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 iswell known that such a group is a semidirect productG = KoH , where K is a nilpotent normal subgroup of G. A second-kind GFrobenius graph is a Cayley graph Γ = Cay(K , aH ∪ (a−1)H) for some a ∈ K with order ≠ 2 and ⟨aH⟩ = K , where H is of odd order and xH 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 fullduplex 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 themessage 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 edgeforwarding 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. © 2012 Elsevier Ltd. All rights reserved. E-mail addresses: [email protected] (X.G. Fang), [email protected] (S. Zhou). 0195-6698/$ – see front matter© 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.ejc.2012.01.008 Author's personal copy 1002 X.G. Fang, S. Zhou / European Journal of Combinatorics 33 (2012) 1001–1014