Distance-regular circulants

A classification of strongly regular Cayley graphs of cyclic group was independently achieved by Bridges and Mena (Ars Combin. 8 (1979) 143), Hughes, van Lint and Wilson (Announcement at the Seventh British Combinatorial Conference (1979)), and Ma (Discrete Math. 52 (1984) 75). Here we discuss their results in a more general setting of distance-regular Cayley graphs, and give the complete classification of distance-regular Cayley graphs of cyclic groups. In particular, we prove that a Cayley graph of a cyclic group is distance-regular if and only if it is isomorphic to a cycle, or a complete graph, or a complete multipartite graph, or a complete bipartite graph on a twice an odd number of vertices with a matching removed, or the Paley graph with a prime number of vertices.