What is a Graph? · Degree Sum Formula
Lesson 2
In the example below, near each point, we show the number of other points it is connected to. There are four points that are joined with five other points.
As it usually happens, playing around with the puzzle will lead us to a solution eventually. To do this, let us add segments one by one.
For each of the five pictures above, the sums of all the integers around the cycle are: \(0, 2, 4, 6, 8\). It seems that each time we add a new segment, the sum increases by two. And this is indeed the case: when we add a segment, the integers next to the endpoints of the segment are increased by one, hence the total is increased by two. In other words, we have just proved the following invariant:
the sum of the integers around the circle is twice the number of segments.This, in particular, means that it is impossible to get all integers equal to five: the sum in this case is equal \(9 \times 5=45\), so one would need \(45/2=22.5\) segments for this.