Cubic Cayley Graphs On PSL (2,p) And Semisymmetric Graphs From Affine Geometries

A simple undirect graph X is said to vertex-transitive, edge-transitive, and arc-transitive, if its automorphism group acts transitively on the vertices, edges and arcs, respectively.Let G be a finite group and S a subset of G not containing the identity element 1 and S-1=S. We define the Cayley graph X=Cay (G, S) of G with respect to 5 by the vertex-set V(X)=G and edge-set E(X)=((g, sg)\g G, s e S).A simple undirected graph is said to be semisymmetric if it is regular and edge-transitive but not vertex-transitive.This thesis consists of two parts. In the first part, we classify the arc-transitive cubic Cayley graphs on the simple groups PSL(2,p), where p is prime at least 5. In the second part, using the affine geometries, we construct an infinite family X(p, n) of semisymmetric graphs of order 2pn and valency p, where p is prime and p > n > 3. Moreover, the smallest member X(3,3) of the family is isomorphic to the Gray graph found in 1932.