Optimal Double-Loop Networks with Non-Unit Steps

A double-loop digraph G(N ; s 1 ; s 2 ) = G(V; E) is de ned by V = ZN and E = f(i; i + s 1 ); (i; i + s 2 )j i 2 V g, for some xed 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 de ne gcd(N;s1 ;s 2 )=1 D(N ; s 1 ; s 2 ); D 1 (N) = min 1<s<N D(N ; 1; s): Some early works about the diameter of these digraphs studied the minimization of D(N ; 1; s), for a xed value N , with 1 < s < N . Although the identity D(N) = D 1 (N) holds for in nite values of N , there are also another in nite set of integers with D(N) < D 1 (N ). These other integral values of N are called non-unit step integers or nus integers

Optimal Double-Loop Networks With Non-Unit Steps

A double-loop digraph G(N ; s 1 ; s 2 ) = G(V; E) is de ned by V = ZN and E = f(i; i + s 1 ); (i; i + s 2 )j i 2 V g, for some xed 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 de ne gcd(N;s1 ;s 2 )=1 D(N ; s 1 ; s 2 ); D 1 (N) = min 1<s<N D(N ; 1; s): Some early works about the diameter of these digraphs studied the minimization of D(N ; 1; s), for a xed value N , with 1 < s < N . Although the identity D(N) = D 1 (N) holds for in nite values of N , there are also another in nite set of integers with D(N) < D 1 (N ). These other integral values of N are called non-unit step integers or nus integers. In thi

Optimal Double-Loop Networks With Non-Unit Steps

A double-loop digraph G(N ; s 1 ,s 2 )=G(V,E) is defined by V = #N and E = {(i, i + s 1 ), (i, i + s 2 )| i V },forsomefixedsteps s 1 <s 2 <Nwith gcd(N,s 1 ,s 2 )=1.LetD(N ; s 1 ,s 2 ) be the diameter of G and let us define D(N ; s 1 ,s 2 ),D 1 (N)= min 1<s<N D(N;1,s). Some early works about the diameter of these digraphs studied the minimization of D(N;1,s), for a fixed value N,with1<s<N. Although the identity D(N)= D 1 (N) holds for infinite values of N , there are also another infinite set of integers with D(N) <D 1 (N ). These other integral values of N are called non-unit step integers or nus integers

On optimal double-loop networks with non-unit steps

This study is an exposition of the article Optimal Double-Loop Networks with Non-Unit Steps by Aguilo, Zaragoza and Simo [3]. A characterization of non-unit steps integers will be presented. In addition, a detailed proof of the theorem which gives different lower bounds of the diameter of an L-title will be presented. This appeared in section 3.4.2, of the book entitled Topological Structure and Analysis of Interconnection Networks by Xu