A Diameter Formula for an Undirected Double-Loop Network

Abstract

Let \(n, s_1\) and \(s_2\) be positive integers such that \(1 \leq s_1 \leq n/2, 1 \leq s_2 \leq n/2, s_1 \neq s_2\) and \(gcd(n, s_1, s_2) = 1\). An undirected double-loop network \(G(n;\pm s_1,\pm s_2)\) is a graph \((V, E)\), where \(V = \mathbb{Z}_n = \{0, 1, 2, \ldots, n-1\}\), and \(E = \{(i \to i+s_1 \mod n), (i\to i-s_1 \mod n), (i\to i+s_2 \mod n), (i\to i-s_2 \mod n) | i = 0, 1, 2, \ldots, n-1\}\). In this paper, a diameter formula is given for an undirected double-loop network \(G(n; \pm s_1, \pm s_2)\). As its application, two new optimal families of undirected double-loop networks are presented.