Equivalence of adjacency and Laplacian spectral gaps

Determine whether, for bounded-degree Borel probability-measure-preserving graphs in the non-regular setting, the adjacency operator having a spectral gap is equivalent to the Laplacian operator having a spectral gap.

Background

The paper defines two potentially different notions of spectral gap for a non-regular bounded-degree Borel pmp graph: an adjacency spectral gap, in which the maximum spectral value of the adjacency operator is isolated, and a Laplacian spectral gap, in which zero is isolated in the spectrum of the Laplacian operator. For regular graphs these notions are equivalent through the relation between the adjacency and Laplacian operators, but that relation does not persist in general for non-regular graphs.

The authors use both notions independently in subsequent results and explicitly leave unresolved whether they are equivalent in the Borel setting. Resolving this would clarify whether the more natural Laplacian formulation can replace the adjacency formulation in spectral characterizations and bipartiteness results.