Let $G$ be a weighted graph with a weight function $w longrightarrow mathbb{R}^{+}$. Let $G’$ denotes the weighted matrix with adjacency matrix

$$A_{G’} = sum_{i=0}^{k} (xA)^{i}$$

where $k$ is integer and $x$ is a variable.

I am not getting what is $A_{G’}$ matrix? Is it contains all walks of length $k$ or is it something else?