Applied Mathematics, 2010, 1, 251-259 doi:10.4236/am.2010.14031 Published Online October 2010 (http://www.SciRP.org/journal/am) Copyright © 2010 SciRes. AM Some Models of Reproducing Graphs: II Age Capped Vertices Richard Southwell, Chris Cannings School of Mathematics and Statistics, University of Sheffield, Sheffield, United Kingdom E-mail: bugzsouthwell@yahoo.com Received June 26, 2010; revised August 10, 2010; accepted August 12, 2010 Abstract In the prequel to this paper we introduced eight reproducing graph models. The simple idea behind these models is that graphs grow because the vertices within reproduce. In this paper we make our models more realistic by adding the idea that vertices have a finite life span. The resulting models capture aspects of sys- tems like social networks and biological networks where reproducing entities die after some amount of time. In the 1940’s Leslie introduced a population model where the reproduction and survival rates of individuals depends upon their ages. Our models may be viewed as extensions of Leslie’s model-adding the idea of net- work joining the reproducing individuals. By exploiting connections with Leslie’s model we are to describe how many aspects of graphs evolve under our systems. Many features such as degree distributions, number of edges and distance structure are described by the golden ratio or its higher order generalisations. Keywords: Reproduction, Graph, Population, Leslie, Golden Ratio 1. Introduction Networks are everywhere, wherever a system can be thought of as a collection of discrete elements, linked up in some way, networks occur. With the acceleration of infor- mation technology more and more attention is being paid to the structure of these networks, and this has led to the proposal of many models [1-3]. In many situations networks grow-expanding in size as material is produced from the inside, not added from outside. To study network growth we introduced a class of pure reproduction models [4,5], where networks grow because the vertices within reproduce. These models can be applied to many situations where entities are intro- duced which derive their connections from pre existing elements. Most obviously they could be used to model social networks, collaboration networks, networks within growing organisms, the internet and protein-protein interaction networks. One of our systems (model 3) has also been introduced independently [6], proposed as a model for the growth of online social networks. In our pure reproduction models networks grow endlessly in a deterministic fashion. This allows a rigo- rous analysis, but costs a degree of realism. Nature in- cludes birth and death and entities may be destroyed for reasons of conflict, crowding or old age. In this paper we consider age; and extend our models by including vertex mortality. 2. The Models In [5] we defined a set { : {0,1, ,7}} m F m   of eight different functions m F which map graphs to graphs. ( ) m F G is the graph obtained by simultaneously giving each of G ’s vertices an offspring vertex and then adding edges according to some rule. The connections given to offspring depend upon the binary representation  of m (i.e. =4 2 m      ) as follows: =1   offspring are connected to their parent’s neighbours, =1   offspring are connected to their parents, =1   offspring are connected to their parent’s neighbour’s offspring. In our age capped reproduction models we think of the vertices as having ages. Graphs grow under these models exactly as before, except that vertices grow and then die when their age exceeds some pre-specified integer Q . Our new update operator , mQ T is defined so that , ( ) mQ T G is the graph obtained by taking the graph G and perfor- ming the following process;