The degree/diameter problem is to determine the largest graphs or digraphs of given maximum degree and given diameter. General upper bounds – called Moore bounds – for the order of such graphs and digraphs are attainable only for certain special graphs and digraphs. Finding better (tighter) upper bounds for the maximum possible number of vertices, given the other two parameters, and thus attacking the degree/diameter problem ‘from above’, remains a largely unexplored area. Constructions producing large graphs and digraphs of given degree and diameter represent a way of attacking the degree/diameter problem ‘from below’. This sur-vey aims to give an overview of the current state-of-the-art of the degree/diameter problem. We focus mainly on the above two streams of research. However, we could not resist mentioning also results on various related problems. These include considering Moore-like bounds for special types of graphs and digraphs, such as vertex-transitive, Cayley, planar, bipartite, and many others, on the one hand, and related properties such as connectivity, regularity, and surface embeddability, on

Let vt(d; 2) be the largest order of a vertex-transitive graph of degree d and diameter two. It is known that vt(d; 2) = d 2 + 1 for d = 1; 2; 3, and 7; for the remaining values of d we have vt(d; 2) d 2 \Gamma 1. The only known general lower bound on vt(d; 2), valid for all d, seems to be vt(d; 2) b d+2 2 cd d+2 2 e. Using voltage graphs, we construct a family of vertex-transitive non-Cayley graphs which shows that vt(d; 2) 8 9 (d + 1 2 ) 2 for all d of the form d = (3q \Gamma 1)=2 where q is a prime power congruent with 1 (mod 4). The construction generalizes to all prime powers and yields large highly symmetric graphs for other degrees as well. In particular, for d = 7 we obtain as a special case the Hoffman-Singleton graph, and for d = 11 and d = 13 we have new largest graphs of diameter two and degree d on 98 and 162 vertices, respectively. 1 Introduction The well-known degree/diameter problem asks for determining the largest possible number n(d; k) of vertic...

We review the status of the Degree#Diameter problem for both, graphs and digraphs and present new Cayley digraphs which yield improvements over some of the previously known largest vertex transitive digraphs of given degree and diameter.

The design of computer networks and parallel processor configurations is a topic of increasing importance. Network designs which efficiently support communications between nodes are crucial for many applications. Cost and physical limitations generally prevent the nodes in a network from having more than a fixed number of hardware connections to other nodes (that is, the nodes must have bounded degree). This fundamental constraint makes the design problem nontrivial. The topic of this thesis is an explanation of ways in which group theory can be used to design bounded-degree communication-efficient networks. Our methods have yielded a number of network designs that are the largest known for networks satisfying specified bounds on node degree and either diameter or broadcast time, for values of these parameters that are in the range of potential engineering significance. Examiners: Dr. Michael R. Fellows, Supervisor (Department of Computer Science) Dr. Wendy Myrvold, Department Member (...

The major aims of this work is to give a comparative survey of static interconnection topologies, and to discuss their properties with respect to their use as interconnection topologies in message passing multi-computer systems, i. e. each processing element has its own local memory, there is no common memory, and the processing elements communicate via message-passing. To this end it was necessary to recall relevant measures on graphs from graph theory, like for example the average distance or the network diameter, and requirements from the parallel processing area, like the reliability or extensibility. Special emphasis has been given to present the construction rules for various graphs, because these seemed -- along with the network characteristics -- most relevant for interconnecting processing elements in reconfigurable multi-computer systems. Critical to applications in these kind of parallel systems is the possibility of exchanging local data among cooperating processing element...

Voltage graphs are a powerful tool for constructing large graphs (called lifts) with prescribed properties as covering spaces of small base graphs. The main objective of the paper is to revisit the classical degree/diameter problem for graphs from this new perspective. We derive a fairly general upper bound on the diameter of a lift in terms of the properties of the base voltage graph, and prove some results on vertex-transitive lifts. The potential of the new method is highlighted by showing that all currently known largest Cayley graphs (of given degree and diameter) for semidirect products of cyclic groups can be described by means of a voltage assignment construction, using simpler groups. This research started when J. Plesn'ik and J. Sir'an were visiting the Department of Computer Science of the University of Newcastle NSW Australia in 1995, supported by small ARC grant. 1 Introduction In the past few decades there has been growing interest in the design of interconnection ...

Introduction The growing demand for more computing power at increasing speed in many scientific and engineering applications made it necessary to develop advanced computer architectures based on the concept of parallel processing. In general a parallel computer system consists of various processing and memory units and other (shared) resources. A critical issue in design and analysis of parallel systems is the way in which the system components are connected together, since this interconnection network determines the performance of the whole system [Bhuy 87]. The network topology, defined as the abstract representation of the connections in the network [?], is a key factor in determing a suitable architectural structure. A lot of criteria for a comparison of interconnection topologies have been proposed, we will focus on the problem of finding topologies with minimum communication delays expressed by the diameter and the average distance of the

The largest known 3-regular planar graph with diameter 3 has 12 vertices. We consider the problem of determining whether there is a larger graph with these properties. We find all nonisomorphic 3-regular, diameter-3 planar graphs, thus solving the problem completely. There are none with more than 12 vertices. An Upper Bound A graph with maximum degree \Delta and diameter D is called a (\Delta; D)-graph. It is easily seen ([9], p. 171) that the order of a (\Delta,D)-graph is bounded above by the Moore bound, which is given by 1+ \Delta + \Delta (\Delta \Gamma 1) + \Delta \Delta \Delta + \Delta(\Delta \Gamma 1) D\Gamma1 = 8 ? ! ? : \Delta(\Delta \Gamma 1) D \Gamma 2 \Delta \Gamma 2 if \Delta 6= 2; 2D + 1 if \Delta = 2: Figure 1: The regular (3,3)-graph on 20 vertices (it is unique up to isomorphism) . For D 2 and \Delta 3, this bound is attained only if D = 2 and \Delta = 3; 7, and (perhaps) 57 [3, 14, 23]. Now, except for the case of C 4 (the cycle on four vertices), the num...

The thesis is concerned with the design of high performance interconnection networks for use predominantly in parallel computing systems and wide area networks. The most important indicating a combined measure of hardware complexity and worst-cast message routing complexity. Furthermore, a high performance network should also have the properties of regular and planar topology, high bisection width and routing simplicity. Specifically, the following problems are studied: (i) constructing the largest possible networks that simultaneously exhibit a number of other properties including a small number of edges, high bisection width and planarity; and (ii) implementing high performance communication networks on a scale comparable to that of the Internet. With respect to specific technology, the thesis addresses the following two questions: (i) exactly how can optical inter-networking be achieved on a world wide scale so as to maximize performance and (ii) just how big can an optical inter-network be, given the present/future technological limits and performance constraints.

Fixed degree network development has received significant attention. This paper proposes fixed degree directed networks. Given the node degree (fan-in/out) and the size of the network (number of nodes) we propose a generic approach of contructing the topology. Our topology has minimum diameter. In addition, we provide an expansion operator Ex(x,n) to scale the network size. Network diameter increases logarithmically in the scale factor. Furhtermore we present fast routing algorithms for these networks having logarithmic time complexity while using constant buffer size. For arbitrary degrees, the routing time complexity on this family of networks is O(n 2 ); for degree 2, it is O(n), which is optimal. This paper answers Maekawas open problem (of constructing a directed degree-p network having minimal diameter for a given number of nodes) partially. 1. Introduction Elspas started researching undirected graphs of fixed degree in 1964 [7]. Given a number of nodes and fixed degree of each n...