Super Ck and Sub-Ck Connectivity of k-Ary n-Cube Networks

2021 ◽  
pp. 1-12
Author(s):  
Yuxing Yang
Keyword(s):  
Interconnection Networks ◽  
Undirected Graph ◽  
Text Structure ◽  
Multiprocessor Systems ◽  
Cube Formula

Let [Formula: see text] be an undirected graph. An H-structure-cut (resp. H-substructure-cut) of [Formula: see text] is a set of subgraphs of [Formula: see text], if any, whose deletion disconnects [Formula: see text], where the subgraphs deleted are isomorphic to a certain graph [Formula: see text] (resp. where for any [Formula: see text] of the subgraphs deleted, there is a subgraph [Formula: see text] of [Formula: see text], isomorphic to [Formula: see text], such that [Formula: see text] is a subgraph of [Formula: see text]). [Formula: see text] is super [Formula: see text]-connected (resp. super sub-[Formula: see text]-connected) if the deletion of an arbitrary minimum [Formula: see text]-structure-cut (resp. minimum [Formula: see text]-substructure-cut) isolates a component isomorphic to a certain graph [Formula: see text]. The [Formula: see text]-ary [Formula: see text]-cube [Formula: see text] is one of the most attractive interconnection networks for multiprocessor systems. In this paper, we prove that [Formula: see text] with [Formula: see text] is super sub-[Formula: see text]-connected if [Formula: see text] and [Formula: see text] is odd, and super [Formula: see text]-connected if [Formula: see text] and [Formula: see text] is odd.

Subgraph-based Strong Menger Connectivity of Hypercube and Exchanged Hypercube

2021 ◽  
pp. 1-26
Author(s):  
Yihong Wang ◽  
Cheng-Kuan Lin ◽  
Shuming Zhou ◽  
Tao Tian
Keyword(s):  
Big Data ◽  
Fault Tolerance ◽  
Connected Graph ◽  
Interconnection Networks ◽  
Large Scale ◽  
Fault Tolerant ◽  
Disjoint Paths ◽  
Multiprocessor Systems ◽  
Edge Disjoint ◽  
Cube Formula

Large scale multiprocessor systems or multicomputer systems, taking interconnection networks as underlying topologies, have been widely used in the big data era. Fault tolerance is becoming an essential attribute in multiprocessor systems as the number of processors is getting larger. A connected graph [Formula: see text] is called strong Menger (edge) connected if, for any two distinct vertices [Formula: see text] and [Formula: see text], there are [Formula: see text] vertex (edge)-disjoint paths between them. Exchanged hypercube [Formula: see text], as a variant of hypercube [Formula: see text], remains lots of preferable fault tolerant properties of hypercube. In this paper, we show that [Formula: see text] [Formula: see text] and [Formula: see text] [Formula: see text] are strong Menger (edge) connected, respectively. Moreover, as a by-product, for dual cube [Formula: see text], one popular generalization of hypercube, [Formula: see text] is also showed to be strong Menger (edge) connected, where [Formula: see text].

Average covering number for some graphs

2019 ◽  
Vol 53 (1) ◽  
pp. 261-268
Author(s):  
D. Doğan Durgun ◽  
Ali Bagatarhan
Keyword(s):  
Interconnection Networks ◽  
Undirected Graph ◽  
Cartesian Product ◽  
Covering Number ◽  
Minimum Cardinality ◽  
Covering Numbers ◽  
Covering Set ◽  
Local Covering

The interconnection networks are modeled by means of graphs to determine the reliability and vulnerability. There are lots of parameters that are used to determine vulnerability. The average covering number is one of them which is denoted by $ \overline{\beta }(G)$, where G is simple, connected and undirected graph of order n ≥ 2. In a graph G = (V(G), E(G)) a subset $ {S}_v\subseteq V(G)$ of vertices is called a cover set of G with respect to v or a local covering set of vertex v, if each edge of the graph is incident to at least one vertex of Sv. The local covering number with respect to v is the minimum cardinality of among the Sv sets and denoted by βv. The average covering number of a graph G is defined as β̅(G) = 1/|v(G)| ∑ν∈v(G)βν In this paper, the average covering numbers of kth power of a cycle $ {C}_n^k$ and Pn □ Pm, Pn □ Cm, cartesian product of Pn and Pm, cartesian product of Pn and Cm are given, respectively.

K-PROCESSOR RELIABILITY OF LARGE-SCALE RING-BASED HIERARCHICAL INTERCONNECTIONS

2002 ◽  
Vol 09 (01) ◽  
pp. 61-77
Author(s):  
M. AL-ROUSAN ◽  
O. AL-JARRAH ◽  
M. MOWAFI
Keyword(s):  
Interconnection Networks ◽  
Large Scale ◽  
Network Reliability ◽  
Multiprocessor Systems ◽  
Hierarchical Systems ◽  
Hierarchical Networks ◽  
Coherent Interface ◽  
Large Acceptance ◽  
Ring Architecture ◽  
Scalable Coherent Interface

Recently, connecting thousands of processors via interconnection networks based on multiple (hierarchical) rings has an increased interest. This is due to the large acceptance and success of the Scalable Coherent Interface (SCI) technology. The inherently weak behavior of ring architecture has led interconnection designers to consider various choices to improve the overall network reliability. An interesting choice is to use braided rings instead of the single (basic) rings in the hierarchy. In this paper, we present new formulas for computing K-processor reliability of SCI ring-based hierarchical networks in the context of large-scale multiprocessor systems. The derived formulas are general and applicable to any given systems size consisting of an arbitrary number of levels. The reliability of hierarchical systems based on the basic and braided rings is evaluated and analyzed using the derived formulas. The results show that hierarchical systems based on braided rings significantly improve the reliability of hierarchies constructed of basic rings. The results are general and not limited to systems of SCI rings; the analysis is valid for any type of rings architecture such as token and slotted rings.

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.

MULTIPROCESSOR INTERCONNECTION NETWORKS WITH SMALL TIGHTNESS

2009 ◽  
Vol 20 (05) ◽  
pp. 941-963 ◽  
Author(s):  
DRAGOŠ CVETKOVIĆ ◽  
TATJANA DAVIDOVIĆ
Keyword(s):  
Interconnection Networks ◽  
Recent Literature ◽  
Vertex Degree ◽  
Multiprocessor Systems ◽  
Undirected Graphs ◽  
Graph Invariants ◽  
Connected Graphs ◽  
Multiprocessor Interconnection Networks ◽  
Related Graph ◽  
Multiprocessor Interconnection

Homogeneous multiprocessor systems are usually modelled by undirected graphs. Vertices of these graphs represent the processors, while edges denote the connection links between adjacent processors. Let G be a graph with diameter D, maximum vertex degree Δ, the largest eigenvalue λ1 and m distinct eigenvalues. The products mΔ and (D+1)λ1 are called the tightness of G of the first and second type, respectively. In recent literature it was suggested that graphs with a small tightness of the first type are good models for the multiprocessor interconnection networks. In a previous paper we studied these and some other types of tightness and some related graph invariants and demonstrated their usefulness in the analysis of multiprocessor interconnection networks. We proved that the number of connected graphs with a bounded tightness is finite. In this paper we determine explicitly graphs with tightness values not exceeding 9. There are 69 such graphs and they contain up to 10 vertices. In addition we identify graphs with minimal tightness values when the number of vertices is n = 2,…, 10.

Relation of Extra Edge Connectivity and Component Edge Connectivity for Regular Networks

2021 ◽  
Vol 32 (02) ◽  
pp. 137-149
Author(s):  
Litao Guo ◽  
Mingzu Zhang ◽  
Shaohui Zhai ◽  
Liqiong Xu
Keyword(s):  
Interconnection Networks ◽  
Reliability Evaluation ◽  
Multiprocessor Systems ◽  
Edge Connectivity ◽  
Two Parameters ◽  
Regular Networks ◽  
Text Component

Reliability of interconnection networks is important to design multiprocessor systems. The extra edge connectivity and component edge connectivity are two parameters for the reliability evaluation. The [Formula: see text]-extra edge connectivity [Formula: see text] is the cardinality of the minimum extra edge cut [Formula: see text] such that [Formula: see text] is not connected and each component of [Formula: see text] has at least [Formula: see text] vertices. The [Formula: see text]-component edge connectivity [Formula: see text] of a graph [Formula: see text] is the minimum edge number of a set [Formula: see text] such that [Formula: see text] is not connected and [Formula: see text] has at least [Formula: see text] components. In this paper, we find the relation of extra edge connectivity and component edge connectivity for regular networks. As an application, we determine the component edge connectivity of BC networks, [Formula: see text]-ary [Formula: see text]-cubes, enhanced hypercubes.

scholarly journals The Edge Connectivity of Expanded k-Ary n-Cubes

2018 ◽  
Vol 2018 ◽  
pp. 1-7
Author(s):  
Shiying Wang ◽  
Mujiangshan Wang
Keyword(s):  
Fault Tolerance ◽  
Interconnection Networks ◽  
Interconnection Network ◽  
Complex Problem ◽  
Multiprocessor System ◽  
Multiprocessor Systems ◽  
Complex Problem Solving ◽  
Edge Connectivity ◽  
Communication Links ◽  
Minimum Number

Mass data processing and complex problem solving have higher and higher demands for performance of multiprocessor systems. Many multiprocessor systems have interconnection networks as underlying topologies. The interconnection network determines the performance of a multiprocessor system. The network is usually represented by a graph where nodes (vertices) represent processors and links (edges) represent communication links between processors. For the network G, two vertices u and v of G are said to be connected if there is a (u,v)-path in G. If G has exactly one component, then G is connected; otherwise G is disconnected. In the system where the processors and their communication links to each other are likely to fail, it is important to consider the fault tolerance of the network. For a connected network G=(V,E), its inverse problem is that G-F is disconnected, where F⊆V or F⊆E. The connectivity or edge connectivity is the minimum number of F. Connectivity plays an important role in measuring the fault tolerance of the network. As a topology structure of interconnection networks, the expanded k-ary n-cube XQnk has many good properties. In this paper, we prove that (1) XQnk is super edge-connected (n≥3); (2) the restricted edge connectivity of XQnk is 8n-2 (n≥3); (3) XQnk is super restricted edge-connected (n≥3).

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