Large Networks of Diameter Two Based on Cayley Graphs

In this contribution we present a construction of large networks of diameter two and of order $\frac{1}{2}d^2$ for every degree $d\geq 8$, based on Cayley graphs with surprisingly simple underlying groups. For several small degrees we construct Cayley graphs of diameter two and of order greater than $\frac23$ of Moore bound and we show that Cayley graphs of degrees $d\in\{16,17,18,23,24,31,\dots,35\}$ constructed in this paper are the largest currently known vertex-transitive graphs of diameter two.Comment: 9 pages, Published in Cybernetics and Mathematics Applications in Intelligent System

Large Cayley graphs of small diameter

The degree-diameter problem seeks to find the largest possible number of vertices in a graph having given diameter and given maximum degree. Very often the problem is studied for restricted families of graph such as vertex-transitive or Cayley graphs, with the goal being to find a family of graphs with good asymptotic properties. In this paper we restrict attention to Cayley graphs, and study the asymptotics by fixing a small diameter and constructing families of graphs of large order for all values of the maximum degree. Much of the literature in this direction is focused on the diameter two case. In this paper we consider larger diameters, and use a variety of techniques to derive new best asymptotic constructions for diameters 3, 4 and 5 as well as an improvement to the general bound for all odd diameters. Our diameter 3 construction is, as far as we know, the first to employ matrix groups over finite fields in the degree-diameter problem

Diameter, Girth And Other Properties Of Highly Symmetric Graphs

We consider a number of problems in graph theory, with the unifying theme being the properties of graphs which have a high degree of symmetry. In the degree-diameter problem, we consider the question of finding asymptotically large graphs of given degree and diameter. We improve a number of the current best published results in the case of Cayley graphs of cyclic, dihedral and general groups. In the degree-diameter problem for mixed graphs, we give a new corrected formula for the Moore bound and show non-existence of mixed Cayley graphs of diameter 2 attaining the Moore bound for a range of open cases. In the degree-girth problem, we investigate the graphs of Lazebnik, Ustimenko and Woldar which are the best asymptotic family identified to date. We give new information on the automorphism groups of these graphs, and show that they are more highly symmetrical than has been known to date. We study a related problem in group theory concerning product-free sets in groups, and in particular those groups whose maximal product-free subsets are complete. We take a large step towards a classification of such groups, and find an application to the degree-diameter problem which allows us to improve an asymptotic bound for diameter 2 Cayley graphs of elementary abelian groups. Finally, we study the problem of graphs embedded on surfaces where the induced map is regular and has an automorphism group in a particular family. We give a complete enumeration of all such maps and study their properties

An investigation into alternative methods for the simulation and analysis of growth models

Complex systems are a rapidly increasing area of research covering numerous disciplines including economics and even cancer research, as such the optimisation of the simulations of these systems is important. This thesis will look specifically at two cellular automata based growth models the Eden growth model and the Invasion Percolation model. These models tend to be simulated storing the cluster within a simple array. This work demonstrates that for models which are highly sparse this method has drawbacks in both the memory consumed and the overall runtime of the system. It demonstrates that more modern data structures such as the HSH tree can offer considerable benefits to these models.Next, instead of optimising the software simulation of the Eden growth model, we detail a memristive-based cellular automata architecture that is capable of simulating the Eden growth model called the MEden model. It is demonstrated that not only is this method faster, up to 12; 704 times faster than the software simulation, it also allows for the same system to be used for the simulation of both EdenB and EdenC clusters without the need to be reconfigured; this is achieved through the use of two different parameters present in the model Pmax and Pchance. Giving the model a broader range of possible clusters which can aid with Monte-Carlo simulations of the model.Finally, two methods were developed to be able to identify a difference between fractally identical clusters; connected component labelling and convolution neural networks are the methods used to achieve this. It is demonstrated that both of these methods allow for the identification of individual Eden clusters able to classify them as either an EdenA, EdenB, or EdenC cluster, a highly nontrivial matter with current methods. It is also able to tell when a cluster was not an Eden cluster even though it fell in the fractal range of an Eden cluster. These features mean that the verification of a new method for the simulation of the Eden model could now be automated