Arc-transitive Regular Coverings Of K_5,5-5K₂

Let X be an automorphism group of Γ, that is, X≤Aut (Γ). If X is transitive on the vertex set VΓ, edge set EΓ or arc set AΓ of Γ, then Γ is called X-vertices-transitive, X-edge-transitive, or X-arc-transitive respectively. Let Γ and Σ be two graphs. If there a surjection p from VΓ to VΣ such that the restriction p|Γ(u) Γ(u)â†'Σ(v) is a bijection for each v∈VΣ, and and each preimage u in Γ of v, then Γ is called a regular cover of1. Further, if there is a semiregular subgroup K such that Γ is isomorphic to the quotient graph Γk, say by Ψ, and the quotient map Γ to Γk is the composition p.Ψ, then Γ is called a regular K-cover of Σ. In particular, if K is cyclic or elementary abelian, then is called a regular cyclic or elementary abelian cover of E.Characterizing regular covers of graphs is one of the most important topics in algebraic graph theory, and many nice results have been obtained, including classifications of cyclic or elementary abelian covers of many symmetric graphs with small valency. Especially, W.Q. Xu and S.F. Du classified2-arc-transitive cyclic regular covers of the graph Kn,n-nK2, while cyclic regular covers of the graph Kn,n-nK2with a’weak symmetry’is still open. In this thesis, we will study arc-transitive regular Zn-covers of K5,5-5K2, by using voltage assignment theory of coverings. The obtained result partially generalizes the result of W.Q. Xu and S.F. Du, and a new class of symmetric graphs CC (n,10, i)(see its construction in the thesis) with valency4is founded. Moreover, there is no arc-transitive regular Zp2-cover of K5,5-5K2with p a prime.