| Let F be an undirected graph with vertex set V, and let X be a group of automorphisms of F, that is, X≤AutT. Forυ∈V, let F(υ) be the set of vertices which are adjacent toυ. Then F is called X-vertex transitive(or simply called vertex transitive) if X is transitive on V, and F is called X-locally prirnitive(or simply called locally primitive) if Xv is primitive on F(υ) for all verticesυ. As usual, the number of vertices of a graph is called the order of the graph, and the size |F(υ)| is called the valency of F atυ. Further,|Γ(v)|is called the valency of F if F is regular.Firstly, we characterization the locally primitive graph of prime- power order, the main result is to extend the result of 2-arc transitive graph of prime power order. This tell us that an undirect locally primitive graph of prime-power order is either a Cayley graph, or a normal cover of a complete bipartite graph. In order to study locally primitive graph In order to Further study the locally primitive graph, we next study the locally primitive graph of order two double prime-power.A framework for studying locally primitive bipartite graphs was established by Giudici, C.H.Li and C.E.Praeger in 2004, which reduces the study to'basic' objects in terms of O'Nan-Scott types. In this thesis, we will study locally primitive graphs based on bi-direct products of graphs. Bi-direct product was used to study homogeneous factorisations of graphs. In particular, we characterize the family of the graphs that are regular, bipartite, locally primitive, and of order 2pe with p prime. Typical examples include the complete bipartite graphs Kpe,pe, the graphs Kpe,Pe-PeK2; D21(11,5) and D21(11,5), which are the incidence graph and non-incidence graph of the 2-(11,5,2)-design, respectively; PH(d,q) and PH(d,q) with d≥3, which are the incidence graph and non-incidence graph of the projective geometry PG(d-1, q), respectively; the standard double cover of the Schlafli graph, other basic graphs are bi-direct powers of these graphs.
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