On the edge-forwarding indices of Frobenius graphs

A G-Frobenius graph Gamma, as defined by Fang, Li, and Praeger, is a connected orbital graph of a Frobenius group G = K x H with Frobenius kernel K and Frobenius complement H. Gamma is also shown to be a Cayley graph, Gamma = Cay(K, S) for K and some subset S of the group K. On the other hand, a network N with a routing function R, written as (N,R), is an undirected graph N together with a routing R which consists of a collection of simple paths connecting every pair of vertices in the graph. The edge-forwarding index pi(N) of a network (N,R), defined by Heydemann, Meyer, and Sotteau, is a parameter to describe the maximum load of edges of N. In this paper, we study the edge-forwarding indices of Frobenius graphs. In particular, we obtain the edge-forwarding index of a G-Frobenius graph Gamma with rank(G) <= 50.