Which finite simple graphs $G$ admit labelings of there vertices $f : V(G) \to \mathbb{Z}$ so that $\det(A) = \pm 1$ where $A = [a_{ij}]$ is the adjacency matrix of $G$ with diagonal $a_{ii} = f(v_i)$. I have found a few such labeled graphs but I am wondering if there is any sort of classification.
Aside on motivation: I am wondering which odd dimensional homology spheres I can get as boundaries of plumbings of sphere bundles. These will exactly correspond to the aforementioned labeled graphs.
Thanks!
If I understood your question correctly, you're looking for a class of unimodular graphs with weighted edges (you can consider linear graph $L(G)$ with assigned weights $f(v_i)$ to get rid of pseudo-loops on the diagonal, which will make the whole thing easier). If you restrict your labeling to $f: e \rightarrow \{-1, 1\}$, certain bipartite graphs can meet these conditions.
From the top of my head, that's one of the few studied cases of unimodular graphs with some sort of labeling (though I might be wrong here).