Finite Order Partial Of This Artwork

This dissertation is devoted to the study of finite locally primitive graphs, including, giving some properties on locally primitive graphs, characterizing some classes of locally primitive graphs, and constructing some new classes of finite locally primitive graphs.Let G be a subgroup of the full automorphisms group of a graphГ, thenГis called G-locally primitive if the stabilizer subgroup Ga acts primitively on the neighbor setГ(α) for any vertices ofГ. Further,if G is transitive on the set of 2-arcs of F, then F is called (G,2)-arc transitive. It is known that a class of 2-arc transitive graphs forms a proper subclass of locally primitive graphs.In 1992, Praeger [65] proved that, for a vertex quasiprimitive (G,2)-arc tran-sitive graph, G is one of types HA, AS, TW, or PA. The first main result of this thesis is to prove that, for a vertex quasiprimitive G-locally primitive graph, G can be each of all the eight types of quasiprimitive permutation groups. More-over, the vertex quasiprimitive locally primitive graphs of type HS, SD, HC, or CD are constructed. This result shows the differences of 2-arc transitive graphs and locally primitive graphs, which extends the result about "basic" 2-arc transitive graphs in [65].A graph is called a cube-free order graph if the order of the graph is cube-free. We study next the vertex transitive G-locally primitive graphs of cube-free order. We obtain that either the group G is almost quasi-simple or the graphs are specifically characterized (see Theorem 1.2 for detail). In particular, the cubic symmetric graphs of cube-free order are specifically characterized (see Theorem 1.3). This result generalizes the result of the cubic symmetric graphs with a given order in [25,26,27].The Cayley graphs of abelian groups have received much attention by many scholars. For instance, the 2-arc transitive Cayley graphs of elementary abelian 2-groups are classified by Ivanov and Praeger in [37]; the classification of 2-arc transitive circulants are given in [2], and finite 2-arc transitive Cayley graphs of abelian groups are characterized in [48]. We give a characterization of locally primitive Cayley graphs of abelian groups, which generalizes the classification results mentioned above.Moreover, some classes of edge transitive Cayley graphs are also studied in this thesis. In specific, we characterize the edge transitive Cayley graphs of valency six of odd order, which generalizes the result in [49] about the tetravalent edge transitive Cayley graphs of odd order, and finally we give a description of locally bi-primitive graphs, and develop some relative results of the locally primitive graphs.