Edge-transitive Cyclic And Z_p~2Cover Of The Three-dimensional Hypercube

The regular coverings of graphs is an important research field of algebraic graph theory. Gross and Tucker established a ’voltage-assignment’method for the studying of regular covering. Since then, covering theory of graphs has been widely used in the study of symmetric graphs, and has became a very important method for constructing and characterizing new classes of symmetric graphs, and a series of results have been obtained in the literature. For example,Fengyanquan obtained some fundamental properties of regular covering, voltage assignment and lifts of automorphisms; He also classified regular coverings of certain symmetric graphs with small valency; And he characterized the regular coverings of some complete bipartite graphs and the three-dimensional hypercube; Dushaofei classified2-arc-transitive cyclic regular coverings and certain elementary abelian coverings of complete graphs. This thesis aims to study edge-transitive cyclic and Zp2-cover of three-dimensional hypercube with p a prime.Graphs considered in the thesis are assumed to be connected, undirected and simple. A graph is called edge-transitive or arc-transitive if its full automorphism group is transitive on its edge set or arc set respectively. Let s be a positive integer. An s arc of a graph F is a sequence of vertices(υ0,υ1,···,υs) such that υi-1～υi for1≤i≤s, and υi-1≠υi+1for1≤i≤s-1. Then F is called s-arc transitive if its full automorphism group is transitive on the set of s-arcs of F. By using voltage assignment theory of coverings and certain theory of permutation groups, a complete classification of edge-transitive cyclic and Zp2-cover of three-dimensional hepercube with p a prime are obtained. Moreover, we prove that all edge-transitive cyclic coverings are arc-transitive, and work out the conditions when the coverings are2-arc-transitive; we also prove that all edge-transitive Zp2-coverings with p a prime are2-arc-transitive.