01121487

j

01121487 Fang, Xin Gui Li, Cai Heng Praeger, Cheryl E. On orbital regular graphs and Frobenius graphs. Discrete Math. 182, No.1-3, 85-99 (1998). 1998 Elsevier Science B.V. (North-Holland), Amsterdam EN orbital regular graph Frobenius group edge forwarding index Zbl 0807.05037

doi:10.1016/S0012-365X(97)00148-9

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 Sol\'{e}'s formula for the edge-forwarding index of orbital-regular graphs, see {\it P. Sol\'e} [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. Herman J.Servatius (Worcester)