Given integers n β₯ 7 and a, b, c with 1 β€ a, b, c β€ n -1 such that a, na, b, nb, c, nc are pairwise distinct, the (undirected) triple-loop network TL n (a, b, c) is the degree-six graph with vertices 0, 1, 2, . . . , n -1 such that each vertex x is adjacent to x Β± a, x Β± b, and x Β± c, where the operation is modulo n. It is known that the maximum order of a connected triple-loop network of the form TL n (a, b, n -(a + b)) with given diameter d β₯ 2 is n d = 3d 2 + 3d + 1, which is achieved by TL n d = TL n d (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 TL n d 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 arcforwarding indices of TL n d , 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 TL n d is a Frobenius graph on a Frobenius group with Frobenius kernel Z n d , and that TL n d is arc-transitive with respect to this Frobenius group. In addition, we show that TL n d admits complete rotations.