On cyclic regular covers of complete graphs of small order

The paper presents classifications of edge-transitive cyclic regular covers of the complete graphs K5 and K6, and arc-transitive cyclic regular covers of the complete graph K7. Two new infinite families of transitive graphs of valency 44 and 66 are found. As an application, tetravalent edge-transitive graphs of order 5p25p2 with pp a prime are classified.