Let $G = (V,E,W)$ be a weighted graph, where each edge $e = (v_i,v_j)$ has weight $w_{ij} \in \mathbb Z^+ \cup \{0\}$. By replacing $e$ with $w_{ij}$ copies of unweighted multiedges, a weighted graph $G$ is transferred as an unweighted multigraph. For example, a $2$-walk $v_0 v_1 v_2$ in the weighted graph version will correspond to $(w_{01} \times w_{12})$ $2$-walks in the unweighted multigraph version.

Let $v \in V(G)$. What can one say about the number of $k$-walks containing $v$, with regard to the types of walks, and the number of walks of each type? For instance, in the case of $2$-walks, there are three types of such walks:- (i) Closed $2$-walks containing the vertex $v$, (ii) Non-closed $2$-walks containing the vertex $v$ as one of the end-vertices, (iii) Non-closed $2$-walks containing the vertex $v$ as the middle point. How does one characterize these types in general for $k$-walks and compute the number of walks of each type?