The Edge Connectivity of Expanded 𝑘 -Ary 𝑛 -Cubes

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 𝑋𝑄 𝑘𝑛 has many good properties. In this paper, we prove that (1) 𝑋𝑄 𝑘𝑛 is super edge-connected ( 𝑛 ≥ 3 ); (2) the restricted edge connectivity of 𝑋𝑄 𝑘𝑛 is 8𝑛 − 2 ( 𝑛 ≥ 3 ); (3) 𝑋𝑄 𝑘𝑛 is super restricted edge-connected ( 𝑛 ≥ 3 ).


Introduction and Terminology
Mass data processing and complex problem solving have higher and higher demands for performance of multiprocessor systems. Many multiprocessor systems have interconnection networks (networks for short) as underlying topologies and a network is usually represented by a graph = ( , ) where nodes (vertices) represent processors and links (edges) represent communication links between processors. The network determines the performance of the system. For the network (graph) , two vertices and V of are said to be connected if there is a ( , V)-path in . Connection is an equivalence relation on the vertex set . Thus there is a partition of into nonempty subsets 1 , 2 , . . . , such that two vertices and V of are connected if and only if both and V belong to the same set . The subgraphs [ 1 ], [ 2 ], . . . , [ ] are called the components of . If has exactly one component, then is connected; otherwise 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.
The fault tolerance of large networks is usually a measure of the extent to which the network can retain its original nature in the event of a certain number of nodes of failure and/or links failure in the network topology. For a connected network (graph) = ( , ), its inverse problem is that − is disconnected, where ⊆ or ⊆ . The connectivity or edge connectivity is the minimum number of | |. Connectivity plays an important role in measuring the fault tolerance of the network. Let = ( , ) be a simple graph. Given a nonempty vertex subset of , the induced subgraph by in , denoted by [ ], is a graph, whose vertex set is and the edge set is the set of all the edges of with both endpoints in . The degree (V) of a vertex V is the number of edges incident with V. We denote by ( ) the minimum degrees of vertices of . For any vertex V, we define the neighborhood (V) of V in to be the set of vertices adjacent to V. If ∈ (V), then is called a neighbor or a neighbor vertex of V. Let ⊆ . We use ( ) to denote the set ∪ V∈ (V) \ . A graph is said to be -regular if for any vertex V, (V) = . For graph-theoretical terminology and notation not defined 2 Discrete Dynamics in Nature and Society here we follow [1]. For a faulty subset of edges in a connected graph , is a -restricted edge cut if − is disconnected and every component of − has at least vertices. If such an edge cut exists, then the -restricted edge connectivity of , denoted by ( ), is defined as the cardinality of a minimum -restricted edge cut. For any positive integer , let ( ) = min{|[ , ]| : | | = , [ ] is connected}. For many graphs, it has been shown that ( ) is an upper bound on ( ) [2][3][4]. A -connected graph is -optimal if ( ) = ( ).
The following is the definition on supergraphs used in our manuscript.
Definition (see [5]). A -connected graph is superrestricted edge-connected (or superfor short) if every minimum -restricted edge cut of isolates one connected subgraph of order .
In the majority of the literature, the 1-restricted edge connectivity of is called the edge connectivity of and is denoted by ( ). The 2-restricted edge connectivity of is called the restricted edge connectivity of and is denoted by ( ). Correspondingly, if is super 1-restricted edgeconnected, then is super edge-connected. If is super 2restricted edge-connected, then is super restricted edgeconnected. The sufficient conditions of superhave been studied by several authors; see [6,7].
The -ary -cube has many desirable properties, such as ease of implementation of algorithms and ability to reduce message latency by exploiting communication locality found in many parallel applications [8][9][10]. Therefore, a number of distributed-memory parallel systems (also known as multicomputers) have been built with a -ary -cube forming the underlying topology, such as the Cray T3D [11], the J-machine [12], the iWarp [13], and the IBM Blue Gene [14]. In 2011, Xiang and Stewart [15] proposed the augmented -arycube. In 2017, Wang et al. [16] proposed the expanded -ary -cube . The following is the definition.

∈ ( [ ]). en four outside neighbor vertices of are in four different
[ ] s.
By Proposition 5 and Theorems 7 and 8, we have the following corollary.

Theorem 10.
Let be the expanded -ary -cube with ≥ 3 and even ≥ 6. en is super edge-connected.
− is connected, a contradiction to that is an edge cut of . Therefore, there is a such that | | = 1.
is only a such that | | = 1. If is incident with each of 0 , then − has two components, one of which is an isolated vertex. If is not incident with each of 0 , then − is connected; a contradiction to that is an edge cut of .
is connected. By Proposition 6, − is connected, a contradiction to that is a edge cut of . By Cases 1-3, is super edge-connected.
[ ] − is connected. By Proposition 6 and 13 < 24, 2 − is connected, a contradiction to that is a cut of 2 .
). When = 2, the result holds by Proposition 14. We proceed by induction on . Our induction hypothesis is that −1 − has two components, one of which is an isolated vertex for | | ≤ 8 − 11 and [0] − 0 is connected or [0] − 0 has two components, one of which is an isolated vertex. Since −1 − 1 > 8 − 13, − is connected (a contradiction) or − has two components, one of which is an isolated vertex.

Case . . ( [0] − 0 is disconnected and
[ 0 ] − 0 is disconnected). By the induction hypothesis, [0] − 0 has two components, one of which is an isolated vertex and [ 0 ] − 0 has two components, one of which is an isolated vertex V. Suppose that is adjacent to V in − . Since |F −1 | ≤ 5, by Proposition 6, is connected; a contradiction to that is an edge cut of . Suppose that is not adjacent to V in − . Since |F −1 | ≤ 5, [0] − 0 has two components, one of which is an isolated vertex or is connected; a contradiction to that is an edge cut of .
Case ( [0] − 0 has two components, one of which is an isolated vertex. Since −1 − 1 > 4 − 5, − is connected (a contradiction) or − has two components, one of which is an isolated vertex.
[ 0 ] − 0 has two components, one of which is an isolated vertex. Therefore, − is connected or − has two components, one of which is an isolated vertex; a contradiction to that is a restricted edge cut of .
[0] − 0 has two components, one of which is an isolated vertex and [ 0 ]− 0 has two components, one of which is an isolated vertex V. Suppose that is adjacent to V in − . Since |F −1 | ≤ 6, by Proposition 6, is connected ( a contradiction) [0]− 0 has two components, one of which is an isolated vertex ( a contradiction) or − has two components, one of which is a 2 . Suppose that is not adjacent to V in − . Since |F −1 | ≤ 6, [0] − 0 has two components, one of which is an isolated vertex or is connected, a contradiction to that is a restricted edge cut of . [0] − 0 has two components, one of which is an isolated vertex. Since −1 −1 > 4 −5, − is connected or − has two components, one of which is an isolated vertex; a contradiction to that is a restricted edge cut of .
A super-graph to be -super-if − is still superfor any edge subset of with | | ≤ . The maximum integer of such , written as ( ), is said to be the edge fault tolerance of G with respect to the super-property.
By Theorems 17 and 18, we have the following theorem.
Remarks on eorem . Note that ( ) ≤ ( ) − 1. Since ( ) = 4 , ( ) is maximum. A spanning subgraph of a graph is a subgraph with ( ) = ( ). A graph is conditional faulty if each vertex of is incident with at least two healthy edges.
By Theorem 21 and Proposition 20, we have the following corollary.

Conclusions
In this paper, we investigate the problem of the super connectivity of the expanded -ary -cube . It is proved that is super edge-connected and super restricted edgeconnected ( ≥ 3). The work will help engineers to develop more different measures of the super connectivity based on application environment, network topology, network reliability, and statistics related to fault patterns.

Data Availability
The data cited in the manuscript titled "the Edge Connectivity of Expanded -Ary -Cubes" are all published articles. There are no other data.

Conflicts of Interest
The authors declare that they have no conflicts of interest.