Triple loop networks with small transmission delay (Q1356442)

The paper studies the triple loop case of the problem of finding multiloop networks with a fixed number of vertices and small diameter. Based on lattice theory and integral circulant matrices, a method is developed to deal with this problem by constructing three infinite families of triple loop networks with large order for the values of the diameter \(D=2, 4, 5\pmod 6\), showing that \(N(3,D)\geq 2D^3/27+ O(D^2)\), where \(N(3,D)\) is the number of vertices for a triple loop network with diameter \(D\). Similar results are also obtained in the more general frame of triple commutative-step digraphs.