Classification Of The Small Valent Cayley Graphs On Some Odd Groups

A Cayley graph Γ of a finite group G is said to be edge-transitive if its full automorphism group acts transitively on E(Γ). A Cayley graph F is said to be normal for a finite group G, if the right regular representa-tion R(G) is a normal subgroup of its full automorphism group Aut(Γ). A2-arc of F is a triple (v0,v1, v2) of vertices such that v0≠v2and v1is adjacent to v0and v2.A graph F is said to be (G,2)-arc transitive if finite group G is a subgroup of Aut(F) and G is transitive on the2-arcs of Γ. In the thesis, small valent Cayley graphs on some odd groups and2-arc transitive graphs admitting a projective unitary group U3(4) are investigated. This thesis can be divided into the following three parts.Firstly, we research the hexavalent edge-transitive Cayley graphs with odd number of vertices, we give a characterization of hexavalent edge-transitive Cayley graphs with odd number of vertices.Secondly, we resolve the normality, arc-transitive property and weak CI-property of connected Cayley graphs of valencies4on a kind of metacyclic groups with order qpn(p. q>3, p≠q are prime, and n≥2), we also determine the complete classification of such graphs.Thirdly, we study the2-arc transitive graphs admitting a projec-tive unitary group U3(4) and give a complete classification of connected graphs Γ which are (U3(4),2)-arc transitive graphs.