph. We also have to add an edge, and Euler’s equation still works.

If we want to add a third vertex to the graph we have two possibilities. We could create a small triangle: this adds one vertex, one face and two edges, so Euler’s equation still works.

Instead we could simply extend the line by one: this adds one vertex and one edge, and Euler’s equation works.

Let’s keep going: if we now create a quadrilateral we add one vertex, two edges and one face. Euler’s equation still works.