Abstract

Connectivity is a significant metric for evaluating the error lenience of an interconnected net . In the paper (Fàbrega and Fiol, 1996), the authors addressed the -extra connectivity which is applied to better measure the consistency and error tolerance of . As a special Cayley graph, the -dimensional wheel network has some desirable features. This paper shows that the -extra connectivity of is when .

1. Introduction

An interconnected net (abbreviated IN) is usually charactered as a graph, whereat each node (vertex) resembles a CPU and an edge resembles the connection between a couple of CPUs, correspondingly. Connectivity of a graph , written as , is defined as the minimum number of nodes whose deletion from implies a detached graph or has only 1 node. Connectivity is an important index in evaluating the dependability and error tolerance of an IN. The greater the connectivity is, the more consistent an IN is. Nevertheless, an understandable drawback of the index is that it adopts that each node neighboring the same node of can flop simultaneously. In fact, it is highly unlikely in applied net uses. Hence, the traditional connectivity is inappropriate for huge computing systems.

To overcome this deficiency, Harary [1] firstly presented the idea of restricted connectivity, which is a more refined index in determining the consistency and error tolerance of INs. Consider to be a linked directionless basic graph, and be a given 1-dimensional topology theory feature. The restricted connectivity, written as , is obtained as the minimum cardinality of a set of nodes, if any, whose deletion detaches and each residual factor has feature . Amid the restricted connectivity, the -extra connectivity has been initially presented by Fàbrega and Fiol [2]. A subset of nodes is known as a cut-set if is detached. For a nonnegative integer , a cut-set is named a -extra cut, if each element of contains nodes, where . The -extra connectivity of , if there must exist 1 -extra cut, written as , is now obtained as the minimum cardinality above all -extra cuts of , namely, , whereat is the feature that each residual element takes at least nodes. Obviously, for each linked incomplete graph . Therefore, the traditional connectivity may be regarded as a simplification of -extra connectivity, and it is able to serve for more precisely measuring the consistency and error lenience for INs. The -extra connectivity of numerous INs was widely investigated (see [217]).

It is worthwhile to mention that the algebraic connectivity [18] is another important metric in measuring how fine a graph is linked and is very significant in controller theory, communications, etc. Further research relating to algebraic connectivity are described in references [19, 20].

The -dimensional wheel network, a promising topology arrangement of INs, has some respectable things. Here, we establish that the -extra connectivity of is , when .

2. Preliminaries

Definitions and notations used through this paper are provided. The -dimensional wheel network and its basic properties are examined. We consider [21] for terms and notations not introduced in this paper.

2.1. Definitions and Notations

Consider a basic undirected graph . Given , , the prompted graph determined by in , written as , is a graph in which set of nodes is and the set of edges contains all the edges of having both endpoints in . For , the degree in is defined as the amount of edges neighboring to , written as . Let the minimum degree in . For arbitrary node , the neighborhood of is determined as the set of nodes neighboring to in a graph . The node is named a neighbor node of , where . For , is applied to represent . If no confusion arises, then , , and can be abbreviated into , , and , correspondingly. The set containing all error nodes of is named as a faulty set in . Each node in is named faulty and any node in is named faulty-free.

2.2. The Wheel Networks

The wheel networks have been known as a desirable topology structure of INs. Here, we recall its definition and some important aspects.

Consider to be a bounded group, to be a spanning set of , where the identity element of cannot be in . The linked Cayley graph has a set of nodes and a set of arcs . The condition that is a spanning set of ensures that is linked. The assumption that the identity element of cannot be in guarantees that is simple. For convenience, let . Here, we concentrate on the Cayley graphs produced by transpositions. We choose the symmetric group on as , and a set of transpositions of as . Notice that contains only transposition, and there exists an arc iff there exists an arc , where are two nodes. Thus, the corresponding Cayley graph can be viewed as an undirected Cayley graph.

Consider a basic linked graph whose set of nodes is . Each edge of is viewed as a transposition of the symmetric graph on ; hence, the set of all edges of resembles a transposition set of . Therefore, is entitled a transposition basic graph, and the resulting Cayley graph is said to be the corresponding Cayley graph of , written as Cay. Akers et al. [22] proved that in Cay.

If mentioned above is a tree (resp. a path, a star), then the resulting Cayley graph is named a transposition tree (resp. a bubble-sort graph, a star graph) [22], written as (resp. , ). When is a sector of nodes, i.e., and , the resulting Cayley graph is named a bubble-sort star graph [23], written as . If is a wheel of nodes, i.e., and , then the resulting Cayley graph is named a -dimensional wheel network [24], written as . In other words, represents a graph with set of nodes  = , where 2 nodes are neighboring iff , , or , , or . Figure 1 illustrates the Cayley graph .

To discuss conveniently, we denote by the permutation , where .

Theorem 1. (see [25]). Each permutation different from identity within the symmetric group is the only (considering the order of the factors) multiply of cycles that are not joined, where every cycle has length greater or equal to 2.

Theorem 2. (see [26]). Suppose that is a basic linked graph where , and and are a pair of diverse graphs gained by labelling with . Then, must be isomorphic to .
By Theorem 1, each permutation must be written as a multiplication of cycles. For instance, . In particular, . The product of 2 permutations and is the composition function trailed by , i.e., . For terms and notations not mentioned here we follow [25].
As a special Cayley graph, owns many attractive properties.

Proposition 1. (see [27]). is -regular, node transitive, .

Proposition 2. (see [27]). is bipartite, .

Proposition 3. (see [28]). The girth of is 4, .
In the next discussion, we often partition into disjoint subgraphs , where each node takes a specified integer in the latter place for . Evidently, each is isomorphic to , where is the bubble-sort star graph. For each node, , , , and are all named external neighbors of , written as , , and , respectively. Any edge is named a cross-edge respectful to a fixed factorization if its 2 nodes are in diverse ’s.

Proposition 4. (see [28]). Let , where is mentioned as previously. Then, , , and are in three diverse ’s .

Proposition 5. (see [29]). For , then , where .

Proposition 6. (see [28]). Let be denoted as previously. There exist exactly autonomous cross-edges among 2 diverse ’s.

Theorem 3 (see [28]). Let be denoted as previously. If 2 nodes are neighboring, no shared neighboring nodes exist of those nodes, that is, . If node is not neighboring to , maximally 3 mutual neighboring nodes exist of these nodes, namely, .

3. κ˜ (2)(CWn)

Here, we establish .

Proposition 7. (see [30]). Suppose that is denoted as previously. Thus, is -regular, node transitive, .

Proposition 8. (see [30]). Suppose that is denoted as previously. The connectivity , .

Lemma 1. (see [30]). For greater or equal to 4 and is a subset of , where . When is detached, a single of these situations for is valid:(1) has 2 constituents, with 1 isolated node(2) has 3 constituents, with 2 isolated nodes

Corollary 1. (see [29]). For greater or equal to 4 and is a subset of , where . When has 3 components, with 2 isolated nodes, it is valid that .

Lemma 2. The 2-extra connectivity .

Proof. Observe and . From Theorem 2, assume that . It is valid: , . From Proposition 2, contains none of 3-cycles. Applying it and Theorem 3, it is valid that . Consider and , and has two parts and (see Figure 2).

Let us now prove that is linked and .

We factorize lengthways the latter location, written as . Recall that each and are isomorphic for . Hence, is -regular using Proposition 7, and by Proposition 8 for each .

Notice that , and , , , , and . Using Proposition 6 and , is linked. Using Proposition 5, every node of is neighboring to 3 faulty-free nodes of , and is linked. Combining and , hence , where and . Therefore, is linked and . Consequently, . has 2 constituents: and . Notice that is valid. Hence, is really a 2-extra cut and .

According to Lemma 2, the next lemma is formulated.

Lemma 3. Let . For greater or equal to 5, , now , is a 2-extra cut of , and must have 2 constituents: and (see Figure 2).

Lemma 4. For any integer , .

Proof. Consider to be a minimum 2-extra cut, and suppose . We shall obtain a contradiction. Assume . Observe 2 situations.

Situation 1. for all .
Notice that ; hence, is linked for each . Using Proposition 6 and for , we have that there must exist at least one edge to connect and , where and . Therefore, is linked, which contradicts the assumption that is a minimum 2-extra cut.

Situation 2. for some .
Assume . Recall that , then we have . For any , . From an analogous statement as in Situation 1, it is valid that is linked, denoted it by .

Situation 2.1. .
In general, suppose . Let is a liked element of . When , must be linked to , since is a 2-extra cut. If , then has a path of three nodes. Suppose and . Thus, and (see Figure 3.). Combining Theorem3 and is -regular, . By Propositions 4 and 5, . Then, there must exist at least one node which has an external neighbor ; thus, is linked to . Using the randomness of , is linked, which contradicts that is a minimum 2-extra cut.

Situation 2.2. .
In general, suppose that and is a linked element of . When , must be linked to , since is an 2-extra cut. If , then has a path of 3 nodes.
First, suppose that one of and , say , has a path on 3 nodes of . Let and . Then, , , and . When , . By Propositions 4and 5, for . Then, there must exist at least one node which has an external neighbor , and is linked to . From the randomness of , is linked, which contradicts that is a minimum 2-extra cut.
If neither nor has a path on three nodes of , then any path of on three nodes has 2 nodes in a single side and 1 node in the different side, say with and . Thus, , from Proposition 2, . By Propositions 4 and 5, we see that ; hence, is linked to ; thus, is also linked to .
In any situations, we have shown that is linked, which contradicts the assumption that is a minimum 2-extra cut.

Situation 2.3. .
In general, suppose . Since and for each , we have that , for , and . Notice that for , combining each is isomorphic to and Lemma 1 and Corollary 1; we have that, for each , if is disconnected, then has exactly 2 constituents, one of which is an inaccessible node which is written by . Notice that and ; hence, there must exist some which does not contain any faulty node. By Proposition 6 and , and are linked for each . Hence, we have that which is written as is linked. Next, we merely need to consider the situation that is disconnected. Let be disconnected and be an arbitrary linked component. If , then is linked to since is a 2-extra cut; thus, must form a path, and the path is the only other linked component of different from . Using Propositions 2, 4, and 5, then does not form a 3-cycle, and each of , and must have an external neighbor in . Notice that ; thus, there must exist at least one node which has an external neighbor ; hence, it is shown that is linked, which contradicts the assumption that is a minimum 2-extra cut.
By Situations 1 and 2, must be not a 2-extra cut of and therefore . Hence, for .
According to obtained Lemmas 2 and 4, the next is valid.

Theorem 4. , .

4. Conclusion

The traditional connectivity may be regarded as a simplification of the -extra connectivity. It is gained that . Notice that when and , When are obtained in [9]. We conclude the paper by summarizing which situations of -extra connectivity of have been not solved in Table 1.

Data Availability

The data used to support the findings of the study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.