\magnification=1200
\hsize=4in
\overfullrule=0pt
\nopagenumbers
\noindent
%
%
{\bf F. Aguil\'o, E. Sim\'o and M. Zaragoz\'a}
%
%
\medskip
\noindent
%
%
{\bf Optimal Double-Loop Networks with Non-Unit Steps}
%
%
\vskip 5mm
\noindent
%
%
%
%
A double-loop digraph $G(N;s_1,s_2)=G(V,E)$ is defined by 
$V={\bf Z}_N$ and $E=\{(i,i+s_1), (i,i+s_2)|\; i\in V\}$, for some  
fixed {\it steps} $1\leq s_1<s_2<N$ with $\gcd(N,s_1,s_2)=1$. 
Let $D(N;s_1,s_2)$ be the diameter of $G$ and let us define 
$$
D(N)=\min_{\scriptstyle1\leq s_1<s_2<N,\atop\scriptstyle\gcd(N,s_1,s_2)=1}D(N;s_1,s_2),\quad D_1(N)=\min_{1<s<N}D(N;1,s).
$$
Some early works about the diameter of these digraphs studied the minimization of 
$D(N;1,s)$, for a fixed value $N$, with $1<
s<N$. 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 {\it non-unit step integers} or {\it nus integers}.

In this work we give a characterization of nus integers and a 
method for finding infinite families of nus integers is developed.
Also the tight nus integers are classified. As a consequence of these results, 
some errata and some flaws in the bibliography 
are corrected.

\bye

.
