Abstract. We investigate an (a, d)-distance antimagic labeling of a graph G = (V, E) of order n. Graph which admits such a labeling is called an (a, d)-distance antimagic graph. We analyze the necessary conditions for the existence of this labeling. We obtain the results that expend a family of not (a, d)-distance antimagic graphs. In particular, we prove that the crown P_n ○ P₁ does not admit an (a, 1)-distance antimagic labeling for n ≥ 2 if a ≤ 2. We determine the values of a at which path P_n can be an (a, 1)-distance antimagic graph. Among regular graphs, we investigate the case of a circulant graph.

Keywords: distance magic labeling, distance antimagic labeling, (a, d)-distance antimagic labeling, path, regular graph, circulant graph.