Isomorphisms of Cubic Cayley Graphs on Dihedral Groups and Sparse Circulant Matrices

We show that, up to isomorphism, there is a unique non-CI connected cubic Cayley graph on the dihedral group of order 2 n for each even number n ≥ 4. This answers in the negative the question of Li whether all connected cubic Cayley graphs are CI-graphs ( Discrete Math. , 256 , 301–334 (2002)). As an application, a formula is derived for the number of isomorphism classes of connected cubic Cayley graphs on dihedral groups, which generalises the earlier formula of Huang et al. dealing with the particular case when n is a prime ( Acta Math. Sin., Engl. Ser. , 33 , 996–1011 (2017)). As another application, a short proof is also given for a result on sparse circulant matrices obtained by Wiedemann and Zieve (arXiv preprint, (2007)).