Thanks in advance!
You do know, I hope, that these definitions are not unique or standard.
So if we mean by a walk a sequence of alternating vertices and edges, beginning with a vertex and ending with a vertex, not necessarily the same.
A path is a walk in which all edges are distinct.
(Again these may not agree with your definitions)
So is a star-like graph with n+1 vertices, 0 to n, both “010” and “102” are walks but only “102” is a path.
Thus the number of walks of length two is . WHY?
But the number of paths of length two is . WHY?
The number of walks of length two is because you either just choose two edges from n (all the paths) or you can walk from 0 to any vertex and back and there are n ways to do this.
and the number of paths of length two is because you just choose two edges from n and because they are all adjacent through 0 you have your path.