Gossiping and routing in second-kind Frobenius graphs

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 Γ = Cay ( K , a H ∪ ( a − 1 ) H ) for some a ∈ K with order ≠ 2 and 〈 a H 〉 = 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 Γ under the store-and-forward, all-port and full-duplex model and prove that Γ 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 Γ 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 Γ , 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.