#### Abstract

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 , two vertices and of are said to be connected if there is a -path in . 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. For a connected network , 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. As a topology structure of interconnection networks, the expanded -ary -cube has many good properties. In this paper, we prove that (1) is super edge-connected (); (2) the restricted edge connectivity of is (); (3) is super restricted edge-connected ().

#### 1. 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 of are said to be connected if there is a -path in . Connection is an equivalence relation on the vertex set . Thus there is a partition of into nonempty subsets such that two vertices and of are connected if and only if both and belong to the same set . The subgraphs 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 of a vertex is the number of edges incident with . We denote by the minimum degrees of vertices of . For any vertex , we define the neighborhood of in to be the set of vertices adjacent to . If , then is called a neighbor or a neighbor vertex of . Let . We use to denote the set . A graph is said to be -regular if for any vertex , . For graph-theoretical terminology and notation not defined here we follow . 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 . For many graphs, it has been shown that is an upper bound on . A -connected graph is -optimal if . The following is the definition on super- graphs used in our manuscript.

Definition 1 (see ). A -connected graph is super -restricted edge-connected (or super- for 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 -restricted edge-connected, then is super edge-connected. If is super -restricted edge-connected, then is super restricted edge-connected. The sufficient conditions of super- have 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 . 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 , the J-machine , the iWarp , and the IBM Blue Gene . In 2011, Xiang and Stewart  proposed the augmented -ary -cube. In 2017, Wang et al.  proposed the expanded -ary -cube . The following is the definition.

Definition 2 (see ). The expanded -ary -cube, denoted by ( and even ), is a graph consisting of vertices . Two vertices and are adjacent if and only if there exists an integer such that, for some , we have (mod ) and for all .

For clarity of presentation, we omit writing “(mod )” in similar expressions for the remainder of the paper. In this paper, we prove that (1) is super edge-connected (); (2) the restricted edge connectivity of is (); (3) is super restricted edge-connected ().

#### 2. The Connectivity of Expanded -Ary -Cubes

We can partition into disjoint subgraphs (abbreviated as , if there is no ambiguity), where every vertex has a fixed integer in the last position for . Let . Then is said to be outside neighbor vertices of .

Proposition 3 (see ). Each is isomorphic to for .

Proposition 4. The expanded -ary -cube is the Cartesian product of expanded -ary -cubes, i.e., .

Proof. We prove by the induction on . Now define a bijection from to given by Note . Let with and . Then (1), if , (mod ) for ; (2), if , (mod ) for . Suppose that , . Then . By the definition of , . Suppose that , . Similarly, .
Conversely, let with and . Then , or , . Suppose that and . By the definition of , (mod ) for . By the definition of , . Suppose that and . Similarly, . This implies that the result is true for . Assume that the result holds for , i.e., for . Then . We will prove that . Now define a bijection from to given by Note . Let with , , , and ,
By Proposition 3, each is isomorphic to ( in the last position ). Suppose that . Then . By the definition of , . Suppose that . Then, by the definition of , and for and . Therefore, and for . By the definition of and .
Conversely, let . Suppose that and . By the definition of , there exists an integer such that (mod ) and , for and . Note that . By the definition of , . Suppose that and . By the definition of , . Note that . By the definition of , .

The automorphism group of a graph is transitive if there exists an automorphism to any pair of vertices in such that . In this case, is called vertex transitive.

Proposition 5 (see ). is -regular, vertex transitive.

Proposition 6 (see ). Let . Then four outside neighbor vertices of are in four different s.

Theorem 7 (see ). Let be the expanded -ary -cube with and even . Then the connectivity .

Theorem 8 (see ). Let be a connected graph. Then .

By Proposition 5 and Theorems 7 and 8, we have the following corollary.

Corollary 9. .

Theorem 10. Let be the expanded -ary -cube with and even . Then is super edge-connected.

Proof. By Corollary 9, . Let with be any minimum edge cut of . We can partition into disjoint subgraphs . By Proposition 3, each is isomorphic to for . Let for with . Let . Then . Since , holds, we consider the following cases.
Case 1 (). Since , . By Corollary 9, is connected. Since and there is a complete matching between and for , is connected; a contradiction to that is an edge cut of .
Case 2 (). Since , holds hence there is only one such that . Therefore, for . By Corollary 9, is connected for . Since and there is a complete matching between and for , is connected.
Case 2.1 ( is connected). Since there is a complete matching between and for , is connected; a contradiction to that is an edge cut of .
Case 2.2 ( is not connected). Since and , holds. Let . By Proposition 6, is connected; a contradiction to that is an edge cut of . Let . Let be the components of . Suppose for . By Proposition 6, . Since , is connected, a contradiction to that is an edge cut of . Therefore, there is a such that . Let . Since , there is only a such that . If is incident with each of , then has two components, one of which is an isolated vertex. If is not incident with each of , then is connected; a contradiction to that is an edge cut of .
Case 3 (). In this case, and is connected. By Proposition 6, is connected, a contradiction to that is a edge cut of .
By Cases 1–3, is super edge-connected.

Lemma 11. Let be the expanded -ary -cube with and even . Then .

Proof. Let and . Then is adjacent to and . Let . Then and for . Therefore, is connected. Let . Then . Therefore, is connected and hence has two components and . By the definition of , .

Proposition 12. Let be the expanded -ary -cube with even , and let with . If is disconnected, then has two components, one of which is an isolated vertex.

Proof. If , by Corollary 9, then is connected, a contradiction. Let . By Proposition 5, is -regular. Therefore, there is one such that has two components, one of which is an isolated vertex. Let . Suppose that has two or more components which have order at least 2. Choose a component of with so that is as small as possible. Then . In view of Proposition 5, we may assume that and , where and . Then and . If , then the edge set (in the case where , we take ), and hence , a contradiction to that . Thus . Suppose that . Since , we have or . We may assume . Then , which implies , and hence , a contradiction to that . Thus . Since , it follows that and . Consequently and , which implies . Therefore , a contradiction to that and hence has two components, one of which is an isolated vertex.

Proposition 13. Let be the expanded -ary -cube with even , and let be a minimum 2-restricted edge cut of . Then .

Proof. By Proposition 12, . Let . Then is a 2-restricted edge cut of and . Therefore, .

Proposition 14. Let be the expanded -ary -cube with even , and let with . If is disconnected, then has two components, one of which is an isolated vertex.

Proof. We can partition into disjoint subgraphs (abbreviated as , if there is no ambiguity), where every vertex has a fixed integer in the last position for . By Proposition 3, each is isomorphic to for . By Corollary 9, . Let for with . Let . Then . We consider the following cases.
Case 1 (). Since , . By Corollary 9, is connected. By Proposition 6 and , is connected, a contradiction to that is a cut of .
Case 2 (). Since , . Since , there are at most three ’s such that for .
Case 2.1 ( and for ). By Corollary 9, is connected for . By Proposition 6 and , is connected. By Theorem 10, is connected or has two components, one of which is an isolated vertex. Since , is connected (a contradiction) or has two components, one of which is an isolated vertex.
Case 2.2 (, and for ). In this case, . Similarly to Case 2.1, is connected (a contradiction) or has two components, one of which is an isolated vertex.
Case 2.3 (, , and for ). In this case, . Similarly to Case 2.1, is connected, a contradiction.
Case 3 (). Since , . Since , there are at most two ’s such that for .
Case 3.1 ( and for ). By Corollary 9, is connected for . By Proposition 6 and , is connected. By Proposition 12, is connected or has two components, one of which is an isolated vertex. Since , is connected (a contradiction) or has two components, one of which is an isolated vertex.
Case 3.2 (, and for ). In this case, . Similarly to Case 3.1, is connected (a contradiction) or has two components, one of which is an isolated vertex.
Case 3.3 (, and for ). In this case, . By Proposition 12, is connected or has two components, one of which is an isolated vertex. Therefore, is connected, a contradiction.
Case 4 (). In this case, and . Therefore, is connected (a contradiction) or has two components, one of which is an isolated vertex.

Proposition 15. Let be the expanded -ary -cube with even , and let with . If is disconnected, then has two components, one of which is an isolated vertex.

Proof. We can partition into disjoint subgraphs (abbreviated as , if there is no ambiguity), where every vertex has a fixed integer in the last position for . By Proposition 3, each is isomorphic to for . Let with and let is disconnected. Let for with . Let . When , the result holds by Proposition 14. We proceed by induction on . Our induction hypothesis is that has two components, one of which is an isolated vertex for and . By Proposition 3, each is isomorphic to for . We consider the following cases.
Case 1 (). Since , for . By Corollary 9, is connected for . Since and there is a complete matching between and , for , is connected; a contradiction to that is a cut of .
Case 2 (). In this case, .
Case 2.1 ( for ). By Corollary 9, is connected for . Since , is connected. By Corollary 9, is connected or has two components, one of which is an isolated vertex. Since , is connected (a contradiction) or has two components, one of which is an isolated vertex.
Case 2.2 ( for ). In this case, and for . Since , . By Corollary 9, is connected for . Therefore, is connected.
Case 2.2.1 ( is connected and