Edge Transitive Complete Bipartite Graphs And Complete Bipartite Maps

The main purpose of this doctoral thesis is to study two important problems associated with complete bipartite graphs in algebraic graph theory and topological graph theory. One is regarding locally s-arc transitive complete bipartite graphs, and the other is about edge-transitive bipartite maps.The study of locally s-arc transitive graphs was initiated by Tutte’s celebrated work, who showed that a vertex-transitive locally s-arc transitive graph of valency three satisfies s≤5,see [88]. Afterwards, Weiss extended this result to the general case that the graphs with valency at least three in1981, which made the study of locally s-arc transitive graphs to be one of the central topics in algebraic graph theory ([89]). So far, there has been a large number of research papers devoted to studying this class of graphs.During the past20years, the leading algebraist Praeger and her colleagues de-veloped the so-called Giudici-Li-Praeger for analyzing locally s-arc transitive graphs. This theory established an efficient method and shaped a new direction for the s-tudy of symmetrical graphs. In this theory, there is a crucial case regarding basic locally s-arc transitive graph unsettled, namely, characterizing finite groups which act on a complete bipartite graph locally2-arc transitively or locally primitively. This problem is solved in the thesis, the main results see Theorem3.2, Corollary3.1, and Theorem4.1and Corollary4.1.Investigating regular maps has been a hot research topic in topological graph theory for many years. The study of complete bipartite maps which are orientable and edge transitive was begun in1970’s when Biggs and White constructed first examples in their monograph "Permutation Groups and Combinatorial Structures" In Grothendieck’s theory of embedding on Riemann surface, orientable bipartite maps corresponds to algebraic curves defined over the field Q of algebraic num-bers. Studying edge-transitive bipartite maps is thus useful for understanding and characterising algebraic curves. The second topic studied in this thesis is regarding edge transitive bipartite maps, and especially edge transitive complete bipartite maps. First, an explicit rep-resentation of orientable edge transitive bipartite maps and their face set is given in terms of the automorphism group and its two distinguished generators, see Theorem5.1. This representation provides a criterion to determine whether an edge transitive bipartite map is a C-map, see Theorem5.2, and offers a nice description of normal quotients of edge transitive bipartite maps, see Theorem5.3. A one-to-one cor-respondence is then established between edge transitive complete bipartite maps, especially complete bipartite C-maps, and bicyclic groups with a pair of distin-guished generators, see Theorem5.4. This translates the edge-transitive embedding problem of complete bipartite graphs into a group factorization problem. This result provides us with a convenient tool for classifying edge transitive complete bipartite graphs proposed by Jones in2007. As an application, a classification is given of complete bipartite graphs which have a unique edge transitive embedding, see The-orem5.5. Also a classification is obtained for orientable edge-transitive C-maps of the complete bipartite graphs Kpe,pf, where p is an odd prime, see Theorem6.1.