More than one definition has been used for the word “normal” as applied to Cayley graphs, but several authors have recently used the following definition: a Cayley graph for a group
is said to be “normal” if
(in its regular action) is normalized by the full automorphism group of the graph. In the present paper the author defines a Cayley graph to be “normal edge-transitive” if its automorphism group contains a subgroup which both normalizes
and acts transitively on the edges. (Throughout the paper the author considers both “graphs,” by which she means directed graphs, and “undirected graphs.”) The author suggests that these form a subfamily of central importance to analyzing the family of all Cayley graphs for a given group. For example, she proves that any Cayley graph for
is an edge-disjoint union of normal edge-transitive Cayley graphs for
. The author also proves results on quotients of Cayley graphs (describing when these are Cayley graphs for a quotient group, etc.), and on constructing all normal edge-transitive Cayley graphs having a given normal edge-transitive Cayley graph as a quotient. The paper concludes with a sample theorem about the full automorphism groups of normal edge-transitive Cayley graphs for finite simple groups.