Combinatorial Group Theory: COMBINATORIAL GROUP THEORY

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 survey 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 the other hand.

Moore Graphs and Beyond: A survey of the Degree/Diameter Problem

This survey provides a comprehensive and unified analysis of symmetry in a wide variety of Cayley graphs of permutation groups. These include the star graph, bubble-sort graph, modified bubble-sort graph, complete-transposition graph, prefix-reversal graph, alternating-group graph, binary and base-b (b ≥ 3) hypercube, cube connected cycles, bisectional graph, folded hypercube and binary orthogonal graph. In addition, we also define a variety of generalizations of the hypercube and orthogonal graphs. The types of symmetry analyzed include vertex and edge transitivity, distance regularity and distance transitivity. Since these notions of symmetry depend on the shortest paths, as a by product we also describe the shortest path routing algorithms for these graphs. We present a number of open problems related to the networks described in this paper.

Symmetry in interconnection networks based on Cayley graphs of permutation groups: a survey

With increasing diverse product demands, the manufacturing paradigm has been transformed into a mass-individualized one, among which one bottleneck is to achieve the interoperability between physical world and the digital world of manufacturing system for the intelligent organizing of resources. This paper presents a digital twin-driven manufacturing cyber-physical system (MCPS) for parallel controlling of smart workshop under mass individualization paradigm. By establishing cyber-physical connection via decentralized digital twin models, various manufacturing resources can be formed as dynamic autonomous system to co-create personalized products. Clarification on the MCPS concept, characteristics, architecture, configuration, operating mechanism and key enabling technologies are elaborated, respectively. A demonstrative implementation of the digital twin-driven parallel controlling of board-type product smart manufacturing workshop is also presented. It addresses a bi-level online intelligence in proactive decision making for the organization and operation of manufacturing resources.

Digital twin-driven manufacturing cyber-physical system for parallel controlling of smart workshop

Due to recent developments of parallel and distributed computing, the design and analysis of various interconnection networks has been a main topic of research for the past few years and is still stimulated by the new technologies of communication networks such as optic fibers. There are many advantages in using Cayley (di)graphs as models for interconnection networks. This work first surveys some classes of Cayley graphs which are well studied as models of interconnection networks. Results and problems related to routings in networks are then presented, with emphasis on loads of nodes and links in routings.

Cayley graphs and interconnection networks

This paper introduces a new class of interconnection scheme based on the Cayley graph of the alternating group. It is shown that this class of graphs are edge symmetric and 2-transitive. We then describe an algorithm for (a) packet routing based on the shortest path analysis, (b) finding a Hamiltonian cycle, (c) ranking and unranking along the chosen Hamiltonian cycle, (d) unit expansion and dilation three embedding of a class of two-dimensional grids, (e) unit dilation embedding of a variety of cycles, and (f) algorithm for broadcasting messages. The paper concludes with a short analysis of contention resulting from a typical communication scheme. Although this class of graphs does not possess many of the symmetry properties of the binary hypercube, with respect to the one source broadcasting, these graphs perform better than does a hypercube, and with respect to the contention problem, these graphs perform better than do the star graphs and are close to the hypercube. © 1993 by John Wiley & Sons, Inc.

A new class of interconnection networks based on the alternating group

The use of the star graph as a viable interconnection scheme for parallel computers has been examined by a number of authors in recent times An attractive feature of this class of graphs is that it has sublogarithmic diameter and has a great deal of symmetry akin to the binary hypercube In this paper we describe a new class of algorithms for embedding (a) Hamiltonian cycle (b) the set of all even cycles and (c) a variety of two- and multi-dimensional grids in a star graph In addition, we also derive an algorithm for the ranking and the unranking problem with respect to the Hamiltonian cycle

Embedding of cycles and grids in star graphs

The use of star graphs as a viable interconnection scheme for parallel computers has been examined by a number of authors. An attractive feature of this class of graphs is that it has sublogarithmic diameter and has a great deal of symmetry akin to the binary hypercube. The authors describe a new class of algorithms for embedding Hamiltonian cycle, the set of all even cycles and a variety of two and multi-dimensional grids in a star graph. They derive an algorithm for the ranking and the unranking problem with respect to the Hamiltonian cycle. >

Enterprise application integration (EAI) is an important issue for a company trying to transform itself into a ubiquitous real-time organization. However, traditional techniques for EAI are only suitable for integrating the virtual information among different applications. The challenge of integrating physical objects with corresponding business processes and applications at the right time and location is a critical concern for companies building a real-time enterprise (RTE) information system. In this research, we proposed an EAI framework based on radio frequency identification (RFID) technology. On the basis of this framework, a prototype system is developed to demonstrate the applicability of the framework in a shop floor environment. This paper also presents a RFID-based standard operation procedure to configure a prototype system for a particular shop floor operation and an operator orientation for performing the corresponding tasks. The findings of this paper demonstrate that the proposed framework is more capable than most current industrial practices in both managing dynamic manufacturing processes and in providing real-time visibility of work-in-process information. Using this framework, an enterprise can easily integrate an RFID-based solution into its IT infrastructure and manufacturing environment to facilitate real-time management of dynamic production operations.

https://ir.nctu.edu.tw:443/bitstream/11536/32129/1/000282840700033.pdf

Jung-Sing Jwo

Papers

Symmetry in interconnection networks based on Cayley graphs of permutation groups: a survey

A new class of interconnection networks based on the alternating group

Embedding of cycles and grids in star graphs

Embedding of cycles and grids in star graphs

An RFID-based enterprise application integration framework for real-time management of dynamic manufacturing processes