Along each

directed cycle C of weighted digraph (G, A), there is

Second, for any node A, the

directed cycle A [right arrow] T [right arrow] S [right arrow] A is of length 3.

The reversal of the orientation of a

directed cycle in an [alpha]-orientation yields another [alpha]-orientation.

An NCM is said to be acyclic if it does not possess any

directed cycle.

For an integer n [greater than or equal to] 3, the

directed cycle of length n is the graph [C.sub.n] whose vertices are 0, 1, ...,n-1 and whose edges are the pairs i, i + 1, where the arithmetic is done modulo n.

We call C a

directed cycle if [i.sub.s] = [i.sub.1].

,[i.sub.r]) associated with the given directed is called a

directed cycle in S.

Each

directed cycle of [D.sub.[phi]] corresponds to a

directed cycle of D.

For example, if D is a simple

directed cycle, then the intervals yield a division of the cycle into two arcs.

Such a condition was first obtained by Robert (1980), who proved that if G(f) has no

directed cycle, then f has a unique fixed point.

A

directed cycle is a sequence of directed edges ([i.sub.1], [i.sub.2]), ([i.sub.2], [i.sub.3]), ..., ([i.sub.k-1], [i.sub.1]) such that all [i.sub.1], ..., [i.sub.k] are distinct.

Matrix of membership (cycles matrix) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] relating the arcs and graph

directed cycles of a network, indicating membership pipes to network ring and having on the lines, order number of rings M, and the columns at the arcs T.