We have completely determine the radio number of duplicated

cycle graphs and triplicated

cycle graphs.

n] is a

cycle graph then the square graph of the

cycle graph [C.

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.

One can decompose the life cycle graph into the loops of interest, as shown in Fig.

Given the pathway structure of the life cycle graph, it makes intuitive sense for all transitions of a loop to have the same elasticity value.

6) that the top-dimensional critical group is cyclic, with order equal to the number of facets, generalizing the corresponding statement [21, 22, 24] for

cycle graphs.

The mathematical basis for the calculation of the elasticities of the loops is rooted in the following two properties: (1) For each stage in the life cycle graph, the summed elasticity of incoming transitions equals the summed elasticity of outgoing transitions (see Appendix for proof).

The following procedure contains the basic steps for performing loop analysis: (1) Construct a life cycle graph and population transition matrix of the population under investigation.

The life cycle graph is presented as the solid lines in Fig.

Gartner's Jackie Fenn, who coined the term Hype Cycle, began publishing Hype

Cycle graphs in 1995 to help present Gartner's view on the prospects for emerging technologies.

Our collection of results on dynamics for graph classes like trees and

cycle graphs follow in Section 4 before we conclude in Section 5.