Gossiping and routing in undirected triple-loop networks

• Authors: Alison Thomson;Sanming Zhou
• Venue: Networks
• Year: 2010

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 = 3d2 + 3d + 1, which is achieved by TLnd = TLnd(1, 3d+ 1, 3d2 - 1). In this article, we study the routing and gossiping problems for such optimal triple-loop 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 edge- and arc-forwarding indices, and the minimal edge- and arc-forwarding 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, 2010