Gossiping and routing in second-kind Frobenius graphs

• Authors: Xin Gui Fang;Sanming Zhou
• Venue: European Journal of Combinatorics
• Year: 2012

A Frobenius group is a permutation group which is transitive but not regular such that only the identity element can fix two points. It is well known that such a group is a semidirect product G=K@?H, where K is a nilpotent normal subgroup of G. A second-kind G-Frobenius graph is a Cayley graph @C=Cay(K,a^H@?(a^-^1)^H) for some a@?K with order 2 and =K, where H is of odd order and x^H denotes the H-orbit containing x@?K. In the case when K is abelian of odd order, we give the exact value of the minimum gossiping time of @C under the store-and-forward, all-port and full-duplex model and prove that @C admits optimal gossiping schemes with the following properties: messages are always transmitted along shortest paths; each arc is used exactly once at each time step; at each step after the initial one the arcs carrying the message originated from a given vertex form a perfect matching. In the case when K is abelian of even order, we give an upper bound on the minimum gossiping time of @C under the same model. When K is abelian, we give an algorithm for producing all-to-all routings which are optimal for both edge-forwarding and minimal edge-forwarding indices of @C, and prove that such routings are also optimal for arc-forwarding and minimal arc-forwarding indices if in addition K is of odd order. We give a family of second-kind Frobenius graphs which contains all Paley graphs and connected generalized Paley graphs of odd order as a proper subfamily. Based on this and Dirichlet's prime number theorem we show that, for any even integer r=4, there exist infinitely many second-kind Frobenius graphs with valency r and order greater than any given integer such that the kernels of the underlying Frobenius groups are abelian of odd order.