Why is the degree of the node pictured 5, not 4? On the Wikipedia page for degree (graph theory)), the bottom-right vertex pictured is said to have a degree of 5. ![]( My understanding is that the degree of a vertex is defined as the number of edges incident to the vertex. In this case, it appears as if the bottom-right vertex has 4 edges incident to it: {2,5}, {3,5}, {2,5}, and {5,5}. So isn't the degree of the lower-right vertex 4, not 5?

But it enters the vertex twice. I'll try to present a few additional reasons to count it twice.

What can you say about the total sum of degrees in a graph? It's an even number, equal to $2 \cdot \\#E$, where $\\#E$ is the number of edges. This would fall apart if you counted such edges only once.

Also, imagine that it was a directed graph. You'd need the info that the edge both enters and exists vertex $5$.