Neighborhood characterization of distance-antimagic graphs

Characterize precisely the finite simple graphs that admit a distance-antimagic labeling by proving or disproving that a graph is distance antimagic if and only if no two distinct vertices have identical neighborhoods.

Background

Distance-antimagic labeling was introduced for undirected graphs by Kamatchi and Arumugam. The conjecture proposes that the only obstruction to such a labeling is the presence of two vertices with identical neighborhoods, since identical neighborhoods necessarily force equal vertex weights under every labeling.

The paper notes that this conjecture had been computationally verified for all graphs of order at most 8, but it remains presented as a conjecture rather than a theorem.