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Indexed by:期刊论文

Date of Publication:2010-10-01

Journal:ARS COMBINATORIA

Included Journals:SCIE、Scopus

Volume:97A

Page Number:81-95

ISSN No.:0381-7032

Key Words:Double-loop network; tight optimal; L-shaped tile; non-unit step integer

Abstract:Double-loop networks have been widely studied as architecture for local area networks. A double-loop network G(N; s(1,)s(2)) is a digraph with N vertices 0, 1, ... , N - 1 and 2N edges of two types:
   s(1)-edge: i -> i + s(1) ( mod N); i = 0, 1, ... , N - 1. s(2)-edge: i -> i + s(2)( mod N); i = 0, 1, ... , N - 1.
   for some fixed steps 1 <= s(1) <= s(2) <= N with gcd(N, s(1), s(2)) = 1. Let D(N; s(1), s(2)) be the diameter of G and let us define D(N) = min{D(N; s(1), s(2))vertical bar 1 <= s(1) < s(2) < N and gcd(N, s(1), s(2)) = 1}, and D(1)(N) = min{D(N; 1, s)vertical bar 1 < s < N}. If N is a positive integer and D(N) < D(1)(N), then N is called a non-unit step integer or a nus integer. Xu and Aguilo et al. gave some infinite families of 0-tight nus integers with D(1)(N) - D(N) >= 1.
   In this work, we give a method for finding infinite families of nus integers. As application examples, we give one infinite family of 0-tight nus integers with D(1)(N) - D(N) >= 5, one infinite family of 2-tight nus integers with D(1)(N) - D(N) >= 1 and one infinite family of 3-tight nus integers with D(1)(N) - D(N) >= 1.