The complete graph [K.sub.2n], complete bipartite graph [K.sub.n.n] and

cycle graph [C.sub.n] are opportunity graph.

We have completely determine the radio number of duplicated

cycle graphs and triplicated

cycle graphs.

If [C.sub.n] is a

cycle graph then the square graph of the

cycle graph [C.sub.n.sup.2] is strongly multiplicative.

The life

cycle graph and the life history pathways not only accurately reflect the biology, but are also the mathematical building blocks of any projection matrix.

Consider the

cycle graph [C.sub.n], labeled as below:

(3) "Decompose the life

cycle graph into branched loops in such a way that all transitions are incorporated at least once into some loop" (van Groenendael et al.

Definition 5 A

cycle graph [C.sub.n], n > 1 have chromatic number

The basic step in loop analysis is to consider elasticities in the context of the underlying life history structure as exemplified by the life

cycle graph (Caswell 1989a).

One of the first such results is due to Lorenzini [21,22] and Merris [24, Example 1(1.4)], who independently noted that the critical group of the

cycle graph on n vertices is Z/nZ, the cyclic group on n elements.

The

cycle graph of S, G = G(S, [[mu].sub.b]), is a graph on n vertices labelled 1, ..., n, with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] if [[pi].sub.j] ([v.sub.1]) = [v.sub.2].

Sample Hamiltonian

Cycle Graphs. The Hamiltonian cycles returned by the proposed algorithm on the collected set of sample graphs are shown in Figure 5.

Next assume that H is the bridge graph B([G.sub.1], [G.sub.2]) [10], where [G.sub.1] and [G.sub.2] are

cycle graphs of lengths 3 and 4, respectively.