DIAMETER FORMULAS FOR A CLASS OF UNDIRECTED DOUBLE-LOOP NETWORKS

Abstract

An n-node network, with nodes numbered 0 to n-1, is an undirected double-loop network with chord lengths 1 and s(2≤s<n/2) when each node i(0≤i<n) is connected to each of the four nodes i±1 and i±s via an undirected link; all node-index expressions are evaluated modulo n. Let n=qs+r, where r(0≤r<s) is the remainder of dividing n by s. Furthermore, let s=ar+b, where b(0≤b<r) is the remainder of dividing s by r. In this paper, we provide closed-form formulas for the diameter of a double-loop network for the case q>r and for a subcase of the case q≤r when b≤aq+1. In the complementary subcase of q≤r, when b>aq+1, network diameter can be derived by applying the O(log n)-time algorithm of Zerovnik and Pisanski (J. Algorithms, Vol. 14, pp. 226-243, 1993). Obtaining a closed-form formula for diameter of the double-loop network in the latter subcase remains an open problem.