Gossiping and routing in undirected triple‐loop networks

Given integers n ≥ 7 and a, b, c with 1 ≤ a, b, c ≤ n − 1 such that a, n − a, b, n − b, c, n − c are pairwise distinct, the (undirected) triple-loop network TLn(a, b, c) is the degree-six graph with vertices 0, 1, 2, . . . , n − 1 such that each vertex x is adjacent to x ± a, x ± b, and x ± c, where the operation is modulo n. It is known that the maximum order of a connected triple-loop network of the form TLn(a, b, n − (a + b)) with given diameter d ≥ 2 is nd = 3d 2 + 3d + 1, which is achieved by TLnd = TLnd (1, 3d +1, 3d 2−1). In this article, we study the routing and gossiping problems for such optimal tripleloop networks under the store-and-forward, all-port, and full-duplex model, and prove that they admit “perfect” gossiping and routing schemes which exhibit many interesting features. Using a group-theoretic approach we develop for TLnd a method for systematically producing such optimal gossiping and routing schemes. Moreover, we determine the minimum gossip time, the edgeand arc-forwarding indices, and the minimal edgeand arcforwarding indices of TLnd , and prove that our routing schemes are optimal with respect to these four indices simultaneously. As a key step towards these results, we prove that TLnd is a Frobenius graph on a Frobenius group with Frobenius kernel Znd , and that TLnd is arc-transitive with respect to this Frobenius group. In addition, we show that TLnd admits complete rotations. © 2009 Wiley Periodicals, Inc. NETWORKS, Vol. 55(4), 341–349 2010