Enumeration of cubic Cayley graphs on dihedral groups (Q2404069)

Finite Cayley graphs are well studied in many aspects. For an odd prime \(p\), the dihedral group \(D_{2p}\) is defined by \(D_{2p}=\langle a,b\mid a^p=b^2=1, bab=a^{-1}\rangle.\) In this paper, the authors attempt to classify the cubic Cayley graphs on \(D_{2p}\) using spectral properties. More specifically, the authors prove that two cubic Cayley graphs on \(D_{2p}\) are isomorphic if and only if they are cospectral. Also, they obtain the number of isomorphic classes of cubic Cayley graphs on \(D_{2p}\) using Gauss's law of quadratic reciprociy.