Quotients of Normal Edge-Transitive Cayley Graphs

The symmetry properties of mathematical structures are often important for understanding these structures. Graphs sometimes have a large group of symmetries, especially when they have an algebraic construction such as the Cayley graphs. These graphs are constructed from abstract groups and are vertex-transitive and this is the reason for their symmetric appearance. Some Cayley graphs have even stronger symmetry properties such as edgetransitivity. In this thesis we will investigate the connections between certain edgetransitive Cayley graphs of a finite group G and edge-transitive Cayley graphs of certain quotient groups of G. The main part of the thesis is a complete account of this new theoretical approach in the case where G is an abelian group of order a product of two primes. Every Cayley graph of a finite group G can be expressed as an edgedisjoint union of edge-transitive Cayley graphs for the same group G, and moreover we may require that the edge-transitive Cayley graphs have a stronger property, that of being “normal edge-transitive”. The family of normal edge-transitive Cayley graphs has the additional property that for each characteristic subgroup M of a group G, each normal edge-transitive Cayley graph for G has a naturally defined quotient graph which is a normal edgetransitive Cayley graph for the quotient group G/M . Thus it is of special interest to study normal edge-transitive Cayley graphs of characteristically simple groups, and the relationship between a given normal edge-transitive Cayley graph for G and its quotients of this type. A major unsolved problem is that of classifying all the normal edgetransitive Cayley graphs for a group G which produce a given normal edgetransitive Cayley graph as a quotient by this method. In this thesis we solve a special case of this problem where G is abelian of order pq (p, q primes)