Triple loop networks with small transmission delay

Abstract

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 ∈ Z N , such that the diagraph 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) = ⌈ 1 3 (D+2) 2⌉ − 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) ⩾ 2 27 D 3 + O(D 2) . Similar results are also obtained in the more general framework of (triple) commutative-step digraphs.