A concentration phenomenon for $h$-extra edge-connectivity reliability analysis of enhanced hypercubes $Q_{n,2}$ with exponentially many faulty links

Reliability evaluation of multiprocessor systems is of significant importance in the design and maintenance of multiprocessor systems. Based on edge-connectivity, more refined quantitative indicators for the reliability of multiprocessor systems have been introduced. The extra edge-connectivity and the component edge-connectivity, as two important parameters to evaluate the robustness of multiprocessor systems, are explored extensively. In this paper, we determine the $h$-extra edge-connectivity and the $(g+1)$-component edge-connectivity of the balanced complete $t$-partite graph $K_{r}^{t}$ for $t, r\geq 2$, where $1\leq h \leq \lfloor tr/2 \rfloor$ and $2\leq g \leq tr-1$.

Reliability measure of multiprocessor system based on enhanced hypercubes

Fault diagnosis of systems is an important aspect of study in the design and maintenance of multiprocessor systems. In order to improve diagnostic ability and strengthen reliability, Fabrega et al. introduced extra connectivity and extra edge-connectivity, which are two important parameters in studying the reliability of multiprocessor systems. The balanced hypercube, denoted by where, is a variant of hypercube network, which has some specific topological properties. This paper studied the reliability of balanced hypercube, and determines its h-extra connectivity and the h-extra edge-connectivity for . Through the above research results, it has very important theoretical value and practical significance for the application and promotion of balanced hypercube in multiprocessor system.

Reliability analysis of bijective connection networks in terms of the extra edge-connectivity

Reliability evaluation of interconnection network is important to the design and maintenance of multiprocessor systems. The extra connectivity and the extra edge-connectivity are two important parameters for the reliability evaluation of interconnection networks. The <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\mbi {n}}$</tex> </formula> -dimensional bijective connection network (in brief, BC network) includes several well known network models, such as, hypercubes, Möbius cubes, crossed cubes, and twisted cubes. In this paper, we explore the extra connectivity and the extra edge-connectivity of BC networks, and discuss the structure of BC networks with many faults. We obtain a sharp lower bound of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${{g}}$</tex> </formula> -extra edge-connectivity of an <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\mbi {n}}$</tex> </formula> -dimensional BC network for <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${{n}} \geq 4$</tex> </formula> and <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$1 \leq { {g}} \leq {2^{[{{{n}} \over 2}]}}$</tex> </formula> . We also obtain a sharp lower bound of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${ {g}}$</tex> </formula> -extra connectivity of an <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${{n}}$</tex> </formula> -dimensional BC network for <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${{n}} \geq 4$</tex> </formula> and <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$1 \leq { {g}} \leq 2{ {n}}$</tex> </formula> which improves the result in [“Reliability evaluation of BC networks,” IEEE Trans. Computers, DOI: 10.1109/tc.2012.106.] for <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$1 \leq { {g}} \leq { {n}} - 3$</tex> </formula> . Furthermore, we give a remark about exploring the <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${ {g}}$</tex> </formula> -extra edge-connectivity of BC networks for the more general <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\mbi {g}}$</tex> </formula> , and we also characterize the structure of BC networks with many faulty nodes or links. As an application, we obtain several results on the <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\mbi {g}}$</tex> </formula> -extra (edge-) connectivity and the structure of faulty networks on hypercubes, Möbius cubes, crossed cubes, and twisted cubes.

Edge connectivity can measure the reliability of a network. Extra edge connectivity and component edge connectivity are the generalization of the classical edge connectivity. The extra component edge connectivity is the combination of the extra edge connectivity and component edge connectivity which can be used to measure the reliability of a network more precisely. The modified bubble-sort graph $MB_n$ is one of the important interconnection network topologies. In this paper, we study the 1-extra 3-component edge connectivity $c\lambda_3^1 (MB_n )$ of the modified bubble-sort graph and prove that $c\lambda_3^1 (MB_n )=4n-6$.

Reliability analysis of an interconnection network is of great significance to the design and maintenance of multiprocessor systems. The $h$ -extra edge-connectivity of a given interconnected network $G$ with $N$ processors, denoted by $\lambda _h(G)$ , is the minimum cardinality of set of faulty links, such that whose removal will disconnect the network with all its resulting components having at least $h$ processors for $h\leq N/2$ . It gives a more refined quantitative analysis of indicators of the robustness of a multiprocessor system in the presence of failing links. The $n$ -dimensional folded hypercube $FQ_n$ , as one of potential interconnected networks, is a well-known variation of the hypercube structure with $N=2^{n}$ processors. In this paper, the $h$ -extra edge-connectivity of the network $FQ_n$ , $\lambda _h(FQ_n)$ , is first investigated for each well-defined positive integer $h\leq N/2$ . We divide the interval $1\leq h\leq N/2$ into some subintervals and obtain some properties of $\lambda _h(FQ_n)$ in these subintervals. Then, we deduce a recursive relation of $\lambda _h(FQ_n)$ . Based on this recursion, an efficient $O({\log}_2(N))$ algorithm is designed to totally determine the exact values and $\lambda _h$ -optimality of $\lambda _h(FQ_n)$ for each $h\leq N/2$ .

Multiprocessor systems are becoming increasingly popular because of the increased throughput possible and the possibility of system availability despite the failure of some of the processing units. This paper describes an innovative concept in multiprocessor system design, and suggests some of the research areas which have not yet been resolved but which can be studied on this system. The system architecture is patterned after Control Data Corporation's peripheral processor barrel except that it possesses much more powerful functional capabilities. In addition, high speed minicomputers were used to handle all of the system I/O requirements. Research areas for which this system is particularly appropriate include computer architecture-operating systems tradeoffs and multiprocessor operating systems design and implementation. The hardware system is entirely microprogrammable, allowing for increased flexibility in evaluating various research strategies.

Multiprocessor systems are becoming increasingly popular because of the increased throughput possible and the possibility of system availability despite the failure of some of the processing units. This paper describes an innovative concept in multiprocessor system design, and suggests some of the research areas which have not yet been resolved but which can be studied on this system. The system architecture is patterned after Control Data Corporation's peripheral processor barrel except that it possesses much more powerful functional capabilities. In addition, high speed minicomputers were used to handle all of the system I/O requirements. Research areas for which this system is particularly appropriate include computer architecture-operating systems tradeoffs and multiprocessor operating systems design and implementation. The hardware system is entirely microprogrammable, allowing for increased flexibility in evaluating various research strategies.

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.

Hybrid fault diagnosis capability analysis of regular graphs

Relationship between extra edge connectivity and component edge connectivity for regular graphs

Connectivity is an important parameter for evaluating the reliability and stability of an interconnection network. Based on the edge connectivity, more refined connectivities have been proposed. The [Formula: see text]-component edge connectivity [Formula: see text] and the [Formula: see text]-extra edge connectivity [Formula: see text] are two important parameters to assess the robustness of an interconnection network, which received attention extensively. In this paper, we determine the [Formula: see text]-component edge connectivity and the [Formula: see text]-extra edge connectivity of bubble-sort star graphs [Formula: see text]. For [Formula: see text]-component edge connectivity, we prove that [Formula: see text], [Formula: see text], and [Formula: see text] for [Formula: see text]. For [Formula: see text]-extra edge connectivity, we prove that [Formula: see text], [Formula: see text], and [Formula: see text] for [Formula: see text].

In the design of multiprocessor systems, evaluating the reliability of interconnection networks is a critical aspect that significantly impacts system performance and functionality. When quantifying the reliability of these networks, extra connectivity and extra diagnosability serve as fundamental metric parameters, offering valuable insights into the network’s resilience and fault-handling capabilities. In this paper, we investigate the 1-extra connectivity and 1-extra diagnosability of the n-dimensional enhanced folded hypercube-like network. Through analysis, we show that the 1-extra connectivity of this network is 2n+2. Moreover, for n&gt;5, we determine its 1-extra diagnosability under both the PMC model and the MM⁢∗ model to be 2n+3. These results show that as the dimension n increases, both the 1-extra connectivity and 1-extra diagnosability of the network approach approximately twice the value of traditional diagnosability metrics. This provides quantitative insights into the reliability properties of the enhanced folded hypercube-like network, contributing to a better understanding of its performance in terms of connectivity and fault diagnosis.

The [formula omitted]-extra connectivity and [formula omitted]-extra conditional diagnosability of Bubble-sort star graphs