Let $G$ be a directed graph with $N$ nodes and weighted adjacency matrix $W $ defined by $$ W_{ij} = \left\{ \begin{array}{cl} w_{ij} & \text{ if } \ i \ \text{ is connected to } j \\ 0 & \text{otherwise} \end{array} \right. $$ A walk of length $\ell$ from $i$ to $j$ on the graph can be represented by a monomial $w_{i i_1} ... w_{i_\ell j}$ of degree $\ell$. Here are two known results:
the entries $W^\ell_{ij}$ of $W^\ell$ are the sum of all connected walks from $i$ to $j$.
the coefficients $\psi_0,...,\psi_N$ of the characteristic polynomial $$\chi_W(\lambda) =\operatorname{det}(\lambda I - W) = \sum_{\ell=0}^N \psi_\ell \lambda^{N-\ell}$$ enumerate the self avoiding cycles (walks from a vertex to itself) in the graph. Precisely, $\psi_\ell$ is equal to the sum of all self avoiding cycles (not necessarily connected) of length $\ell$ with an even number of components minus the sum of all self avoiding cycles of length $\ell$ with an odd number of components.
Now, I was doing some simulations and I found the following formula: for $1 \leq \ell \leq N$, the entries of the matrix $$ X^{(\ell)} = \sum_{k=0}^{\ell-1} \psi_k W^{\ell-k} $$ enumerate all self avoiding walks on the graph. Precisely, $X^{(\ell)}_{ij}$ is the sum of all self avoiding walks from $i$ to $j$ of length $\ell$ (with a plus sign if there is an even number of connected components and a minus sign otherwise). I was wondering if this result is known by the graph theory community? I find this result interesting but I couldn't find any paper mentionning it. I would be grateful if anyone can give me some references on this. Thanks.