Arc Transitive Cyclic Regular Covers Of The Complete Bipartite Graph K_4,4

Characterizing regular covers of symmetric graphs is one of the most important topics in algebraic graph theory, and has received much attention in the literature. By using covering theory, cyclic and elementary abelian covers of many symmetric graphs with small valency have been classified. For example, see Feng and Kwak for s-egular cubic graphs as coverings of the complete bipartite graph K3,3, see Wang and Chen for semisymmetric cubic graphs as regular covers of K3,3, see Feng and Wang for s-egular cyclic covering of K4,4-4K2, see Chen wen for arc-transitive regular Zp-overs with p a prime of K4,4, arc-transitive elementary abelian covers of the complete graph K5is obtained, see Malnic, Marusic and Potocnik for elementary abelian covers of graphs, see Du, Marusic and Waller for on2-arc-transitive covers of complete graphs Kn.In this thesis, we will characterize arc-transitive regular cyclic covers of K4,4. We first determine all minimal arc-transitive automorphism groups of K4,4, which exactly consist of six conjugate classes. Then, by using voltage assignment theory, we investigate the lift problem of the corresponding automorphisms of basic graph, and determine the coverings for each of the three minimal arc-transitive automorphism groups up to isomorphism. Let Γ be a connected arc-transitive regular Zn-over of, where n is a positive integer. It is proved that either Γ≌R₈（6,5）, CC（2;1）, CC（n;k） with k even, CC（n;k）, or CC（n;I,m,p）（see the specific constructions of these graphs in the thesis）. The result of Chen wen is generalized, and three new classes of symmetric graphs are founded.