Chordal Ring Research Articles

Structural and Spectral Properties of Chordal Ring, Multi-Ring, and Mixed Graphs

The chordal ring (CR) graphs are a well-known family of graphs used to model some interconnection networks for computer systems in which all nodes are in a cycle. Generalizing the CR graphs, in this paper, we introduce the families of chordal multi-ring (CMR), chordal ring mixed (CRM), and chordal multi-ring mixed (CMRM) graphs. In the case of mixed graphs, we can have edges (without direction) and arcs (with direction). The chordal ring and chordal ring mixed graphs are bipartite and 3-regular. They consist of a number r (for r≥1) of (undirected or directed) cycles with some edges (the chords) joining them. In particular, for CMR, when r=1, that is, with only one undirected cycle, we obtain the known families of chordal ring graphs. Here, we used plane tessellations to represent our chordal multi-ring graphs. This allowed us to obtain their maximum number of vertices for every given diameter. Additionally, we computationally obtained their minimum diameter for any value of the number of vertices. Moreover, when seen as a lift graph (also called voltage graph) of a base graph on Abelian groups, we obtained closed formulas for the spectrum, that is, the eigenvalue multi-set of its adjacency matrix.

Studies of Chordal Ring Networks via Double Metric Dimensions

Locating the sources of information spreading in networks including tracking down the origins of epidemics, rumors in social networks, and online computer viruses, has a wide range of applications. In many situations, identifying where an epidemic started, or which node in a network served as the source, is crucial. But it can be difficult to determine the root of an outbreak, especially when data are scarce and noisy. The goal is to find the source of the infection by analysing data provided by only a limited number of observers, such as nodes that can indicate when and where they are infected. Our goal is to investigate where the least number of observers should be placed, which is similar to how to figure out the minimal doubly resolving sets in the network. In this paper, we calculate the double metric dimension of chordal ring networks C R n 1,3,5 by describing their minimal doubly resolving sets.

IMPLEMENTATION OF CHORDAL RING NETWORK TOPOLOGY TO ENHANCE THE PERFORMANCE OF WIRELESS BROADBAND NETWORK

The expansion of networks involved higher jump on the users utilizing the networks resources, which may require extra higher bandwidth. Due to the development of technology especially those folded under the Internet of Things (IoT), the new demand of higher data rate is been witnesses among the users. In order to feed the demand of users with high data rate, broadband networks are required where high data rate can be ensured for each user. Broadband networks can be established using optical network that carries the data through wide broadband. Areas such as rural and forests sides which are witnessing plenty of natural obstacles such as mountains, trees, seas, etc. are forming big challenge for propagating a cable (wire) or optical network.
 Due to the limitations of the wire network, World Interoperability for Microwave Access (WiMAX) technology has been introduced as substitution for the broadband network. Such kind of alternative can be deployed through any geographical area without concerning on the wire paths. WiMAX preserved large coverage area and hence it may not suffer from the signal hand-off as in the case of another wireless network. In this project, Chordal Ring network topology is implemented to enhance the performance of wireless broadband network. With different routing protocols such as Destination Sequenced Distance Vector Protocol (DSDV) and Ad hoc On Demand Distance Vector (AODV), the network performance was examined for various Chordal Ring degree (e.g. fourth degree and fifth degree). Performance metrics such as number of transmitted packets, number of received packers, delivered packet rate (PDR), total number of drop packers (DP) and average queuing delay (been measured). Finally, the obtained results had been shown that fifth-degree Chordal Ring network is outperforming through DSDV routing protocol

Improving the efficiency of cleaning gas emissions from foundry

The work is devoted to the topical problem of increasing the efficiency of cleaning process of gases in packed-type scrubbers in order to reduce emissions of harmful substances and solid impurities into the atmosphere and improve the technical and economic indicators of wet gas cleaning systems. For this, a triangular cross-cut packing has been developed. Due to the size of the holes and their specific shape, the triangular packing provides longer contact of the packaging with water, which reduces water consumption. The packing is made of alloy steel AISI 316, so it can be used even in the most aggressive environment. In the work, the efficiency of cleaning with a scrubber with this packing is experimentally determined. The tests were performed at an ambient temperature of 23 °C. Several measurements were made for each type of packing, in a typical scrubber design. To conduct these industrial tests, a thermohygrometer-carbon dioxide gas analyzerAZ-7797 was used with high-precision NDIR (infrared) sensor, automatic background calibration (W/ABC), and compensation for deviations during long-term operation. The concentration measurement range is 0-9999 ppm with an error of ± 5 %. The gases were cooled in two stages. At first, they were cooled with ammonia water from 750 to 80 °C. Then they were fed to a primary gas refrigerator (basic type) of direct action, where its temperature were falled to 15 °C. Carbon dioxide concentration was measured after the first stage of cooling, after the final cooling, and after the cleaning process. The comparison is carried out with the most effective types of packing for wet gas cleaning today – chordal wood and Raschig rings. It is shown that the developed triangular packing with cross-shaped notches during the purification of coke oven gas from carbon dioxide improves the purification by 25 % in comparison with the wooden chord one, and by 15 % - with Raschig rings. This allows us to recommend it for widespread implementation.

Multicasting in modified chordal ring of degree six network topology

Wired networking environment has presented some interesting challenges to the study of network casting. Hence, graphs are usually used to represent networks of communication. In practice, the topology should be well-designed to meet future reliability demands. Therefore, Modified Chordal Ring of Degree Six (CHRm6) topology has been proposed as mathematical model to represent a network. This paper discussed about multicasting scheme focusing on even and odd nodes. CHRm6 structure has been used to derive results in multicasting scheme. In this type of network, CHRm6 involved a high total number of nodes to deliver a message to multiple destinations simultaneously in terms of multicasting.

An Algorithmic Approach to Compute the Metric Index of Chordal Ring Networks

The metric is a non-negative assignment to the pairs of nodes in a connected network N, which assigns the number of links lying in a smallest path between the nodes in the pairs. A pair (a, b) of nodes in N is said to be uniquely identified by a node c of N if the metric assigned to the pair (a, c) is different from the metric assigned to the pair (b, c). The metric index of N is the minimum number of nodes in N chosen in such a manner that every two nodes in N are uniquely identified by a chosen node. It is said to be constant for a family of networks if it remains unchange with the extension in the networks. In this paper, we consider a family of chordal ring networks and propose an algorithm which assistances in proving, with the aid of mathematical induction and the concept of good nodes, that there is no change in the metric index of chordal ring networks with the extension in the networks.

Reliable networking in Ethernet ring mesh networks using regular topologies

Reliable networking is an important factor in Ethernet ring mesh networks (ERMs) with ITU-T G.8032 Ethernet ring protection recommendation or with the IEC 62439-3 high-availability seamless redundancy protocol. However, there are two major challenges for this purpose: (1) Hitherto, irregular topologies are used in ERMs that it causes difficulty in analyzing and improving reliability. (2) A topology is appropriate for practical implementation, if it is low cost, simple to implement, and simple to develop for large scale-systems (i.e. scalable). However, irregular topologies are extremely difficult to implement and develop. To deal with these challenges, this paper introduces two regular topologies called chordal ring and k-cube networks, for the first time in the area of ERMs. In addition, the proposed topologies are carefully analyzed in terms of reliability. These analyzes prove that the proposed regular topologies outperform existing irregular multiple-ring networks namely shared link, shared node, complex shared link, redundant link, and 3-connected network.

Analyzing methods of network topologies based on chordal rings

This paper presents a methodology for the transmission properties research of networks patterned according to the third-degree chordal graphs. When designing and analyzing ICT systems, it is vital to consider the topology of network components. The telecommunication network structure can be described by a graph. Computing modes or specialized computers are the vertexes and the edges are bidirectional, independent transmission channels that represent the connected nodes. The algorithm is potentially useful for the determination of the impact of diameter and an average path length of the chordal rings on the properties of the transmission network. It is also useful for the indications of alternate paths in the event of node or link failure. Two types of tests were conducted. The first of them is realized by the HTTP protocol and the second by the SSIM method. The obtained results confirmed that these parameters are decisive in the flow capacity and delaying the transferred data.

A hierarchical architecture based on traveling salesman problem for hybrid wireless network-on-chip

The increased latency and power consumption are the major challenges of traditional wired Network-on-Chip (NoC). The Wireless NoC (WNoC) architecture is regarded as a novel approach to solve NoC problems. In this paper, hybrid hierarchical structures are studied. The considered topologies for the first level of the hierarchy include: Chordal Ring of degree 3 (CR) and Ring-Connected Cycles (RCC). On the second level of the hierarchy, inspired by the Traveling Salesman Problem (TSP), a new method is proposed to form the topology. Considering the NP-Hard nature of the problem, the hybrid PS-ACO algorithm is used to obtain the desired tour. In this paper, the uniform random traffic is used as the synthetic traffic pattern and the 3-tuple traffic is used as a real application traffic pattern. The simulation results show that the proposed structure has fewer wired links and considerably increased efficiency compared with a wired mesh topology in NoC.

A New Degree Six Modified Chordal Ring Network Topology

The performance of a parallel or distributed network depends on the design of its interconnection topology. One good network interconnection topology that has been the focus of various researchers is the chordal ring. In this paper, a new degree six modified chordal ring is presented, the optimised degree six 3-modified chordal ring (CHR6o3), with the aim of comparing its performance parameters in terms of optimal diameter and optimal average path length to existing degree six chordal rings. Formulae to generate the data for each different chordal ring were derived from its corresponding tree visualisation or used from previous work. Network sizes tested were from 1200 to 12000 nodes. Large networks of CHR6o3 were shown to perform better than those of previous degree six chordal rings. This gives CHR6o3 an added advantage for its implementation in large distributive networks, such as Fibre to the Home (FTTH) networks, since it offers redundancies at higher network layers.

Chromatic numbers and indices of the optimised degree six 3-modified chordal ring network topology

Graph colouring is the labelling of the elements of a graph subject to certain constraints. It is divided into vertex and edge colouring. In both cases, the assignment of labels, traditionally called colours is such that two vertices or edges must not have the same colour. This has various applications, especially in parallel computing. This paper introduces the degree six 3-modified chordal ring, CHR6o3, a parallel network interconnection topology model, and discusses its node and link colouring. Despite its asymmetry, the chromatic number of CHR6o3 cannot be generalised and must be determined specifically based on the combination of chords in each case. Cases where the chromatic index of CHR6o3 is its degree and where it is its one colour more than its degree were generalised and proven. Chromatic numbers are important in minimising the completion times of processes in parallel processing by reducing points of synchronisation, and link colouring aids processor scheduling.

Some graph properties of the optimised degree six 3-modified chordal ring network

The interconnection topology of a parallel or distributed network is pivotal in ensuring good system performance. It can be modelled by a graph, where its edges represent the links between processor nodes represented by vertices. One such graph model that has gained attention by researchers since its founding is the chordal ring, based on an undirected circulant graph. This paper discusses the degree six 3-modified chordal ring, CHR6o3, and presents its graph theoretical properties of symmetry and Hamiltonicity. CHR6o3 is shown to be asymmetric, and can be decomposed into similar subgraphs, each consisting of only one type of node in its class if ring links are ignored. These properties aid both the development of a routing scheme and also determining lower bounds for its chromatic number. Conditions for the existence of a Hamiltonian Circuit within CHR6o3 are also discussed. The existence of a Hamiltonian Circuit within a network simplifies parallel processing as the processors can be arranged to work on a task in a linear array. An Eulerian Circuit was shown to exist in CHR6o3. The existence of an Eulerian Circuit plays a role in routing in optical networks.

Completely Independent Spanning Trees on 4-Regular Chordal Rings

Completely Independent Spanning Trees on 4-Regular Chordal Rings

Hypercube emulation of interconnection networks topologies

We address various topologies (de Bruijn, chordal ring, generalized Petersen, and meshes) in various ways (isometric embedding, embedding up to scale, and embedding up to a distance) in a hypercube or a half‐hypercube. Example of obtained embeddings: infinite series of hypercube embeddable bubble sort and double chordal rings topologies, as well as of regular maps. Copyright © 2016 John Wiley & Sons, Ltd.

推广立方连通圈网络的Hamilton分解的算法

立方连通圈网络是超立方体的有界度变形，它具有超立方体几乎所有的优良性质，而且克服了超立方体顶点度随网络规模增大而增大的缺点，是代替超立方体的一个具有强大竞争力的网络结构。但立方连通圈网络的结构是简单还是复杂呢？这是一个悬而未决的问题。带弦环网络是一类经典的互连网络，该网络具有结构简单等优点。在这篇文章中利用师海忠提出的正则图连通圈网络模型设计出了包含立方连通圈网络的一类网络——推广立方连通圈网络GCCC(n) (n > 2)，证明了GCCC(n) (n > 2)可分解为边不交的一个Hamilton圈和一个完美对集的并，即GCCC(n) (n > 2)是带弦环网络。并给出推广立方连通圈网络分解为边不交的一个Hamilton圈和一个完美对集的并的算法。 The cube-connected cycles network is a hypercube of bounded-degree derivative. It has almost all of the excellent properties of hypercube. It also overcomes the shortcoming of hypercube vertices of degrees which will increase with the augment of the network size. But the cube-connected cycles network is simple or complex? This is an open question. Chordal ring network is a kind of classical interconnection network, which has the advantages of simple structure and so on. In this article, a model of the regular graph connected cycle network which is proposed by Haizhong Shi is used to design a kind of network which contains the cube-connected cycles network—the generalized cube-connected network GCCC(n) (n > 2). The authors prove that GCCC(n) (n > 2) can be decomposed into union of edge-disjoint a Hamiltonian cycle and a perfect matching, namely, GCCC(n) (n > 2) is a chordal ring network. They also make a algorithm for GCCC(n) (n > 2) can be decomposed into union of edge-disjoint a Hamiltonian cycle and a perfect matching.

The crossing number of chordal ring networks

The chordal ring network of order n, denoted by $$CR_n(x,y,z)$$ , is the graph with vertex set $$Z_n$$ , an additive group of integers modulo n, and adjacencies given by $$i\sim i+x, i\sim i+y, i\sim i+z$$ for all even vertex i and distinct odd integers x, y, z in $$[1, n-1]$$ . The crossing number of a network(graph) is closely related to the minimum layout area required for the implementation of a VLSI circuit for that network. From this perspective, the analysis of crossing numbers of graphs makes most sense when one focuses on networks(graphs) that have good properties as interconnection networks. In this paper, we study the crossing number of the chordal ring networks $$CR_n(1,3,9)$$ for all even $$n\ge 10$$ .

Broadcasting Communication in High Degree Modified Chordal Rings Networks

The design of the interconnection network is one of the main research issues in distributed computing with regard to some specific parameters. This paper works on Modified Chordal Rings Degree Six (CHRm6) topology. Two broadcasting schemes for CHRm6 are proposed. The first scheme is for even source nodes and the second scheme is for odd source nodes. The proposed broadcasting schemes give upper and lower bound of broadcasting in CHRm6 depends on total number of nodes. We prove the lower bound on the broadcast time is d+2 where d is the diameter of the CHRm6.

L-Networks: A Topological Model for Regular 2D Interconnection Networks

A complete family of Cayley graphs of degree four, denoted as L-networks, is considered in this paper. L-networks are 2D mesh-based topologies with wrap-around connections. L-networks constitute a graph-based model which englobe many previously proposed 2D interconnection networks. Some of them have been extensively used in the industry as the underlying topology for parallel and distributed computers of different scales. Tori, twisted and doubly twisted tori, toroidal diagonal meshes, chordal rings, and circulant graphs are, among others, members of the L-network family. Therefore, many results obtained in previous studies on these networks can be deduced from the general framework presented in this work. In addition, the network model presented in this work allows for new results on the domain of low-degree interconnection networks. Particularly, closed expressions for the graph distance properties have been derived and an optimal routing algorithm of constant complexity is provided. Since symmetry has a big impact on network performance, we have also identified which L-networks are symmetric by studying their group of automorphisms. Finally, a very simple model that predicts the performance of L-networks is also presented. Such model has been contrasted with empirical evaluation.

A Survey on Chordal Rings, N2R and Other Related Topologies

Abstract In this short paper we summarize the main results of 10 years of research on network topologies carried out jointly at Aalborg University (AAU), Aalborg, Denmark, and University of Life Science (UTP), Bydgoszcz, Poland. The starting points are the Chordal Rings, which were mainly studied in UTP, and the N2R which were studied at AAU. From here the universities have collaborated intensely, and studied a number of alternatives and variations of the original topologies, based on methodologies developed both independently and jointly over time. This paper gives an overview of the work done, and provides appropriate references to the work for further details. The paper is presented as a tribute to Professor and Dean Antoni Zabłudowski who passed away during 2012.

Utility of three-dimensional echocardiography for assessment of double-orifice mitral valve

Double-orifice mitral valve is a rare congenital malformation of the mitral valve. A range of structural abnormalities of the subvalvular apparatus are found including a chordal ring, accessory papillary muscle, crossing/fused chordae tendineae, and a central fibrous subdivision creating two separate mitral valve orifices. More commonly, an iatrogenic double-orifice mitral valve is formed during …