Triple loop networks with small transmission delay

The problem of finding multiloop networks with a fixed number of vertices and small diameter has been widely studied. In this work, we study the triple loop case of the problem by using a geometrical approach which has been already used in the double loop case. Given a fixed number of vertices N, the general problem is to find 'steps' s(1), s(2), ..., s(d) is an element of Z(N), such that the digraph G(N; s(1), s(2), ..., s(d)), with set of vertices V = Z(N) and adjacencies given by v --> v + s(i) (mod N), i = 1, 2, ..., d, has minimum diameter D(N). A related problem is to maximize the number of vertices N(d,D) when the degree d and the diameter D are given. In the double loop case (d = 2) it is known that N(2,D) = inverted right perpendicular 1/3(D + 2)(2) inverted left perpendicular - 1. Here, a method based on lattice theory and integral circulant matrices is developed to deal with the triple loop case (d = 3). This method is then applied for constructing three infinite families of triple loop networks with large order for the values of the diameter D - 2, 4, 5 (mod 6), showing that N(3, D) greater than or equal to 2/27 D-3 + O(D-2). Similar results are also obtained in the more general framework of(triple) commutative-step digraphs.