Circulant double coverings of a circulant graph of valency five (Q856812)

In the present paper, the authors carry on the study of isomorphism classes of circulant double coverings of circulant graphs (where a covering is called circulant if its covering graph is circulant). In [Discrete Math. 277, 73--85 (2004; Zbl 1033.05081)], they already enumerated the isomorphism classes of circulant double coverings of a certain kind, called typical, and showed as a consequence that no double covering of a circulant graph of valency three is circulant. Further, in [Graphs Comb. 21, 385-400 (2005; Zbl 1090.05056)], the isomorphism classes of connected circulant double coverings of a circulant graph of valency four were enumerated. Now, they obtain the following result on the number of isomorphism classes of circulant double coverings of a circulant graph of valency five: Let \(n\) be even and \(G=\text{Cay}(\mathbb Z_n, \{\pm i,\pm j, \frac n 2\})\) be a connected circulant graph of valency 5 with order \(n,\) where \(0 < i \neq j \leq \frac n 2.\) Then the number of isomorphism classes of circulant double coverings of \(G\) is 1 if \(i + j = \frac n 2,\) and 0 otherwise.