Alternating Group Networks Research Articles

A recurrence for the surface area of the (n,k)$$ \left(n,k\right) $$‐star graph

SummaryWe present a simple recurrence for the surface area of the ‐star graph, , that is, the number of nodes at a certain distance from the identity node in the graph, an important parameter for interconnection networks in parallel computing. The family of the ‐star graphs includes several popular interconnection networks such as the star graph and the alternating group network. Previously, a surface area recurrence has been obtained for a special case, for example, when , in the family of ‐star. Our recurrence gives one single recurrence for all graphs in the family, thus completely solving the surface area of ‐star for all . Compared to explicit surface area formulas previously obtained through complicated and involved combinatorial analysis and generating function approach, our derivation is more elementary and our recurrence gives a way to compute the surface area of the ‐star efficiently.

Generalized 4-connectivity of alternating group networks

Generalized 4-connectivity of alternating group networks

Enhancing fault tolerance of balanced hypercube networks by the edge partition method

Interconnection networks are key component of parallel and distributed systems. As the network scale increases, the growing number of link failures becomes inevitable. Therefore, how to enhance the fault tolerance of the network has become a research issue in parallel and distributed systems. Recently, two new metrics based on edge partition in networks have been introduced, called matroidal connectivity and conditional matroidal connectivity. Existing research indicates that they significantly enhance the fault tolerance of alternating group networks and star networks compared to other indicators such as g-component edge connectivity and g-extra edge connectivity. In this work, we consider the matroidal connectivity and conditional matroidal connectivity of the n-dimensional balanced hypercube BHn, a class of networks with vertex transitivity and edge transitivity, and theoretically demonstrate their high fault tolerance in BHn. Furthermore, through numerical simulation, it is observed that the matroidal connectivity and conditional matroidal connectivity of BHn are greater than the g-component edge connectivity and g-extra edge connectivity of BHn, which implies that (conditional) matroidal connectivity can further improve BHn' fault tolerance.

The Structure Fault Tolerance of Alternating Group Networks

The alternating group network [Formula: see text] can be used to model the topology structure of a large-scale parallel network system. In this work, the structure fault tolerance of alternating group networks based on the star and path structures is investigated. For a graph [Formula: see text] and its a connected subgraph [Formula: see text], the [Formula: see text]-structure connectivity [Formula: see text] (resp. [Formula: see text]-substructure connectivity [Formula: see text]) of [Formula: see text] is the cardinality of a minimum family [Formula: see text] whose every element is isomorphic to [Formula: see text] (resp. isomorphic to a subgraph of [Formula: see text]) such that [Formula: see text] is disconnected. Specifically, we determine [Formula: see text] and [Formula: see text] for [Formula: see text].

Component Connectivity of Alternating Group Networks and Godan Graphs

Connectivity is an important index to evaluate the reliability and fault tolerance of a graph. As a natural extension of the connectivity of graphs, the [Formula: see text]-component connectivity of a graph [Formula: see text], denoted by [Formula: see text], is the minimum number of vertices whose removal from [Formula: see text] results in a disconnected graph with at least [Formula: see text] components. It is a scientific issue to determine the exact values of [Formula: see text] for distinguishing the fault tolerability of networks. However, [Formula: see text]-component connectivity of many well-known interconnection networks has not been explored even for small [Formula: see text]. For the [Formula: see text]-dimensional alternating group networks [Formula: see text] and [Formula: see text]-dimensional godan graphs [Formula: see text], we show that [Formula: see text] for [Formula: see text], and [Formula: see text] for [Formula: see text] and [Formula: see text].

Conditional Diagnosability of Alternating Group Networks Under the PMC Model

Fault diagnosis of processors has played an essential role when evaluating the reliability of multiprocessor systems. In many novel multiprocessor systems, their diagnosability has been extensively explored. Conditional diagnosability is a useful measure for evaluating diagnosability by adding a further condition that all neighbors of every node in the system do not fail at the same time. In this paper, we study the conditional diagnosability of $n$ -dimensional alternating group networks $AN_{n}$ under the PMC model, and obtain the results $t_{c}(AN_{4})=5$ , and $t_{c}(AN_{n})=6n-17$ for $n\ge 5$ . In addition, for the isomorphism property between $AN_{n}$ and $S_{n,k}$ with $k=n-2$ , namely $(n,n-2)$ -star graphs $S_{n,n-2}$ , the above results can be extended to $S_{n,n-2}$ , and we have $t_{c}(S_{4,2})=5$ and $t_{c}(S_{n,n-2})=6n-17$ for $n\ge 5$ . It is worth noting that the conditional diagnosability is about six times the degree of $AN_{n}$ and $S_{n,n-2}$ , which is very different from general networks with a multiple of four.

Recursive and parallel constructions of independent spanning trees in alternating group networks

In this paper, we propose a recursive algorithm and a parallel algorithm for constructing independent spanning trees in ANn. The main ideas of the recursive algorithm are to construct large trees from small trees, use triangle breadth-first search (TBFS) to build a frame of an IST, and use breadth-first search (BFS) to link the rest of the nodes. The main ideas of the parallel algorithm are to build frames through TBFS in parallel, and to use the specific rules to link the rest of the nodes in parallel. We prove the correctness of both algorithms for constructing ISTs in AN n . Both algorithms are accurate; furthermore, the parallel algorithm is more ecient than the recursive algorithm. (An extended abstract of this paper appeared in: Jie-Fu Huang and Sun-Yuan Hsieh ‘Two methods for constructing independent spanning trees in alternating group networks’ Proceedings of International Conference on Creative Lifestyle Computing (ICCLC), 2020.)

Constructing dual-CISTs of pancake graphs and performance assessment of protection routings on some Cayley networks

For a connected graph $$G=(V,E)$$ , two spanning trees $$T_1$$ and $$T_2$$ of G are said to be a pair of completely independent spanning trees (or a dual-CIST for short) if for any two vertices $$u,v\in V$$ , the paths joining u and v in the two trees have no common vertex except for u and v. Although the existence of a dual-CIST in the underlying graph of a network has the practical application of protection routing on fault-tolerance, it has been proved that determining whether a graph G admits a dual-CIST is NP-complete. As we know that Cayley graphs are a large family of graphs, some of its subclasses have been attracted and thus graphs in these subclasses have been adopted as the topologies of interconnection networks, such as the n-dimensional star graphs $$S_n$$ , bubble sort graphs $$BS_n$$ , pancake graph $$P_n$$ , alternating group networks $$AGN_n$$ and so on. Pai and Chang (IEEE/ACM Trans Netw 27(3): 1112–1123, 2019) recently showed that there exist dual-CISTs in $$S_n$$ , $$BS_n$$ , $$AGN_n$$ for $$n\geqslant 5$$ and provided their corresponding protection routings. So far, the problem of constructing dual-CISTs on $$P_n$$ has not been dealt with yet. In this sequel, we continue the investigation of the construction of dual-CISTs in pancake graphs as a complementary result. Since $$P_n$$ , $$S_n$$ , and $$BS_n$$ are with the same scale, we experimentally assess the performance of protection routing through simulation results for comparing them when $$n=5,6,7$$ .

Top-Down Construction of Independent Spanning Trees in Alternating Group Networks

A set of spanning trees in a graph G is called independent spanning trees (ISTs) if they are rooted at the same vertex r, and for each vertex v(= r) in G, the two paths from v to r in any two trees share no common vertex expect for v and r. ISTs can be applied in many research fields, such as fault-tolerant broadcasting and secure message distribution in reliable communication networks. Since Cayley graphs have been widely used to design interconnection networks, constructing ISTs on cayley graphs is worth studying. The alternating group network is a subclass of Cayley graphs, and the approach of constructing ISTs in alternating group networks is still unknown. In this paper, we propose a novel and simple top-down approach for constructing ISTs in alternating group networks. The main ideas of the algorithm are to use induction to develop small trees to big trees, to use a triangle breadth-first search (TBFS) process to create a backbone of an IST, and to use breadth-first search (BFS) process to connect the rest of nodes. Compared to other methods of different interconnection networks in the literature, the uniqueness of our method is that it does not need to determine the parent of one node by any rule, on the contrary others determine that by rules. The time complexity in n-dimensional alternating group network AN n is O(n 2 × n!), where n! is twice the number of nodes of AN n ; hence it is polynomial time. We implement the algorithm in PHP and run cases from AN 3 to AN 10. The results reveal that all spanning trees of AN 3 to AN 10 are independent and that our algorithm is accurate and efficient. INDEX TERMS Independent spanning trees, alternating group networks, triangle breadth-first search, interconnection networks, Cayley graph.

Reliability Assessment of Some Regular Networks

Abstract The generalized $k$-connectivity of a graph $G$ is a parameter that can measure the reliability of a network $G$ to connect any $k$ vertices in $G$, which is a generalization of traditional connectivity. Let $S\subseteq V(G)$ and $\kappa _{G}(S)$ denote the maximum number $r$ of edge-disjoint trees $T_{1}, T_{2}, \cdots , T_{r}$ in $G$ such that $V(T_{i})\bigcap V(T_{j})=S$ for any $i, j \in \{1, 2, \cdots , r\}$ and $i\neq j$. For an integer $k$ with $2\leq k\leq n$, the generalized $k$-connectivity of a graph $G$ is defined as $\kappa _{k}(G)= min\{\kappa _{G}(S)|S\subseteq V(G)$ and $|S|=k\}$. In this paper, we introduce a family of regular graph $G_{n}$ that can be constructed recursively and each vertex with exactly one outside neighbor. The generalized $3$-connectivity of the regular graph $G_{n}$ is studied, which attains a previously proven upper bound on $\kappa _{3}(G)$. As applications of the main result, the generalized $3$-connectivity of some important networks including some known results such as the alternating group network $AN_{n}$, the star graph $S_{n}$ and the pancake graphs $P_{n}$ can be obtained directly.

Some properties of alternating group networks

Let An be the alternating group of degree n. Set S={(123),(132),(12)(3i):4≤i≤n}. The alternating group network, denoted by ANn, is defined as the Cayley graph on An with respect to S. The fixing number of a graph Γ is the minimum size of a subset T⊆V(Γ) such that only the identity automorphism of Γ fixes every vertex in T. In this paper, we completely determine the full automorphism group of ANn. Based on this result, we further show that the fixing number of ANn is 2. In addition, we also obtain the independence number and the domination number of ANn.

Dual-CISTs: Configuring a Protection Routing on Some Cayley Networks

A set of $k~(\geqslant 2)$ spanning trees in the underlying graph of a network topology is called completely independent spanning trees, (CISTs for short), if they are pairwise edge-disjoint and inner-node-disjoint. Particularly, if $k=2$ , the two CISTs are called a dual-CIST. However, it has been proved that determining if there exists a dual-CIST in a graph is an NP-hard problem. Kwong et al. [IEEE/ACM Transactions Networking 19(5) 1543–1556, 2011] defined that a routing is protected, if there is an alternate with loop-free forwarding, when a single link or node failure occurs. Shortly afterward, Tapolcai [Optim. Lett. 7(4) 723–730, 2013] showed that a network possessing a dual-CIST suffices to establish a protection routing. It is well-known that Cayley graphs have a large number of desirable properties of interconnection networks. Although many results of constructing dual-CISTs on interconnection networks have been proposed in the literature, so far, the work has not been dealt with on Cayley graphs due to that their expansions are in exponential scalability. In this paper, we try to make a breakthrough of this work on some famous subclasses of Cayley graphs, including alternating group networks, bubble-sort network, and star networks. We first propose tree searching algorithms for helping the construction of dual-CISTs on low-dimensional networks. Then, by inductive construction, we show that dual-CISTs on high-dimensional networks can also be constructed agreeably. As a result, we can configure protection routings by using the constructed dual-CISTs. In addition, we complement some analysis with a simulation study of the proposed construction to evaluate the corresponding performance.

The generalized connectivity of alternating group graphs and [formula omitted]-star graphs

Let S⊆V(G) and κG(S) denote the maximum number r of edge-disjoint trees T1,T2,…,Tr in G such that V(Ti)⋂V(Tj)=S for any i,j∈{1,2,…,r} and i≠j. For an integer k with 2≤k≤n, the generalizedk-connectivity of a graph G is defined as κk(G)=min{κG(S)|S⊆V(G) and |S|=k}. The generalized k-connectivity is a generalization of traditional connectivity. In this paper, we focus on the alternating group graphsand (n,k)-star graphs, denoted by AGn and Sn,k, respectively. We study the generalized3-connectivity of the two kinds of graphs and show that κ3(AGn)=2n−5 for n≥4 and κ3(Sn,k)=n−2 for n≥k+1 and k≥4, which generalize the known result about star graphs given by Li et al. (2016). In addition, as the alternating group network ANn is isomorphic to Sn,k for k=n−2, the generalized 3-connectivity of ANn for n≥6 can be obtained directly.

The 4-component connectivity of alternating group networks

The ℓ-component connectivity (or ℓ-connectivity for short) of a graph G, denoted by κℓ(G), is the minimum number of vertices whose removal from G results in a disconnected graph with at least ℓ components or a graph with fewer than ℓ vertices. This generalization is a natural extension of the classical connectivity defined in term of minimum vertex-cut. As an application, the ℓ-connectivity can be used to assess the vulnerability of a graph corresponding to the underlying topology of an interconnection network, and thus is an important issue for reliability and fault tolerance of the network. So far, only a little knowledge of results have been known on ℓ-connectivity for particular classes of graphs and small ℓ's. In a previous work, we studied the ℓ-connectivity on n-dimensional alternating group networks ANn and obtained the result κ3(ANn)=2n−3 for n⩾4. In this sequel, we continue the work and show that κ4(ANn)=3n−6 for n⩾4.

A general approach to deriving the g-good-neighbor conditional diagnosability of interconnection networks

We discuss a general approach to deriving the g-good-neighbor conditional diagnosability of interconnection networks. As demonstrative examples, we derive the 1- and 2-good-neighbor conditional diagnosabilities of the arrangement graphs under both the commonly adopted PMC and MM⁎ model. We also derive the general g-good-neighbor conditional diagnosability of the (n,k)-star graphs under the PMC model for g∈[1,n−k], and under the MM⁎ model for g∈[2,n−k], as well as that of the related graphs, such as the star graph, the alternating group graph, and the alternating group network.

Reliability of (n,k)-star network based on g-extra conditional fault

In the wake of rapid development of multiprocessor systems, fault tolerance of processors plays an even more vital role in measuring the reliability of a multiprocessor system. The g-extra connectivity of a multiprocessor system modeled by graph G, denoted by κo(g)(G), is the minimum number of nodes whose deletion will disconnect the network and every remaining component has more than g vertices. The g-extra conditional diagnosability of multiprocessor system G, denoted by tg˜(G), is the maximum number of faulty vertices that the system can guarantee to identify under the condition that every fault-free component contains at least g+1 vertices. Only a few achievements have been established on κo(g)(G) for some special graphs with small g. In this paper, we first determine that the g-extra connectivity of (n,k)-star network Sn,k is κo(g)(Sn,k)=n+g(k−2)−1 for 2≤k≤n−1 and 0≤g≤n−k and then show that the g-extra conditional diagnosability of Sn,k under the PMC model (n≥4, 2≤k≤n−1 and 1≤g≤n−k) and the MM* model (n≥6, 3≤k≤n−3 and 1≤g≤min⁡{n−k+14,k−2}) is tg˜(Sn,k)=n+g(k−1)−1, respectively. Meanwhile, we also show that Sn,k is (n+m(k−2)−1)/m-diagnosable under the pessimistic diagnostic strategy. As by-products, we derive the g-extra conditional diagnosability of n-dimensional star graph Sn (n≥4, g=1) and n-dimensional alternating group network ANn (n≥4, 1≤g≤2) under the PMC model.

Neighbor Connectivity of Two Kinds of Cayley Graphs

In this paper, we determine the neighbor connectivity κNB of two kinds of Cayley graphs: alternating group networks AN n and star graphs S n ; and give the exact values of edge neighbor connectivity λNB of ANn and Cayley graphs generated by transposition trees Γ n . Those are κNB(AN n ) = n−1, λNB(AN n ) = n−2 and κNB(S n ) = λNB(Γ n ) = n−1.

Equal relation between the extra connectivity and pessimistic diagnosability for some regular graphs

Extra connectivity and the pessimistic diagnosis are two crucial subjects for a multiprocessor system's ability to tolerate and diagnose faulty processor. The pessimistic diagnosis strategy is a classic strategy based on the PMC model in which isolates all faulty vertices within a set containing at most one fault-free vertex. In this paper, the result that the pessimistic diagnosability tp(G) equals the extra connectivity κ1(G) of a regular graph G under some conditions are shown. Furthermore, the following new results are gotten: the pessimistic diagnosability tp(Sn2)=4n−9 for split-star networks Sn2; tp(Γn)=2n−4 for Cayley graphs generated by transposition trees Γn; tp(Γn(Δ))=4n−11 for Cayley graph generated by the 2-tree Γn(Δ); tp(BPn)=2n−2 for the burnt pancake networks BPn. As corollaries, the known results about the extra connectivity and the pessimistic diagnosability of many famous networks including the alternating group graphs, the alternating group networks, BC networks, the k-ary n-cube networks etc. are obtained directly.

Generalized 3-Connectivity of Alternating Group Networks

For strengthening the classical connectivity of networks, Hager (1985) first introduced the concept of generalized connectivity of networks, which was also called the tree-connectivity by Okamoto and Zhang (2010). In this paper, we prove that the generalized 3-connectivity of the n-dimensional alternating group network is n-2.

On computing of a conditional edge connectivity of alternating group network

An edge subset F of a connected graph G is called a k-restricted edge cut if is disconnected and every vertex in has at least k neighbours in . The cardinality of a minimum k-restricted edge-cut of G is called the k-restricted edge connectivity of G, denoted by . The alternating group network () is defined as the Cayley graph on the alternating group of degree n with respect to . In this paper, we prove that for each .