Abstract
Toeplitz networks are used as interconnection networks due to their smaller diameter, symmetry, simpler routing, high connectivity, and reliability. The edge metric dimension of a network is recently introduced, and its applications can be seen in several areas including robot navigation, intelligent systems, network designing, and image processing. For a vertex and an edge of a connected graph , the minimum number from distances of with and is called the distance between and . If for every two distinct edges , there always exists , such that ; then, is named as an edge metric generator. The minimum number of vertices in is known as the edge metric dimension of . In this study, we consider four families of Toeplitz networks , , , and and studied their edge metric dimension. We prove that for all , , for , , and for , . We further prove that for all , , and hence, it is bounded.
1. Introduction and Preliminaries
Computer networking provides a technique of communication between a number of processors connected in a network. An interconnection network is a structure of links that joins one or more computers to each other for communication purposes. In the framework of computer networking, an interconnection network is used mainly to attach processors to processors or to permit several processors to access one or more common memory disks. Often, they are used to attach processors with nearby attached memories to each other. The approach in which these processors/memories are attached to each other have a major effect on the cost, applicability, consistency, scalability, and performance of a computer networking. It is always desirable for an interconnection network to have a smaller diameter, alternate links among the processors, a higher level of symmetry, and simpler routing. Toeplitz networks are the finest example of such an interconnection network [1].
Let be a connected and undirected graph. Let represents the degree of a vertex which is the total number of vertices adjacent to . Also, the minimum degree of graph is , and the maximum degree of is represented by . The number of edges on a shortest path from vertex to vertex is called the distance between them, which is denoted by . Let be an arbitrary edge of graph and belongs to ; then, the distance between them is represented and defined by .
Metric dimension introduced by Slater introduced the metric dimension in [2], and he used it to address the challenge of locating an intruder in a network. Slater worked on the application of robot navigation and coast guard loran in [2, 3]. Melter and Harary introduced the term resolving set by expanding Slater’s concept in [4]. Melter and Tomescu studied the metric dimension’s role in pattern recognition and image processing issues [5]. Sebo and Tannier studied the metric dimension in combinatorial optimization in [6]. Caceres et al. worked on the mastermind and coin weighing games through metric dimension in [7]. Chartrand et al. computed the resolvability of graphs in [8]. Khuller et al. studied the application of metric dimension in navigation systems [9]. Salman et al. calculated the metric dimension of circulant graphs in [10]. A vertex distinguishes two vertices and if . We assume is a metric generator of graph , if each pair of elements of can be distinguished by some vertex of . The metric dimension of graph is the smallest cardinality of the metric generator of .
Kelenc et al. in [11] introduced the idea of edge metric dimension as follows. A vertex distinguishes any two edges and , if . We assume is an edge metric generator of graph , if each pair of elements of can be distinguished by some vertex of . The edge metric dimension of graph is the smallest cardinality of the edge metric generator of graph . The smallest edge metric generator is called the edge basis (edge metric basis). Furthermore, Kelenc et al. in [11] compared the metric dimension with the edge metric dimension and also discussed some useful results for paths , cycles , complete graphs , and wheel graphs. Zubrilina computed the edge metric dimension of a graph with relation to the total number of vertices of graph in [12]. Filipovi et al. computed for and found the lower bound for all other values of in [13]. Mufti et al. calculated in [14]. Ahsan et al. computed for in [15]. Fang et al. discussed the application of networks in electrical engineering in [16]. Chen et al. studied the application in chemical graphs in [17]. Yang et al. calculated the edge dimension of some families of wheel-related graphs in [18]. Wei et al. studied the edge dimension of some complex convex polytopes in [19]. Deng et al. computed the edge dimension of triangular, square, and hexagonal Mobius ladder networks in [20]. Ahmad et al. calculated the edge dimension of the benzenoid tripod structure in [21]. Moreover, Ahsan et al. calculated the edge dimension of convex polytopes in [22]. Xing et al. computed the vertex edge resolvability of the wheel graphs in [23]. Some useful lemmas are given.
Lemma 1 (See [11]). For any , , , and . Moreover, is is path.
Lemma 2 (See [11]). For a simple, connected graph ,(i).(ii).The rest of the study is organized as follows. The exact edge metric dimension of the families of Toeplitz networks , , and are computed in Sections 2, 3, and 4, respectively. In Section 5, we will calculate the upper bound of the family of Toeplitz networks . Last, the conclusion of the article is given.
2. Edge Metric Dimension of Toeplitz Networks
In this section, we will find . It has and .
The Toeplitz network for is shown in Figure 1. The metric dimension of is given.
Theorem 1 (See [24, 25]). If be a graph of the Toeplitz network with , then .
In the next theorem, we will find .
Theorem 2. Let be the Toeplitz network. Then, , where .
Proof. We have the following cases in order to determine . Case (i): let , , and ; we will prove that is an edge basis of . Now, representations of each edge of are given by Since representations of every two edges are different, it shows that . Now, we will prove that the edge metric generator of cardinality three does not exist. Suppose contrarily that and let . Now, Table 1 provides conditions on , , and all edges for which . As a result, there is no generator with three vertices, showing that for , . Case (ii): let , , and ; we will prove that is an edge basis of . Now, representations of each edge of are given bySince representations of every two edges are different, it shows that .
Now, we will prove that the edge metric generator of cardinality three does not exist. Suppose contrarily that and let . Now, Table 2 provides conditions on , , and all edges for which .
Hence, there is no generator having three vertices which prove that for , .
3. Edge Metric Dimension of Toeplitz Networks
Now, we will find . It has and . Figure 2 shows the Toeplitz network for . The metric dimension of is given.
Theorem 3 (See [24]). If be a graph of the Toeplitz network with , then .
In the next theorem, we will find .
Theorem 4. Let be the Toeplitz network. Then, , where .
Proof. We have the following cases in order to compute . Case (i): let , , and ; we will prove that is an edge basis of . Now, representations of each edge of are given by Since representations of every two edges are different, it shows that . Now, we will prove that the edge metric generator of cardinality two does not exist. Suppose contrarily that and let . Now, Table 3 provides conditions on , , and all edges for which . Hence, there is no generator having two vertices which prove that for , . Case (ii): let , , and ; we will prove that is an edge basis of . Now, representations of each edge of are given by Since representations of every two edges are different, it shows that . Now, we will prove that the edge metric generator of cardinality two does not exist. Suppose contrarily that and let . Now, Table 4 provides conditions on , , and all edges for which . Hence, there is no generator having two vertices which prove that for , . Case (iii): let , and ; we will prove that is an edge basis of . Now, representations of each edge of are given bySince representations of every two edges are different, it shows that .
Now, we will prove that the edge metric generator of cardinality two does not exist. Suppose contrarily that and let . Now, Table 5 provides conditions on , , and all edges for which .
Hence, there is no generator having two vertices which prove that for , .
4. Edge Metric Dimension of Toeplitz Networks
Now, we will determine . It has and . The Toeplitz network for is shown in Figure 3. has the following metric dimension.
Theorem 5 (See [24]). If be a graph of the Toeplitz network with , then .
In next theorem, we will find .
Theorem 6. Let be the Toeplitz network. Then , where .
Proof. We have the following cases in order to determine . Case (i): let , , and , we will prove that is an edge basis of . Now, representations of each edge of are given by Since representations of every two edges are different, it shows that . Now, we will prove that the edge metric generator of cardinality two does not exist. Suppose contrarily that and let . Now, Table 6 provides conditions on , and all edges for which . Hence, there is no generator having two vertices which prove that for , . Case (ii): let , , and ; we will prove that is an edge basis of . Now, representations of each edge of are given by Since representations of every two edges are different, it shows that . Now, we will prove that the edge metric generator of cardinality two does not exist. Suppose contrarily that and let . Now, Table 7 provides conditions on , , and all edges for which . Hence, there is no generator having two vertices which prove that for , . Case (iii): let , , and ; we will prove that is an edge basis of . Now, representations of each edge of are given by Since representations of every two edges are different, it shows that . Now, we will prove that the edge metric generator of cardinality two does not exist. Suppose contrarily that and let . Now, Table 8 provides conditions on , , and all edges for which . Hence, there is no generator having two vertices which prove that for , . Case (iv): let , , and ; we will prove that is an edge basis of . Now, representations of each edge of are given bySince representations of every two edges are different, it shows that .
Now, we will prove that the edge metric generator of cardinality two does not exist. Suppose contrarily that and let . Now, Table 9 provides conditions on , , and all edges for which .
Hence, there is no generator having two vertices which prove that for , .
5. Upper Bounds for the Edge Metric Dimension of Toeplitz Networks
We shall compute the upper bound of in this section. It has and . Figure 4 shows the Toeplitz network . The metric dimension of is stated.
Theorem 7 (See [24]). If be a graph of the Toeplitz network with , then .
In the next result, we will find the upper bound of .
Theorem 8. Let be the Toeplitz networks. Then, , where .
Proof. We have the following cases to calculate the upper bound of : Case (i): let , , and ; we will prove that is an edge basis of . Now, representations of each edge of are given by Since representations of every two edges are different, it shows that . Case (ii): let , , and ; we will prove that is an edge metric generator of . Now, representations of each edge of are given by Since representations of every two edges are different, it shows that . Case (iii): let , , and ; we will prove that is an edge metric generator of . Now, representations of each edge of are given bySince representations of every two edges are different, it shows that .
6. Conclusion
In this article, we have calculated the exact value of edge metric dimension of Toeplitz networks , , and and the upper bound of the Toeplitz network . We conclude that the edge metric dimension of these Toeplitz networks is constant and does not depend on the number of vertices of the graph. Here, we end with the following open problem.
Open problem 1. Calculate for .
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
Zohaib Zahid, Muhammad Ahsan, and Imran Siddique conceptualized the study. Dalal Alrowaili, Sohail Zafar, and Imran Siddique developed methodology. Muhammad Ahsan, Imran Siddique, and Dalal Alrowaili wrote the original draft. Sohail Zafar and Zohaib Zahid wrote and reviewed the study. Zohaib Zahid supervised the study. All authors have read and agreed to the published version of the manuscript.
Acknowledgments
This work was funded by the Deanship of Scientific Research at Jouf University (DSR-2021-03-03 115).