Are beta index, and average degree of a graph the same?

Question

It seems that a beta index, and an average degree of a graph are the same. They both are defined as the ratio of the number of edges to vertices of the graph. Please see definitions here: beta index in graph

and average vertex degree

If both mean the same thing, what is the need of different nomenclatures?

Answer 1

The average vertex degree is always exactly twice the beta index, because each edge connects two vertices. This is a version of the handshaking lemma.

For example, your link states that the beta index of a cycle is $1$, and every vertex of a cycle has degree $2$.