THE Qn,k,m GRAPH: A COMMON GENERALIZATION OF VARIOUS POPULAR INTERCONNECTION NETWORKS

2013 ◽  
Vol 23 (03) ◽  
pp. 1350011 ◽  
Author(s):  
EDDIE CHENG ◽  
NART SHAWASH
Keyword(s):  
Interconnection Networks ◽  
Cayley Graphs ◽  
Alternating Group ◽  
Star Graph ◽  
Strong Connectivity ◽  
Common Generalization ◽  
Arrangement Graph ◽  
Special Case ◽  
Alternating Group Graph

The star graph and the alternating group graph were introduced as competitive alternatives to the hypercube, and they are indeed superior over the hypercube under many measures. However, they do suffer from scaling issues. To address this, different generalizations, namely, the (n,k)-star graph and the arrangement graph were introduced to address this shortcoming. From another direction, the star graph was recognized as a special case of Cayley graphs whose generators can be associated with a tree. Nevertheless, all these networks appear to be very different and yet share many properties. In this paper, we will solve this mystery by providing a common generalization of all these networks. Moreover, we will show that these networks have strong connectivity properties.

A Note of Independent Number and Domination Number of Qn,k,m-Graph

2019 ◽  
Vol 29 (03) ◽  
pp. 1950011
Author(s):  
Jiafei Liu ◽  
Shuming Zhou ◽  
Zhendong Gu ◽  
Yihong Wang ◽  
Qianru Zhou
Keyword(s):  
Interconnection Networks ◽  
Interconnection Network ◽  
Domination Number ◽  
Multiprocessor System ◽  
Star Graph ◽  
Minimum Cardinality ◽  
Arrangement Graph ◽  
Independent Number ◽  
Potential Applications ◽  
By Products

The independent number and domination number are two essential parameters to assess the resilience of the interconnection network of multiprocessor systems which is usually modeled by a graph. The independent number, denoted by [Formula: see text], of a graph [Formula: see text] is the maximum cardinality of any subset [Formula: see text] such that no two elements in [Formula: see text] are adjacent in [Formula: see text]. The domination number, denoted by [Formula: see text], of a graph [Formula: see text] is the minimum cardinality of any subset [Formula: see text] such that every vertex in [Formula: see text] is either in [Formula: see text] or adjacent to an element of [Formula: see text]. But so far, determining the independent number and domination number of a graph is still an NPC problem. Therefore, it is of utmost importance to determine the number of independent and domination number of some special networks with potential applications in multiprocessor system. In this paper, we firstly resolve the exact values of independent number and upper and lower bound of domination number of the [Formula: see text]-graph, a common generalization of various popular interconnection networks. Besides, as by-products, we derive the independent number and domination number of [Formula: see text]-star graph [Formula: see text], [Formula: see text]-arrangement graph [Formula: see text], as well as three special graphs.

A Note on the Nature Diagnosability of Alternating Group Graphs Under the PMC Model and MM* Model

2018 ◽  
Vol 18 (01) ◽  
pp. 1850005 ◽  
Author(s):  
SHIYING WANG ◽  
LINGQI ZHAO
Keyword(s):  
Interconnection Networks ◽  
Interconnection Network ◽  
Alternating Group ◽  
Multiprocessor System ◽  
Multiprocessor Systems ◽  
Free Node ◽  
Pmc Model ◽  
Important Study ◽  
Communication Links ◽  
Alternating Group Graph

Many multiprocessor systems have interconnection networks as underlying topologies and an interconnection network is usually represented by a graph where nodes represent processors and links represent communication links between processors. No faulty set can contain all the neighbors of any fault-free node in the system, which is called the nature diagnosability of the system. Diagnosability of a multiprocessor system is one important study topic. As a favorable topology structure of interconnection networks, the n-dimensional alternating group graph AGn has many good properties. In this paper, we prove the following. (1) The nature diagnosability of AGn is 4n − 10 for n − 5 under the PMC model and MM* model. (2) The nature diagnosability of the 4-dimensional alternating group graph AG4 under the PMC model is 5. (3) The nature diagnosability of AG4 under the MM* model is 4.

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.

Strong Matching Preclusion of Arrangement Graphs

2016 ◽  
Vol 16 (02) ◽  
pp. 1650004 ◽  
Author(s):  
EDDIE CHENG ◽  
OMER SIDDIQUI
Keyword(s):  
Interconnection Networks ◽  
Perfect Matchings ◽  
Alternating Group ◽  
Star Graphs ◽  
Common Generalization ◽  
Minimum Number

The strong matching preclusion number of a graph is the minimum number of vertices and edges whose deletion results in a graph with neither perfect matchings nor almost-perfect matchings. This is an extension of the matching preclusion problem that was introduced by Park and Ihm. The class of arrangement graphs was introduced as a common generalization of the star graphs and alternating group graphs, and to provide an even richer class of interconnection networks. In this paper, the goal is to find the strong matching preclusion number of arrangement graphs and to categorize all optimal strong matching preclusion sets of these graphs.

The number of shortest paths in the (n, k)-star graph

2014 ◽  
Vol 06 (04) ◽  
pp. 1450051 ◽  
Author(s):  
Eddie Cheng ◽  
Ke Qiu ◽  
Zhizhang Shen
Keyword(s):  
Shortest Paths ◽  
The Other ◽  
Alternating Group ◽  
Star Graph ◽  
Arrangement Graph

We count the shortest paths, not necessarily disjoint, between any two vertices in an (n, k)-star graph by counting the minimum factorizations of a permutation in terms of the transpositions corresponding to edges in that graph. This result generalizes a previous one for the star graph, and can be applied to obtain the number of the shortest paths between a pair of vertices in some of the other structures closely related to the (n, k)-star graph, such as the alternating group networks. Furthermore, the techniques made use of in this paper can be applied to solve the same problem for some of the other structures such as the arrangement graph.

Subgraph Reliability of Alternating Group Graph With Uniform and Nonuniform Vertex Fault-Free Probabilities

2020 ◽  
Author(s):  
Yanze Huang ◽  
Limei Lin ◽  
Li Xu
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Fault Model ◽  
Alternating Group ◽  
Multiprocessor System ◽  
Single Vertex ◽  
Arrangement Graph ◽  
Uniform Probability ◽  
Better Than ◽  
Alternating Group Graph

Abstract As the size of a multiprocessor system grows, the probability that faults occur in this system increases. One measure of the reliability of a multiprocessor system is the probability that a fault-free subsystem of a certain size still exists with the presence of individual faults. In this paper, we use the probabilistic fault model to establish the subgraph reliability for $AG_n$, the $n$-dimensional alternating group graph. More precisely, we first analyze the probability $R_n^{n-1}(p)$ that at least one subgraph with dimension $n-1$ is fault-free in $AG_n$, when given a uniform probability of a single vertex being fault-free. Since subgraphs of $AG_n$ intersect in rather complicated manners, we resort to the principle of inclusion–exclusion by considering intersections of up to five subgraphs and obtain an upper bound of the probability. Then we consider the probabilistic fault model when the probability of a single vertex being fault-free is nonuniform, and we show that the upper bound under these two models is very close to the lower bound obtained in a previous result, and it is better than the upper bound deduced from that of the arrangement graph, which means that the upper bound we obtained is very tight.

A UNIFIED APPROACH TO THE CONDITIONAL DIAGNOSABILITY OF INTERCONNECTION NETWORKS

2012 ◽  
Vol 13 (03n04) ◽  
pp. 1250007 ◽  
Author(s):  
EDDIE CHENG ◽  
LÁSZLÓ LIPTÁK ◽  
KE QIU ◽  
ZHIZHANG SHEN
Keyword(s):  
Interconnection Networks ◽  
Ad Hoc ◽  
Cayley Graphs ◽  
Alternating Group ◽  
Unified Approach ◽  
Star Graphs ◽  
Conditional Diagnosability

The conditional diagnosability of interconnection networks has been studied in a number of ad-hoc methods resulting in various conditional diagnosability results. In this paper, we utilize these existing results to give an unified approach in studying this problem. Following this approach, we derive the exact value of the conditional diagnosability for a number of interconnection networks including Cayley graphs generated by 2-trees (which generalize alternating group graphs), arrangement graphs (which generalize star graphs and alternating group graphs), hyper Petersen networks, and dual-cube like networks (which generalize dual-cubes.)

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.

scholarly journals Borel Cayley Graph-Based Topology Control for Consensus Protocol in Wireless Sensor Networks

2013 ◽  
Vol 2013 ◽  
pp. 1-15 ◽  
Author(s):  
Junghun Ryu ◽  
Jaewook Yu ◽  
Eric Noel ◽  
K. Wendy Tang
Keyword(s):  
Wireless Sensor Networks ◽  
Sensor Networks ◽  
Interconnection Networks ◽  
Topology Control ◽  
Cayley Graphs ◽  
Small Diameter ◽  
Wireless Sensor ◽  
Control Algorithms ◽  
Transmission Range ◽  
Consensus Protocol

Borel Cayley graphs have been shown to be an efficient candidate topology in interconnection networks due to their small diameter, short path length, and low degree. In this paper, we propose topology control algorithms based on Borel Cayley graphs. In particular, we propose two methods to assign node IDs of Borel Cayley graphs as logical topologies in wireless sensor networks. The first one aims at minimizing communication distance between nodes, while the entire graph is imposed as a logical topology; while the second one aims at maximizing the number of edges of the graph to be used, while the network nodes are constrained with a finite radio transmission range. In the latter case, due to the finite transmission range, the resultant topology is an “incomplete” version of the original BCG. In both cases, we apply our algorithms in consensus protocol and compare its performance with that of the random node ID assignment and other existing topology control algorithms. Our simulation indicates that the proposed ID assignments have better performance when consensus protocols are used as a benchmark application.

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