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?