Distance antimagic labeling of circulant graphs

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A distance antimagic labeling of graph $ G = (V, E) $ of order $ n $ is a bijection $ f:V(G)\rightarrow \{1, 2, \ldots, n\} $ with the property that any two distinct vertices $ x $ and $ y $ satisfy $ \omega(x)\ne\omega(y) $, where $ \omega(x) $ denotes the open neighborhood sum $ \sum_{a\in N(x)}f(a) $ of a vertex $ x $. In 2013, Kamatchi and Arumugam conjectured that a graph admits a distance antimagic labeling if and only if it contains no two vertices with the same open neighborhood. A circulant graph $ C(n; S) $ is a Cayley graph with order $ n $ and generating set $ S $, whose adjacency matrix is circulant. This paper provides partial evidence for the conjecture above by presenting distance antimagic labeling for some circulant graphs. In particular, we completely characterized distance antimagic circulant graphs with one generator and distance antimagic circulant graphs $ C(n; \{1, k\}) $ with odd $ n $.