Suppose there is a finite undirected graph $G(V,E)$ having $n$ vertices and $m$ edges. The non-backtracking matrix $B$ is indexed by $2m$ directed edges and defined as $$ B(a \to b, c \to d) = \delta_{bc} (1 - \delta_{ad}) \, . $$ The adjacency matrix $A$ is defined as usual.
I learned that the Ihara–Bass formula was established in the context of graph zeta functions. At present I only make sense of an elementary proof without diving into graph zeta functions. By this proof, I learned that if $u^2 \neq 1$ then $$ \det(I-uB) = \left(1-u^2\right)^{m-n} \det\left(I - u A + u^2\left[D-I\right]\right) \, , $$ where $D$ is the degree matrix.
As a corollary, all roots of the polynomial $\det(B - \lambda I)$, except those being $\pm 1$, are roots of the polynomial $\det\left(\lambda^2 I - \lambda A + \left[D-I\right]\right)$.
For all simple graphs, we have $ \det(D - A) = 0 $. But for trees, the non-backtracking matrix has no eigenvalues other than $0$, which leads to $ \det(I - B) = \det(I) = 1 $. Thus the polynomial $\det\left(\lambda^2 I - \lambda A + \left[D-I\right]\right)$ can not be used to search the eigenvalues of $B$ being $1$.
Can the corollary be extended to search the eigenvalues of $B$ being $\pm 1$?
The motivation behind this question is to get the whole spectrum of the non-backtracking matrix of a graph with a much smaller matrix.
