Locally Primitive Graphs Of Four Times A Prime

Graphs studied in this thesis are connected, undirected and simple. Let T be a graph with vertex set V, T(υ) be the set of all vertices which adjacent to v, X≤AυtT be a group of automorphisms of T. If for each vertex υ∈V of T, the vertex stabilizer Xv:={x∈X|υx=υ} acts primitively on Tυ, then TF is called X-Iocally primitive. Locally-primitive graphs contains rich families of interesting graphs, such as edge-transitive graphs with prime valency, s-arc-transitive graphs with s≥2).In the literature. locally primitive graphs and2-arc-transitive graphs have re-ceived much attention. Fore example, Ivanov and Praeger classified2-arc-transitive Cayley graphs of elementary abelian p-groups; a characterization of2-arc transitive graphs of prime-power order is given by Li; a characterization of vertex transitive lo-cally primitive graphs of prime-power order is given by L. M; edge transitive graphs of order two times a prime is classified by Cheng. The main purpose of this paper is to characterize locally primitive graphs of order four times a prime.A permutation group is quasiprimitive if each of its nontrivial normal subgroups is transitive; a permutation group G is bi-quasiprimitive if each nontrivial normal subgroups of G has at most two orbits, and there is at least one which has exactly two orbits. The main method using in this thesis is to characterize quasiprimitive and bi-quasiprimitive permutation groups of degree four times a prime, and then by studying their orbital graphs, we reach a characterization of locally primitive graphs of order four times a prime. The result obtained generalizes certain previous results, and some new interesting graphs are founded.