UNIDIRECTIONAL (n, k)-STAR GRAPHS

2002 ◽  
Vol 03 (01n02) ◽  
pp. 19-34 ◽  
Author(s):  
EDDIE CHENG ◽  
MARC J. LIPMAN
Keyword(s):  
Directed Graph ◽  
Connected Graph ◽  
Interconnection Networks ◽  
Routing Algorithm ◽  
Optimal Routing ◽  
Star Graph ◽  
Star Graphs

Arrangement graphs14 and (n, k)-star graphs11 were introduced as generalizations of star graphs1. They were introduced to provide a wider set of choices for the order of topologically attractive interconnection networks. Unidirectional interconnection networks are more appropriate in many applications. Cheng and Lipman5, and Day and Tripathi17 studied the unidirectional star graphs, and Cheng and Lipman7 studied orientation of arrangement graphs. In this paper, we show that every (n, k)-star graph can be oriented to a maximally arc-connected graph when the regularity of the graph is even. If the regularity is odd, then the resulting directed graph can be augmented to a maximally arc-connected directed graph by adding a directed matching. In either case, the diameter of the resulting directed graph is small. Moreover, there exists a simple and near-optimal routing algorithm.

scholarly journals Design and Analysis of a Symmetric Log Star Graph with a Smaller Network Cost Than Star Graphs

Electronics ◽  
2021 ◽  
Vol 10 (8) ◽  
pp. 981
Author(s):  
Jung-Hyun Seo ◽  
Hyeong-Ok Lee
Keyword(s):  
Parallel Computing ◽  
Computer Science ◽  
Regular Graph ◽  
Connected Graph ◽  
Routing Algorithm ◽  
Star Graph ◽  
Topological Properties ◽  
Star Graphs ◽  
Network Cost ◽  
Network Costs

Graphs are used as models to solve problems in fields such as mathematics, computer science, physics, and chemistry. In particular, torus, hypercube, and star graphs are popular when modeling the connection structure of processors in parallel computing because they are symmetric and have a low network cost. Whereas a hypercube has a substantially smaller diameter than a torus, star graphs have been presented as an alternative to hypercubes because of their lower network cost. We propose a novel log star (LS) that is symmetric and has a lower network cost than a star graph. The LS is an undirected, recursive, and regular graph. In LSn, the number of nodes is n! while the degree is 2log2n − 1 and the diameter is 0.5n(log2n)2 + 0.75nlog2n. In this study, we analyze the basic topological properties of LS. We prove that LSn is a symmetrical connected graph and analyzed its subgraph characteristics. Then, we propose a routing algorithm and derive the diameter and network cost. Finally, the network costs of the LS and star graph-like networks are compared.

Optimal Neighborhood Broadcast in Star Graphs

2003 ◽  
Vol 04 (04) ◽  
pp. 419-428 ◽  
Author(s):  
Satoshi Fujita
Keyword(s):  
Interconnection Networks ◽  
Completion Time ◽  
Single Port ◽  
Multicast Tree ◽  
Communication Model ◽  
Star Graph ◽  
Star Graphs ◽  
Multicast Trees ◽  
Multicast Scheme ◽  
Special Case

In this paper, we consider the problem of constructing a multicast tree in star interconnection networks under the single-port communication model. Unlike previous schemes for constructing space-efficient multicast trees, we adopt the completion time of each multicast as the objective function to be minimized. In particular, we study a special case of the problem in which all destination vertices are immediate neighbors of the source vertex, and propose a multicast scheme of [Formula: see text] time units for the star graph of dimension n.

The Generalized Connectivity of Bubble-Sort Star Graphs

2019 ◽  
Vol 30 (05) ◽  
pp. 793-809
Author(s):  
Shu-Li Zhao ◽  
Rong-Xia Hao
Keyword(s):  
Fault Tolerance ◽  
Upper Bound ◽  
Interconnection Networks ◽  
Interconnection Network ◽  
Star Graph ◽  
Important Indicator ◽  
Star Graphs ◽  
Generalized Connectivity ◽  
Internally Disjoint Trees

The connectivity plays an important role in measuring the fault tolerance and reliability of interconnection networks. The generalized [Formula: see text]-connectivity of a graph [Formula: see text], denoted by [Formula: see text], is an important indicator of a network’s ability for fault tolerance and reliability. The bubble-sort star graph, denoted by [Formula: see text], is a well known interconnection network. In this paper, we show that [Formula: see text] for [Formula: see text], that is, for any three vertices in [Formula: see text], there exist [Formula: see text] internally disjoint trees connecting them in [Formula: see text] for [Formula: see text], which attains the upper bound of [Formula: see text] given by Li et al. for [Formula: see text].

Fault Tolerance on Star Graphs

1997 ◽  
Vol 08 (02) ◽  
pp. 127-142 ◽  
Author(s):  
Shuo-Cheng Hu ◽  
Chang-Biau Yang
Keyword(s):  
Fault Tolerance ◽  
Fault Tolerant ◽  
Routing Algorithm ◽  
Constant Time ◽  
Multiprocessor Systems ◽  
Star Graph ◽  
Cycle Structure ◽  
Star Graphs ◽  
Broadcasting Algorithm

The capability of fault tolerance is one of the advantages of multiprocessor systems. In this paper, we prove that the fault tolerance of an n-star graph is 2n-5 with restriction to the forbidden faulty set. And we propose an algorithm for examining the connectivity of an n-star graph when there exist at most 2n - 4 faults. The algorithm requires O(n2 log n) time. Besides, we improve the fault-tolerant routing algorithm proposed by Bagherzadeh et al. by calculating the cycle structure of a permutation and the avoidance of routing message to a node without any nonfaulty neighbor. This calculation needs only constant time. And then, we propose an efficient fault-tolerant broadcasting algorithm. When there is no fault, our broadcasting algorithm remains optimal. The penalty is O(n) if there exists only one fault, and the penalty is O(n2) if there exist at most n - 2 faults.

ANALYSIS OF A CLASS OF NETWORKS BASED ON CUBIC STAR GRAPHS

1994 ◽  
Vol 04 (02) ◽  
pp. 191-222
Author(s):  
S.V.R. MADABHUSHI ◽  
S. LAKSHMIVARAHAN ◽  
S.K. DHALL
Keyword(s):  
Interconnection Networks ◽  
Fault Tolerant ◽  
Cayley Graphs ◽  
Binary Trees ◽  
Star Graph ◽  
Cubic Graphs ◽  
Topological Properties ◽  
Star Graphs ◽  
New Class ◽  
Edge Transitive

A new class of interconnection networks based on a family of graphs, called cubic graphs are introduced. These latter graphs arise as Cayley graphs of certain subgroups of the symmetric group. It turns out that these Cayley graphs are a hybrid between the binary hypercube and the star graph, and hence are called cubic star graphs, and are denoted by CS(m, n), m≥1 and n≥1. CS(m, n) inherits several of the properties of the hypercube and the star graph. In this paper, we present an analysis of the symmetric and topological properties. In particular, it is shown that CS(m, n) is edge transitive and hence maximally fault tolerant. We give an algorithm for finding the shortest path and provide an enumeration of the node disjoint paths. Optimal algorithms for single source and all-source broadcasting (also called gossiping) are derived. It is shown that CS(m, n) is Hamiltonian and interesting embeddings of several cycles, grids, and binary trees are derived. The paper concludes with a comparison of CS(m, n) with the binary hypercube and the star graph.

LINEARLY MANY FAULTS IN (n, k)-STAR GRAPHS

2011 ◽  
Vol 22 (07) ◽  
pp. 1729-1745 ◽  
Author(s):  
ALLEN YUAN ◽  
EDDIE CHENG ◽  
LÁSZLÓ LIPTÁK
Keyword(s):  
Interconnection Networks ◽  
Structural Integrity ◽  
Star Graph ◽  
Vertex Connectivity ◽  
Large Component ◽  
Star Graphs

The star graph proposed by [1] has many advantages over the n-cube. However it suffers from having large gaps in the number of possible vertices. The (n,k)-star graph was proposed in [18] to address this issue. Since it is a generalization of the star graph, it retains many of the nice properties of the star graph. There are many different measures of structural integrity of interconnection networks. In this paper, we prove results of the following type for the (n,k)-star graph. If n + (r - 1)k - g(r) vertices are deleted from an (n,k)-star graph, the resulting graph will either be connected or has a large component and small components having at most r - 1 vertices in total. Additional results on conditional vertex connectivity and cycle connectivity will also be given.

scholarly journals On the Generalized Distance Energy of Graphs

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 17 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Connected Graph ◽  
Distance Matrix ◽  
Wiener Index ◽  
Diagonal Matrix ◽  
Upper And Lower Bounds ◽  
Star Graph ◽  
Extremal Graphs ◽  
Generalized Distance

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

Researching, building the routing algorithm in Zigbee networks

2021 ◽  
pp. 32-41
Author(s):  
Dao Xuan Uoc
Keyword(s):  
Ieee 802.15.4 ◽  
Mesh Networks ◽  
Routing Algorithm ◽  
Optimal Routing ◽  
Star Network ◽  
Transmission Distance ◽  
Zigbee Networks ◽  
Iot Devices ◽  
Ieee 802.15.4 Standard ◽  
Better Than

Zigbee wireless network built on IEEE 802.15.4 standard is becoming one of the most popular wireless networks in modern IoT devices. One of the disadvantages of Zigbee networks is the short transmission distance between devices. This paper focuses on researching and comparing routing algorithms in Zigbee networks, thereby building the optimal routing algorithm in the existing system. The paper’s objective is to form the basis for making Zigbee tree and mesh networks, which improves the transmission distance for Zigbee networks better than the star network.

Research on Optimal Routing Algorithm for Public Transit Transfer Based on the Adjacency Matrix

2010 ◽  
Vol 40-41 ◽  
pp. 341-346
Author(s):  
Cai Xia Zhang ◽  
Liang Liang Zhuang ◽  
Xiang Dong Wang ◽  
Hai Wu Rong
Keyword(s):  
Data Processing ◽  
Adjacency Matrix ◽  
Public Transit ◽  
Routing Algorithm ◽  
Optimal Routing ◽  
Optimal Route ◽  
Multi Objective Optimization ◽  
Multi Objective

This paper introduces the Adjacency Matrix at the very beginning, the least transfer between two nodes can be obtained by using the Adjacency Matrix, and then Z matrix is introduced to achieve optimal routing algorithm for public transit transfer and to obtain optimal route by using the “two-step-descending-proliferation” algorithm. Through the "two-step" approach, efficiency and feasibility of data processing was increased. The algorithm focus on multi-objective optimization - takes the least transfer, the least cost, the shortest time, and so on.

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