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Let $\Gamma=(V,E)$ be a graph of order $n$. A {\em closed distance magiclabeling} of $\Gamma$ is a bijection $\ell : V \to \{1,2, \ldots, n\}$ forwhich there exists a positive integer $r$ such that $\sum_{x \in N[u]} \ell(x)= r$ for all vertices $u \in V$, where $N[u]$ is the closed neighborhood of$u$. A graph is said to be {\em closed distance magic} if it admits a closeddistance magic labeling.  In this paper, we classify all connected closed distance magic circulantswith valency at most $5$, that is, Cayley graphs$\operatorname{Cay}(\mathbb{Z}_n;S)$ where $|S| \le 5$ and $S$ generates$\mathbb{Z}_n$.

On closed distance magic circulants of valency up to $5$