On Regular Covers Of Several Classes Of Symmetric Graphs And Related Research

This doctoral thesis is mainly devoted to study arc-transitive cyclic and meta-cyclic regular covers of certain families of symmetric graphs, and some relative prob-lems.Characterizing regular covers of symmetric graphs is one of the fundamental topics in the field of algebra graph theory, and is often a key step for approaching general symmetric graphs (see definition of regular cover in Chapter 2). With the efforts of many researchers, a theory of voltage graphs for studying regular covers has been obtained, which is generally powerful for determining cyclic and elementary abelian regular covers of symmetric graphs with ’small order’. By the theory, edge-transitive or arc-transitive cyclic and elementary abelian regular covers of a lot of small symmetric graphs have been classified. Moreover,2-arc-transitive cyclic and partial elementary abelian regular covers of two typical graphs Kn and Kn,n - nK2 have been classified. However, the results appearing in the literature regarding regular covers have the following limitations:(1) most contributions are treating ’abelian’(mostly cyclic and elementary abelian) regular covers of symmetric graphs with ’small order’.(2) characterizations of regular covers of infinite families of graphs are rare, and are almost dealing with the regular covers under the assumption of high symmetry (2-arc-transitive).Therefore, characterizing ’nonabelian’ regular covers of small symmetric graphs and regular covers of infinite families of graphs with ’weaker’ symmetry (for example, edge-transitive or arc-transitive) are definitely interesting topics. This thesis will study some relative problems. In precise, we obtain the following results.1. Classifications of arc-transitive cyclic and certain metacyclic regular covers of prime-valent symmetric graphs with order twice a prime are obtained. We remark that the family of symmetric prime-valent graphs of order twice a prime contain the Petersen graph, Heawood graph and some other small symmetric graphs, and two infinite families of graphs:the complete graph K2P with p and 2p-1 primes, and the complete bipartite graph Kp,p with p a prime, and a class of normal Cayley graphs of a dihedral group.2. Edge-transitive metacyclic regular covers of the Petersen graph have been determined, which exactly consist of seven specific symmetric graphs.We notice that the results obtained in above 1 and 2 generalize a series of previous results in the literature.3. For the better understanding of Cayley graphs of abelian groups, we com-pletely determine almost simple and M-transitive permutation groups containing a transitive abelian subgroup (this topic has independent interest in permutation group theory), which generalizes corresponding results of Praeger and Li. A transi-tive permutation group is called M-transitive if it has a transitive minimal normal subgroup.4. A characterization of pentavalent symmetric graphs of order Apn with p a prime is achieved. In particular, it is proved that there is no such graph if p> 3, thus reducing the study to the special cases where p=2 and 3.