On orbital regular graphs and Frobenius graphs (Q1379826)

A group is a Frobenius group if it acts transitively but not freely on a set such that no two elements are fixed by a non-trivial element of the group. An orbital-regular graph is a finite graph whose automorphism group has a subgroup which is transitive on the edges and contains no element which fixes two vertices. The authors show that every connected orbital-regular graph is either a cycle, a star, or a Frobenius graph, that is connected orbital-regular graph corresponding to a Frobenius group, and that every Frobenius graph is a Cayley graph. This more precise group theoretical description allows them to improve the computability of Patrick Solé's formula for the edge-forwarding index of orbital-regular graphs, see \textit{P. Solé} [Discrete Math. 130, No. 1-3, 171-176 (1994; Zbl 0807.05037)]. They also examine the structure of quotients of orbital-regular graphs using the Sylow theorems.