# Edge Metric Dimension of Some Classes of Toeplitz Networks

Authors: Dalal Alrowaili; Zohaib Zahid; Muhammad Ahsan; Sohail Zafar; Imran Siddique
Journal: Journal of Mathematics
 (2021)

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## 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 vertexs and an edge g=s1s2 of a connected graph G, the minimum number from distances of s with s1 and s2 is called the distance between s and g. If for every two distinct edges s1,s2∈EG, there always exists w1ɛWE⊆VG, such that ds1,w1≠ds2,w1; then, WE is named as an edge metric generator. The minimum number of vertices in WE is known as the edge metric dimension of G. In this study, we consider four families of Toeplitz networks Tn1,2, Tn1,3, Tn1,4, and Tn1,2,3 and studied their edge metric dimension. We prove that for all n≥4, edimTn1,2=4, for n≥5, edimTn1,3=3, and for n≥6, edimTn1,4=3. We further prove that for all n≥5, edimTn1,2,3≤6, and hence, it is bounded.


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## Body

## 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].LetG=VG,EG be a connected and undirected graph. Let dv represents the degree of a vertex v which is the total number of vertices adjacent to v∈VG. Also, the minimum degree of graph G is δ=δG, and the maximum degree of G is represented by Δ=ΔG. The number of edges on a shortest path from vertex x1 to vertex x2 is called the distance between them, which is denoted by dx1,x2. Let e1=y1y2 be an arbitrary edge of graph G and x1 belongs to VG; then, the distance between them is represented and defined by dx1,e1=mindx1,y1,dx1,y2.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 x1 distinguishes two vertices v1 and v2 if dx1,v1≠dx1,v2. We assume W⊆VG is a metric generator of graph G, if each pair of elements of VG can be distinguished by some vertex of W. The metric dimension dimG of graph G is the smallest cardinality of the metric generator of G.Kelenc et al. in [11] introduced the idea of edge metric dimension as follows. A vertex x1 distinguishes any two edges f1 and f2, if df1,x1≠df2,x1. We assume WE⊆VG is an edge metric generator of graph G, if each pair of elements of EG can be distinguished by some vertex of WE. The edge metric dimension edimG of graph G is the smallest cardinality of the edge metric generator of graph G. 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 Pn, cycles Cn, complete graphs Kn, and wheel graphs. Zubrilina computed the edge metric dimension of a graph with relation to the total number of vertices of graph G in [12]. Filipovi et al. computed edimGPn,k for k=1,2 and found the lower bound for all other values of k in [13]. Mufti et al. calculated edimBSCayZn⊕Z2 in [14]. Ahsan et al. computed edimCn1,k for k=2,3 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 anyn≥2, edimPn=dimPn=1, edimCn=dimCn=2, and edimKn=dimKn=n−1. Moreover, edimG is 1⇔G is path.Lemma 2 (See [11]).
For a simple, connected graphG,(i)
edimG≥log2ΔG.(ii)
edimG≥1+⌈log2δG⌉.
The rest of the study is organized as follows. The exact edge metric dimension of the families of Toeplitz networksTn1,2, Tn1,3, and Tn1,4 are computed in Sections 2, 3, and 4, respectively. In Section 5, we will calculate the upper bound of the family of Toeplitz networks Tn1,2,3. Last, the conclusion of the article is given.


## 2. Edge Metric Dimension of Toeplitz NetworksTn1,2

In this section, we will findedimTn1,2. It has VTn1,2=v1,v2,v3,…,vn and ETn1,2=vξvξ+1:1≤ξ≤n−1∪vξvξ+2:1≤ξ≤n−2.The Toeplitz network forn=10 is shown in Figure 1. The metric dimension of Tn1,2 is given.Figure 1
Toeplitz networkT101,2.Theorem 1 (See [24, 25]).
IfTn1,2 be a graph of the Toeplitz network with n≥4, then dimTn1,2=2.
In the next theorem, we will findedimTn1,2.Theorem 2.
LetTn1,2 be the Toeplitz network. Then, edimTn1,2=4, where n≥4.Proof.
We have the following cases in order to determineedimTn1,2.Case (i): letn=2ρ, ρ≥2, and WE=v1,v2,vn−1,vn⊂VTn1,2; we will prove that WE is an edge basis of Tn1,2. Now, representations of each edge of Tn1,2 are given by(1)rv2ξ−1v2ξ+1|WE=0,1,ρ−2,ρ−1,ifξ=1,ξ−1,ξ−1,ρ−ξ−1,ρ−ξ,if2≤ξ≤ρ−1,rv2ξv2ξ+2|WE=ξ,ξ−1,ρ−ξ−1,ρ−ξ−1,if1≤ξ≤ρ−2,ρ−1,ρ−2,1,0,ifξ=ρ−1,rv2ξ−1v2ξ|WE=ξ−1,ξ−1,ρ−ξ,ρ−ξ,for1≤ξ≤ρ,rv2ξv2ξ+1|WE=ξ,ξ−1,ρ−ξ−1,ρ−ξ,for1≤ξ≤ρ−1.Since representations of every two edges are different, it shows thatedimTn1,2≤4.Now, we will prove that the edge metric generator of cardinality three does not exist. Suppose contrarily thatedimTn1,2=3 and let WE=v1,vα,vβ. Now, Table 1 provides conditions on α, β, and all edges e1,e2 for which re1|WE=re2|WE.As a result, there is no generator with three vertices, showing thatedimTn1,2=4 for n=2ρ, ρ≥2.Case (ii): letn=2ρ+1, ρ≥2, and WE=v1,v2,vn−1,vn⊂VTn1,2; we will prove that WE is an edge basis of Tn1,2. Now, representations of each edge of Tn1,2 are given by(2)rv2ξ−1v2ξ+1|WE=0,1,ρ−1,ρ−1,ifξ=1,ξ−1,ξ−1,ρ−ξ,ρ−ξ,if2≤ξ≤ρ−1,ρ−1,ρ−1,1,0,ifξ=ρ,rv2ξ−1v2ξ|WE=ξ−1,ξ−1,ρ−ξ,ρ−ξ+1,for1≤ξ≤ρ,rv2ξv2ξ+1|WE=ξ,ξ−1,ρ−ξ,ρ−ξ,for1≤ξ≤ρ,rv2ξv2ξ+2|WE=ξ,ξ−1,ρ−ξ−1,ρ−ξ,for1≤ξ≤ρ−1.
Since representations of every two edges are different, it shows thatedimTn1,2≤4.
Now, we will prove that the edge metric generator of cardinality three does not exist. Suppose contrarily thatedimTn1,2=3 and let WE=v1,vα,vβ. Now, Table 2 provides conditions on α, β, and all edges e1,e2 for which re1|WE=re2|WE.
Hence, there is no generator having three vertices which prove thatedimTn1,2=4 for n=2ρ+1, ρ≥2.Table 1
e1,e2 for which re1|WE=re2|WE.
Conditions onαandβe1,e22≤α≤n−3,3≤β≤n−2vn−2vn−1,vn−2vnα=n−2,β=n−1vn−3vn−2,vn−2vn−4α=n−1,β=nv3v4,v2v4Table 2
e1,e2 for which re1|WE=re2|WE.
Conditions onαandβe1,e22≤α≤n−3,3≤β≤n−2vn−2vn−1,vn−2vnα=n−2,β=n−1vn−3vn−4,vn−3vn−5α=n−1,β=nv3v4,v2v4


## 3. Edge Metric Dimension of Toeplitz NetworksTn1,3

Now, we will findedimTn1,3. It has VTn1,3=v1,v2,…,vn and ETn1,3=vξvξ+1:1≤ξ≤n−1∪vξvξ+3:1≤ξ≤n−3. Figure 2 shows the Toeplitz network for T181,3. The metric dimension of Tn1,3 is given.Figure 2
Toeplitz networkT181,3.Theorem 3 (See [24]).
IfTn1,3 be a graph of the Toeplitz network with n≥5, then dimTn1,3=3.
In the next theorem, we will findedimTn1,3.Theorem 4.
LetTn1,3 be the Toeplitz network. Then, edimTn1,3=3, where n≥5.Proof.
We have the following cases in order to computeedimTn1,3.Case (i): letn=3ρ, ρ≥2, and WE=v1,v2,vn−1⊂VTn1,3; we will prove that WE is an edge basis of Tn1,3. Now, representations of each edge of Tn1,3 are given by(3)rv3ξ+1v3ξ+2|WE=ξ,ξ,ρ−ξ−1,for0≤ξ≤ρ−1,rv3ξ+2v3ξ+3|WE=ξ+1,ξ,ρ−ξ−1,for0≤ξ≤ρ−1,rv3ξ+3v3ξ+4|WE=ξ+1,ξ+1,ρ−ξ−1,for0≤ξ≤ρ−2,rv3ξ+1v3ξ+4|WE=ξ,ξ+1,ρ−ξ−1,for0≤ξ≤ρ−2,rv3ξ+2v3ξ+5|WE=ξ+1,ξ,ρ−ξ−2,for0≤ξ≤ρ−2rv3ξ+3v3ξ+6|WE=ξ+2,ξ+1,ρ−ξ−1,for0≤ξ≤ρ−2.Since representations of every two edges are different, it shows thatedimTn1,3≤3.Now, we will prove that the edge metric generator of cardinality two does not exist. Suppose contrarily thatedimTn1,3=2 and let WE=v1,vβ. Now, Table 3 provides conditions on α, β, and all edges e1,e2 for which re1|WE=re2|WE.Hence, there is no generator having two vertices which prove thatedimTn1,3=3 for n=3ρ, ρ≥2.Case (ii): letn=3ρ+1, ρ≥2, and WE=v1,v2,vn−2⊂VTn1,3; we will prove that WE is an edge basis of Tn1,3. Now, representations of each edge of Tn1,3 are given by(4)rv3ξ+1v3ξ+2|WE=ξ,ξ,ρ−ξ−1,for0≤ξ≤ρ−1,rv3ξ+2v3ξ+3|WE=ξ+1,ξ,ρ−ξ−1,for0≤ξ≤ρ−1,rv3ξ+3v3ξ+4|WE=ξ+1,ξ+1,ρ−ξ−1,if0≤ξ≤ρ−2,ρ,ρ,1,ifξ=ρ−1,rv3ξ+1v3ξ+4|WE=ξ,ξ+1,ρ−ξ−1,if0≤ξ≤ρ−2,ρ−1,ρ,1,ifξ=ρ−1,rv3ξ+2v3ξ+5|WE=ξ+1,ξ,ρ−ξ−2,for0≤ξ≤ρ−2,rv3ξ+3v3ξ+6|WE=ξ+2,ξ+1,ρ−ξ−1,for0≤ξ≤ρ−2.Since representations of every two edges are different, it shows thatedimTn1,3≤3.Now, we will prove that the edge metric generator of cardinality two does not exist. Suppose contrarily thatedimTn1,3=2 and let WE=v1,vβ. Now, Table 4 provides conditions on α, β, and all edges e1,e2 for which re1|WE=re2|WE.Hence, there is no generator having two vertices which prove thatedimTn1,3=3 for n=3ρ+1, ρ≥2.Case (iii): letn=3ρ+2, ρ≥1 and WE=v1,v2,vn⊂VTn1,3; we will prove that WE is an edge basis of Tn1,3. Now, representations of each edge of Tn1,3 are given by(5)rv3ξ+1v3ξ+2|WE=ξ,ξ,ρ−ξ,for0≤ξ≤ρ,rv3ξ+2v3ξ+3|WE=ξ+1,ξ,ρ−ξ,for0≤ξ≤ρ−1,rv3ξ+3v3ξ+4|WE=ξ+1,ξ+1,ρ−ξ,for0≤ξ≤ρ−1,rv3ξ+1v3ξ+4|WE=ξ,ξ+1,ρ−ξ,for0≤ξ≤ρ−1,rv3ξ+2v3ξ+5|WE=ξ+1,ξ,ρ−ξ−1,for0≤ξ≤ρ−1,rv3ξ+3v3ξ+6|WE=ξ+2,ξ+1,ρ−ξ,for0≤ξ≤ρ−2.
Since representations of every two edges are different, it shows thatedimTn1,3≤3.
Now, we will prove that the edge metric generator of cardinality two does not exist. Suppose contrarily thatedimTn1,3=2 and let WE=v1,vβ. Now, Table 5 provides conditions on α, β, and all edges e1,e2 for which re1|WE=re2|WE.
Hence, there is no generator having two vertices which prove thatedimTn1,3=3 for n=3ρ+2, ρ≥1.Table 3
e1,e2 for which re1|WE=re2|WE.
Conditions onβe1,e2β=3α,1≤α≤ρv2v3,v3v4β=3α+1,1≤α≤ρ−1v3v4,v4v5β=3α+2,1≤α≤ρ−1v2v3,v3v4β=2v3v4,v4v5Table 4
e1,e2 for which re1|WE=re2|WE.
Conditions onβe1,e2β=3α,1≤α≤ρv2v3,v3v4β=3α+1,1≤α≤ρv3v4,v4v5β=3α+2,1≤α≤ρ−1v2v3,v3v4β=2v3v4,v4v5Table 5
e1,e2 for which re1|WE=re2|WE.
Conditions onβe1,e2β=3α,1≤α≤ρv2v3,v3v4β=3α+1,1≤α≤ρv3v4,v4v5β=3α+2,1≤α≤ρ−1v2v3,v3v4β=2v3v4,v4v5


## 4. Edge Metric Dimension of Toeplitz NetworksTn1,4

Now, we will determineedimTn1,4. It has VTn1,4=v1,v2,v3,…,vn and ETn1,4=vξvξ+1:1≤ξ≤n−1∪vξvξ+4:1≤ξ≤n−4. The Toeplitz network for n=20 is shown in Figure 3. Tn1,4 has the following metric dimension.Figure 3
Toeplitz networka+c.Theorem 5 (See [24]).
IfTn1,4 be a graph of the Toeplitz network with n≥6, then dimTn1,4=2.
In next theorem, we will findedimTn1,4.Theorem 6.
LetTn1,4 be the Toeplitz network. Then edimTn1,4=3, where n≥6.Proof.
We have the following cases in order to determineedimTn1,4.Case (i): letn=4ρ, ρ≥2, and WE=v1,v2,vn−2⊂VTn1,4, we will prove that WE is an edge basis of Tn1,4. Now, representations of each edge of Tn1,4 are given by(6)rv4ξ+4v4ξ+8|WE=ξ+2,ξ+2,ρ−ξ−1,if0≤ξ≤ρ−3,ρ,ρ,2,ifξ=ρ−2,rv4ξ+1v4ξ+2|WE=ξ,ξ,ρ−ξ−1,for0≤ξ≤ρ−1,rv4ξ+2v4ξ+3|WE=ξ+1,ξ,ρ−ξ−1,for0≤ξ≤ρ−1,rv4ξ+3v4ξ+4|WE=ξ+2,ξ+1,ρ−ξ,for0≤ξ≤ρ−1,rv4ξ+4v4ξ+5|WE=ξ+1,ξ+2,ρ−ξ−1,for0≤ξ≤ρ−2,rv4ξ+1v4ξ+5|WE=ξ,ξ+1,ρ−ξ−1,for0≤ξ≤ρ−2,rv4ξ+2v4ξ+6|WE=ξ+1,ξ,ρ−ξ−2,for0≤ξ≤ρ−2,rv4ξ+3v4ξ+7|WE=ξ+2,ξ+1,ρ−ξ−1,for0≤ξ≤ρ−2.Since representations of every two edges are different, it shows thatedimTn1,4≤3.Now, we will prove that the edge metric generator of cardinality two does not exist. Suppose contrarily thatedimTn1,4=2 and let WE=v1,vβ. Now, Table 6 provides conditions on α, β and all edges e1,e2 for which re1|WE=re2|WE.Hence, there is no generator having two vertices which prove thatedimTn1,4=3 for n=4ρ, ρ≥2.Case (ii): letn=4ρ+1, ρ≥2, and WE=v1,v4,vn⊂VTn1,4; we will prove that WE is an edge basis of Tn1,4. Now, representations of each edge of Tn1,4 are given by(7)rv4ξ+1v4ξ+2|WE=0,2,ρ,ifξ=0,ξ,ξ,ρ−ξρ,if1≤ξ≤ρ−1,rv4ξ+2v4ξ+3|WE=ξ+1,ξ+1,ρ−ξ+1,for0≤ξ≤ρ−1,rv4ξ+3v4ξ+4|WE=ξ+2,ξ,ρ−ξ,for0≤ξ≤ρ−1,rv4ξ+4v4ξ+5|WE=ξ+1,ξ,ρ−ξ−1,for0≤ξ≤ρ−1,rv4ξ+1v4ξ+5|WE=0,1,ρ−1,ifξ=0,ξ,ξ,ρ−ξ−1,if1≤ξ≤ρ−1,rv4ξ+2v4ξ+6|WE=1,2,ρ,ifξ=0,ξ+1,ξ+1,ρ−ξ,if1≤ξ≤ρ−2,rv4ξ+3v4ξ+7|WE=ξ+2,ξ+1,ρ−ξ,for0≤ξ≤ρ−2,rv4ξ+4v4ξ+8|WE=ξ+2,ξ,ρ−ξ−1,for0≤ξ≤ξ−2.Since representations of every two edges are different, it shows thatedimTn1,4≤3.Now, we will prove that the edge metric generator of cardinality two does not exist. Suppose contrarily thatedimTn1,4=2 and let WE=v1,vβ. Now, Table 7 provides conditions on α, β, and all edges e1,e2 for which re1|WE=re2|WE.Hence, there is no generator having two vertices which prove thatedimTn1,4=3 for n=4ρ+1, ρ≥2.Case (iii): letn=4ρ+2, ρ≥1, and WE=v1,v2,vn⊂VTn1,4; we will prove that WE is an edge basis of Tn1,4. Now, representations of each edge of Tn1,4 are given by(8)rv4ξ+1v4ξ+2|WE=ξ,ξ,ρ−ξ,for0≤ξ≤ρ,rv4ξ+2v4ξ+3|WE=ξ+1,ξ,ρ−ξ,for0≤ξ≤ρ−1,rv4ξ+3v4ξ+4|WE=ξ+2,ξ+1,ρ−ξ+1,for0≤ξ≤ρ−1,rv4ξ+4v4ξ+5|WE=ξ+1,ξ+2,ρ−ξ,for0≤ξ≤ρ−1,rv4ξ+1v4ξ+5|WE=ξ,ξ+1,ρ−ξ,for0≤ξ≤ρ−1,rv4ξ+2v4ξ+6|WE=ξ+1,ξ,ρ−ξ−1,for0≤ξ≤ρ−1,rv4ξ+3v4ξ+7|WE=ξ+2,ξ+1,ρ−ξ,for0≤ξ≤ρ−2,rv4ξ+4v4ξ+8|WE=ξ+2,ξ+2,ρ−ξ,for0≤ξ≤ρ−2.Since representations of every two edges are different, it shows thatedimTn1,4≤3.Now, we will prove that the edge metric generator of cardinality two does not exist. Suppose contrarily thatedimTn1,4=2 and let WE=v1,vβ. Now, Table 8 provides conditions on α, β, and all edges e1,e2 for which re1|WE=re2|WE.Hence, there is no generator having two vertices which prove thatedimTn1,4=3 for n=4ρ+2, ρ≥1.Case (iv): letn=4ρ+3, ρ≥1, and WE=v1,v2,vn−1⊂VTn1,4; we will prove that WE is an edge basis of Tn1,4. Now, representations of each edge of Tn1,4 are given by(9)rv4ξ+1v4ξ+2|WE=ξ,ξ,ρ−ξ,for0≤ξ≤ρ,rv4ξ+2v4ξ+3|WE=ξ+1,ξ,ρ−ξ,for0≤ξ≤ρ,rv4ξ+3v4ξ+4|WE=ξ+2,ξ+1,ρ−ξ+1,for0≤ξ≤ρ−1,rv4ξ+4v4ξ+5|WE=ξ+1,ξ+2,ρ−ξ,for0≤ξ≤ρ−1,rv4ξ+1v4ξ+5|WE=ξ,ξ+1,ρ−ξ,for0≤ξ≤ρ−1,rv4ξ+2v4ξ+6|WE=ξ+1,ξ,ρ−ξ−1,for0≤ξ≤ρ−1,rv4ξ+3v4ξ+7|WE=ξ+2,ξ+1,ρ−ξ,for0≤ξ≤ρ−1,rv4ξ+4v4ξ+8|WE=ξ+2,ξ+2,ρ−ξ,for0≤ξ≤ρ−2.
Since representations of every two edges are different, it shows thatedimTn1,4≤3.
Now, we will prove that the edge metric generator of cardinality two does not exist. Suppose contrarily thatedimTn1,4=2 and let WE=v1,vβ. Now, Table 9 provides conditions on α, β, and all edges e1,e2 for which re1|WE=re2|WE.
Hence, there is no generator having two vertices which prove thatedimTn1,4=3 for n=4ρ+3, ρ≥1.Table 6
e1,e2 for which re1|WE=re2|WE.
Conditions onβe1,e2β=4α,1≤α≤ρv2v3,v5v6β=4α+1,1≤α≤ρ−1v4v5,v5v6β=4α+2,1≤α≤ρ−1v2v3,v4v5β=4α+3,1≤α≤ρ−1v2v3,v5v6β=3v5v6,v5v9Table 7
e1,e2 for which re1|WE=re2|WE.
Conditions onβe1,e2β=4α,1≤α≤ρv2v3,v5v6β=4α+1,1≤α≤ρv4v5,v5v6β=4α+2,1≤α≤ρ−1v2v3,v4v5β=4α+3,1≤α≤ρ−1v2v3,v5v6β=2,3v3v4,v3v7Table 8
e1,e2 for which re1|WE=re2|WE.
Conditions onβe1,e2β=4α,1≤α≤ρv2v3,v5v6β=4α+1,1≤α≤ρv4v5,v5v6β=4α+2,1≤α≤ρv2v3,v4v5β=4α+3,1≤α≤ρ−1v2v3,v5v6β=2,3v3v4,v3v7Table 9
e1,e2 for which re1|WE=re2|WE.
Conditions onβe1,e2β=4α,1≤α≤ρv2v3,v5v6β=4α+1,1≤α≤ρv4v5,v5v6β=4α+2,1≤β≤ρv2v3,v4v5β=4α+3,1≤α≤ρv2v3,v5v6β=2,3v3v4,v3v7


## 5. Upper Bounds for the Edge Metric Dimension of Toeplitz NetworksTn1,2,3

We shall compute the upper bound ofedimTn1,2,3 in this section. It has VTn1,2,3=v1,v2,…,vn and ETn1,2,3=vξvξ+1:1≤ξ≤n−1∪vξvξ+2:1≤ξ≤n−2∪vξvξ+3:1≤ξ≤n−3. Figure 4 shows the Toeplitz network T191,2,3. The metric dimension of Tn1,2,3 is stated.Figure 4
Toeplitz networkT191,2,3.Theorem 7 (See [24]).
IfTn1,2,3 be a graph of the Toeplitz network with n≥5, then dimTn1,2,3=3.
In the next result, we will find the upper bound ofedimTn1,2,3.Theorem 8.
LetTn1,2,3 be the Toeplitz networks. Then, edimTn1,2,3≤6, where n≥5.Proof.
We have the following cases to calculate the upper bound ofedimTn1,2,3:Case (i): letn=3ρ, ρ≥2, and WE=v1,v2,v3,vn−2,vn−1,vn⊂VTn1,2,3; we will prove that WE is an edge basis of Tn1,2,3. Now, representations of each edge of Tn1,2,3 are given by(10)rv3ξ+1v3ξ+2|WE=0,0,1,ρ−1,ρ−1,ρ,ifξ=0,ξ,ξ,ξ,ρ−ξ−1,ρ−ξ−1,ρ−ξ,if1≤ξ≤ρ−1,rv3ξ+2v3ξ+3|WE=ξ+1,ξ,ξ,ρ−ξ−1,ρ−ξ−1,ρ−ξ−1,if0≤ξ≤ρ−2,ρ,ρ−1,ρ−1,1,0,0,ifξ=ρ−1,rv3ξ+3v3ξ+4|WE=ξ+1,ξ+1,ξ,ρ−ξ−2,ρ−ξ−1,ρ−ξ−1,for0≤ξ≤ρ−2,rv3ξ+1v3ξ+4|WE=0,1,1,ρ−2,ρ−1,ρ−1,ifξ=0,ξ,ξ,ξ,ρ−ξ−2,ρ−ξ−1,ρ−ξ−1,if1≤ξ≤ρ−2,rv3ξ+2v3ξ+5|WE=1,0,1,ρ−2,ρ−2,ρ−1,ifξ=0,ξ+1,ξ,ξ,ρ−ξ−2,ρ−ξ−2,ρ−ξ−1,if1≤ξ≤ρ−3,ρ−1,ρ−2,ρ−2,1,0,1,ifξ=ρ−2,rv3ξ+3v3ξ+6|WE=ξ+1,ξ+1,ξ,ρ−ξ−2,ρ−ξ−2,ρ−ξ−2,if0≤ξ≤ρ−3,ρ−1,ρ−1,ρ−2,1,1,0,ifξ=ρ−2,r=0,1,0,ρ−1,ρ−1,ρ−1,ifξ=0,ξ,ξ,ξ,ρ−ξ−1,ρ−ξ−1,ρ−ξ−1,if1≤ξ≤ρ−2,ρ−1,ρ−1,ρ−1,0,1,0,ifξ=ρ−1,rv3ξ+2v3ξ+4|WE=1,0,1,ρ−2,ρ−1,ρ−1,ifξ=0,ξ+1,ξ,ξ,ρ−ξ−2,ρ−ξ−1,ρ−ξ−1,if1≤ξ≤ρ−2,rv3ξ+3v3ξ+5|WE=ξ+1,ξ+1,i,ρ−ξ−2,ρ−ξ−2,ρ−ξ−1,if0≤ξ≤ρ−3,ρ−1,ρ−1,ρ−2,1,0,1,ifξ=ρ−2.Since representations of every two edges are different, it shows thatedimTn1,2,3≤6.Case (ii): letn=3ρ+1, ρ≥2, and WE=v1,v2,v3,vn−2,vn−1,vn⊂VTn1,2,3; we will prove that WE is an edge metric generator of Tn1,2,3. Now, representations of each edge of Tn1,2,3 are given by(11)rv3ξ+1v3ξ+2|WE=0,0,1,ρ−1,ρ,ρ,ifξ=0,ξ,ξ,ξ,ρ−ξ−1,ρ−ξ,ρ−ξ,if1≤ξ≤ρ−1,rv3ξ+2v3ξ+3|WE=ξ+1,ξ,ξ,ρ−ξ−1,ρ−ξ−1,ρ−ξ,for0≤ξ≤ρ−1,rv3ξ+3v3ξ+4|WE=ξ+1,ξ+1,ξ,ρ−ξ−1,ρ−ξ−1,ρ−ξ−1,if0≤ξ≤ρ−2,ρ,ρ,ρ−1,1,0,0,ifξ=ρ−1,rv3ξ+1v3ξ+4|WE=0,1,1,ρ−1,ρ−1,ρ−1,ifξ=0,ξ,ξ,ξ,ρ−ξ−1,ρ−ξ−1,ρ−ξ−1,if1≤ξ≤ρ−2,ρ−1,ρ−1,ρ−1,1,1,0,ifξ=ρ−1,rv3ξ+2v3ξ+5|WE=1,0,1,ρ−2,ρ−1,ρ−1,ifξ=0,ξ+1,ξ,ξ,ρ−ξ−2,ρ−ξ−1,ρ−ξ−1,if1≤ξ≤ρ−2,rv3ξ+3v3ξ+6|WE=ξ+1,ξ+1,ξ,ρ−ξ−2,ρ−ξ−2,ρ−ξ−1,if0≤ξ≤ρ−3,ρ−1,ρ−1,ρ−2,1,0,1,ifξ=ρ−2,rv3ξ+1v3ξ+3|WE=0,1,0,ρ−1,ρ−1,ρ,ifξ=0,ξ,ξ,ξ,ρ−ξ−1,ρ−ξ−1,ρ−ξ,if1≤ξ≤ρ−2,ρ−1,ρ−1,ρ−1,1,0,1,ifξ=ρ−1,rv3ξ+2v3ξ+4|WE=1,0,1,ρ−1,ρ−1,ρ−1,ifξ=0,ξ+1,ξ,ξ,ρ−ξ−1,ρ−ξ−1,ρ−ξ−1,if1≤ξ≤ρ−2,ρ,ρ−1,ρ−1,0,1,0,ifξ=ρ−1,rv3ξ+3v3ξ+5|WE=ξ+1,ξ+1,ξ,ρ−ξ−2,ρ−ξ−1,ρ−ξ−1,for0≤ξ≤ρ−2.Since representations of every two edges are different, it shows thatedimTn1,2,3≤6.Case (iii): letn=3ρ+2, ρ≥1, and WE=v1,v2,v3,vn−2,vn−1,vn⊂VTn1,2,3; we will prove that WE is an edge metric generator of Tn1,2,3. Now, representations of each edge of Tn1,2,3 are given by(12)rv3ξ+1v3ξ+2|WE=0,0,1,ρ,ρ,ρ,ifξ=0,ξ,ξ,ξ,ρ−ξ,ρ−ξ,ρ−ξ,if1≤ξ≤ρ−1,ρ,ρ,ρ,1,0,0,ifξ=ρ,rv3ξ+2v3ξ+3|WE=ξ+1,ξ,ξ,ρ−ξ−1,ρ−ξ,ρ−ξ,for0≤ξ≤ρ−1,rv3ξ+3v3ξ+4|WE=ξ+1,ξ+1,ξ,ρ−ξ−1,ρ−ξ−1,ρ−ξ,for0≤ξ≤ρ−1,rv3ξ+1v3ξ+4|WE=0,1,1,ρ−1,ρ−1,ρ,ifξ=0,ξ,ξ,ξ,ρ−ξ−1,ρ−ξ−1,ρ−ξ,if1≤ξ≤ρ−2,ρ−1,ρ−1,ρ−1,1,0,1,ifξ=ρ−1,rv3ξ+2v3ξ+5|WE=1,0,1,ρ−1,ρ−1,ρ−1,ifξ=0,ξ+1,ξ,ξ,ρ−ξ−1,ρ−ξ−1,ρ−ξ−1,if1≤ξ≤ρ−2,ρ,ρ−1,ρ−1,1,1,0,ifξ=ρ−1,rv3ξ+3v3ξ+6|WE=ξ+1,ξ+1,ξ,ρ−ξ−2,ρ−ξ−1,ρ−ξ−1,for0≤ξ≤ρ−2,rv3ξ+1v3ξ+3|WE=0,1,0,ρ−1,ρ,ρ,ifξ=0,ξ,ξ,ξ,ρ−ξ−1,ρ−ξ,ρ−ξ,if1≤ξ≤ρ−1,rv3ξ+2v3ξ+4|WE=1,0,1,ρ−1,ρ−1,ρ,ifξ=0,ξ+1,ξ,ξ,ρ−ξ−1,ρ−ξ−1,ρ−ξ,if1≤ξ≤ρ−2,ρ,ρ−1,ρ−1,1,0,1,ifξ=ρ−1,rv3ξ+3v3ξ+5|WE=ξ+1,ξ+1,ξ,ρ−ξ−1,ρ−ξ−1,ρ−ξ−1,if0≤ξ≤ρ−2,ρ,ρ,ρ−1,0,1,0,ifξ=ρ−1.
Since representations of every two edges are different, it shows thatedimTn1,2,3≤6.


## 6. Conclusion

In this article, we have calculated the exact value of edge metric dimension of Toeplitz networksTn1,2, Tn1,3, and Tn1,4 and the upper bound of the Toeplitz network Tn1,2,3. 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.
CalculateedimTn1,2,t for n≥t+3.



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