The question is: how can we recover the graph Laplacian or its spectrum from the Hashimoto Matrix (also commonly called the Non-backtracking operator)?

To make the question as self-contained as possible, here I give the main definitions:

Let $W$ be a given symmetric adjacency matrix of graph $G = (V,E)$.

Let $D$ be a diagonal matrix such that $D_{ii} = d_i$, where $d_i$ is the degree of vertex $i$, i.e. $d_i = \sum_i^n w_{ij}$

Then,

The Laplacian matrix $L$ is defined as $$ L=D-W $$

The Hashimoto matrix $B$ is defined as $$ B_{(i \rightarrow j), (k \rightarrow l)} = \begin{cases} 1 & j=k \text{ and } i\neq l \\ 0 & \text{ otherwise } \end{cases} $$

**Observation:**See that the Hashimoto Matrix is a kind of directed linear graph of $G$.

Is there an operator such that we can recover $L$ from $B$, i.e. an operator $\mathcal{M}$ such that $\mathcal{M}(B) = L$?

Is there any relation between the spectrum of $B$ and $L$?

Answers with references are highly appreciated.