Locally Primitive Graphs Of Six Times A Prime

Graphs considered in this thesis are connected、 undirected and simple. Let Γ be a graph with vertex set VT and let v∈VT. Denote by Γ(v) the neighbor set of v in Γ, that is, Γ(v)={u～v|u∈VΓ}. Then Γ is called X-locally-primitive, with X≤AutΓ, if the vertex stabilizer Xv:={x|vx=x} acts primitively on Γ(v) for each v∈VΓ. Clearly, esge-transitive graphs with prime valency is locally-primitive.The class of locally-primitive graphs is one of most important classes in algebraic graph theory, and it contains a lot of important subfamilies of graphs (for example, s-arc-transitive graphs with s≥2) and has nice properties, and thus received much attention in the literature. The main aim of this thesis is to characterize locally-primitive graphs of order6p with p is a prime. Suppose Γ is a vertex-transitive X-locally-primitive graph of order6p. Our discussion can be divided into three cases:(1) X is quasiprimitive on VΓ, that is, each nontrivial normal subgroup of X is transitive;(2) X is bi-quasiprimitive on VΓ, that is, X is not quasiprimitive, and each nontrivial normal subgroup of X has at most two orbits on VΓThe study of cases (1-2) depends on the characterization of X by using permu-tation group theory. Using well-known classification of Praeger on quasiprimitive permutation groups, most subcases can be reduced to almost simple group case. By determine these simple groups and analyze the corresponding orbital graphs, the graphs Γ are determined. The main results in this thesis generalize certain previous results, and some new classes of locally-primitive graphs are constructed.