The new soliton solution types to the Myrzakulov-Lakshmanan-XXXII-equation

Emad H. M. Zahran; Ahmet Bekir; Reda A. Ibrahim; Ratbay Myrzakulov; Emad H. M. Zahran; Ahmet Bekir; Reda A. Ibrahim; Ratbay Myrzakulov

doi:10.3934/math.2024300

AIMS Mathematics

2024, Volume 9, Issue 3: 6145-6160. doi: 10.3934/math.2024300

Previous Article Next Article

Research article

The new soliton solution types to the Myrzakulov-Lakshmanan-XXXII-equation

1.
Department of Basic Science, Benha University, Faculty of Engineering, Shubra, Egypt
2.
Neighbourhood of Akcaglan, Imarli Street, Number: 28/4, 26030, Eskisehir, Turkey
3.
Departments of Basic Science, Benha University, Faculty of Engineering, Shubra, Egypt
4.
Ratbay Myrzakulov Eurasian International Centre for Theoretical Physics, Astana, Kazakhstan

Received: 13 December 2023 Revised: 19 January 2024 Accepted: 23 January 2024 Published: 02 February 2024
MSC : 35C07, 35Q51, 83C15

Our attention concenters on deriving diverse forms of the soliton arising from the Myrzakulov-Lakshmanan XXXII (M-XXXII) that describes the generalized Heisenberg ferromagnetic equation. This model has been solved numerically only using the N-fold Darboux Transformation method, not solved analytically before. We will derive new types of the analytical soliton solutions that will be constructed for the first time in the framework of three impressive schemas that are prepared for this target. These three techniques are the Generalized Kudryashov scheme (GKS), the (G'/G)-expansion scheme and the extended direct algebraic scheme (EDAS). Moreover, we will establish the 2D, 3D graphical simulations that clear the new dynamic properties of our achieved solutions.

Keywords:

Citation: Emad H. M. Zahran, Ahmet Bekir, Reda A. Ibrahim, Ratbay Myrzakulov. The new soliton solution types to the Myrzakulov-Lakshmanan-XXXII-equation[J]. AIMS Mathematics, 2024, 9(3): 6145-6160. doi: 10.3934/math.2024300

Related Papers:

[1]	Chenggang Huo, Humera Bashir, Zohaib Zahid, Yu Ming Chu . On the 2-metric resolvability of graphs. AIMS Mathematics, 2020, 5(6): 6609-6619. doi: 10.3934/math.2020425
[2]	Zahid Iqbal, Muhammad Ishaq . Depth and Stanley depth of edge ideals associated to some line graphs. AIMS Mathematics, 2019, 4(3): 686-698. doi: 10.3934/math.2019.3.686
[3]	Tongzhu Li, Ruiyang Lin . Classification of Möbius homogeneous curves in $\mathbb{R}^4$ . AIMS Mathematics, 2024, 9(8): 23027-23046. doi: 10.3934/math.20241119
[4]	Bangchao Yin, Shujie Zhai . Classification of Möbius minimal and Möbius isotropic hypersurfaces in $\mathbb{S}^{5}$ . AIMS Mathematics, 2021, 6(8): 8426-8452. doi: 10.3934/math.2021489
[5]	Hongzhe Cao . Two meromorphic functions on annuli sharing some pairs of small functions or values. AIMS Mathematics, 2021, 6(12): 13311-13326. doi: 10.3934/math.2021770
[6]	Yajun Xie, Changfeng Ma, Qingqing Zheng . On the nonlinear matrix equation $X^{s}+A^{H}F(X)A = Q$ . AIMS Mathematics, 2023, 8(8): 18392-18407. doi: 10.3934/math.2023935
[7]	Dalal Awadh Alrowaili, Uzma Ahmad, Saira Hameeed, Muhammad Javaid . Graphs with mixed metric dimension three and related algorithms. AIMS Mathematics, 2023, 8(7): 16708-16723. doi: 10.3934/math.2023854
[8]	Ian Marquette . Representations of quadratic Heisenberg-Weyl algebras and polynomials in the fourth Painlevé transcendent. AIMS Mathematics, 2024, 9(10): 26836-26853. doi: 10.3934/math.20241306
[9]	Ahmed Alamer, Hassan Zafar, Muhammad Javaid . Study of modified prism networks via fractional metric dimension. AIMS Mathematics, 2023, 8(5): 10864-10886. doi: 10.3934/math.2023551
[10]	Muhammad Ahmad, Muhammad Faheem, Sanaa A. Bajri, Zohaib Zahid, Muhammad Javaid, Hamiden Abd El-Wahed Khalifa . Optimizing SNARK networks via double metric dimension. AIMS Mathematics, 2024, 9(8): 22091-22111. doi: 10.3934/math.20241074

Abstract

1. Introduction

Let $G = (V, E)$ be a simple, connected graph. For $x_1, x_2\in V,$ the distance $d(x_1, x_2)$ between vertices $x_1$ and $x_2$ is the count of edges between $x_1$ and $x_2.$ A vertex $v\in V$ is said to distinguish two vertices $x_1$ and $x_2,$ if $d(v, x_1)\neq d(v, x_2).$ A set $Q\subset V$ is called a metric generator for $G,$ if any pair of distinct vertices of $G$ is distinguished by some element of $Q.$ A metric generator of minimum cardinality is named as metric basis, and its cardinality is the metric dimension of $G,$ denoted by $dim(G).$ For a vertex $v$ from the vertex set and an edge $e = x_1x_2$ from its corresponding edge set, the distance betwixt $v$ and $e$ is measured as $d(e, v) = min\{d(x_1, v), d(x_2, v)\}.$ A vertex $x\in V$ categorizes two edges $e_1, e_2\in E,$ if $d(x, e_1)\neq d(x, e_2).$ A subset $Q_e$ having minimum vertices from a graph $G$ is an edge metric generator for $G,$ if each pair of two distinct edges of $G$ are distinguished by some vertex of $Q_e.$ The minimum cardinality of an edge metric generator for a graph $G$ is called the edge metric dimension and is denoted by $dim_e(G).$ A vertex $x$ of a graph $G$ distinguishes two elements (vertices or edges) $u, v$ of $G,$ if $d(u, x)\neq d(v, x).$ A subset $Q_m$ is a mixed metric generator, if there exist any two distinct elements (vertices or edges) of $G$ are distinguished by some vertex of $Q_m.$ The minimum numbers of elements in a mixed metric generator for a graph, is called the mixed metric dimension and is denoted by $dim_m(G).$ Moreover, mixed metric dimension is a blended version of metric and edge metric dimension.

The authors in ^[1] introduced the concept of metric dimension, where the metric generators were referred to as locating sets due to some connections with the problem of uniquely identifying the position of intruders in networks. Furthermore, it was studied separately in ^[2], where metric generators were named resolving sets. Aligning with the same pattern of recognizing the vertices of a graph by subset (resolving set) of vertex set, in ^[3] introduced the edge metric dimension and later analogous to this definition ^[4] proposed mixed metric dimension in which both concepts are together (recognizing vertices and edges), adding one more condition that no two elements (vertices and edges) have identical representation with respect to the chosen subset.

The standard metric generator is based on the concept of uniquely recognizing the entire vertex set of a graph, such as how to represent different vertices of a graph with respect to the metric generator. This concept might fail when an unknown problem arises in edges instead of vertices, then edge metric generator parameter take attention of entire working, but still, a question can arise, what happened if both of the behaviors came across with an issue? Then recognizing vertices and edges simultaneously to overcome the failure and both elements (vertices and edges) can be correctly recognized the entire structure of graphs with respect to some chosen vertices, which is named mixed metric generator.

The idea of metric dimension has numerous applications in different fields of science, regarding chemical structure ^[5] robot navigation ^[6], connections of a metric dimension of Hamming graphs with understanding and analysis of the mastermind of games ^[7] and variety of coin weighing problems linked with this concept in ^[8,9]. Resolving sets have proven worthwhile in a variety of other applications, and they also served as a motivation for the theoretical study of metric dimension.

The metric dimension of different classes of graphs has been extensively studied for the last few decades. It is proved that the metric and one of its generalization named as edge metric, of ladder network is two ^[10,11], metric dimension of Möbius ladder network is three ^[12], while edge metric is increased by one and proved four in ^[11], it is proved that both parameters of dimension for the hexagonal Möbius ladder network is three ^[11,13], similarly, metric dimension of triangular ladder network is three and edge metric dimension increased by one proved in ^[11,14]. In ^[15], discussed vertex metric-based dimension of generalized perimantanes diamondoid structure, some extended topics of metric dimension are discussed in ^[16,17,18]. In ^[19], measured the metric-based resolvability of polycyclic aromatic hydrocarbons.

Moreover, metric dimension of some cycle related graphs studied in ^[20], detailed analysis of the exact metric dimension of newly designed hexagonal Möbius ladder network can be seen in ^[13]. The local edge metric dimension is discussed for some graphs, including the ladder network in ^[21]. Convex polytopes graph discussed in ^[22] with the concept of edge metric dimension. Mixed metric dimension of generalized Peterson graph discussed in ^[23], some families of rotationally symmetric graph discussed in ^[24] with the concept of mixed metric dimension, some lower and upper bounds are discussed in ^[4] with respect to the girth of graph. Limited to the topic, one can find interesting research on metric dimension ^[25,26,27], on edge metric dimension ^[28,29].

Theorem 1.1. ^[4] Let $G$ be a simple connected graph $G.$ Then $dim_m(G) \geq max\{dim(G), dim_e(G)\}.$

Definition 1.1. A size $h$ -gap between two vertices $x_1$ and $x_2$ is a path of length $h$ between them.

In ^[4], researcher proved that the mixed metric dimension of a graph is an NP-hard problem. Our emphasis is to provide the exact mixed metric dimension of some families of ladder networks. We investigate the mixed metric dimension of hexagonal Möbius ladder, triangular ladder, triangular Möbius ladder and simple ladder network. We prove that these ladder network families lie in the constant mixed metric dimension category. A comparative analysis between metric, edge metric, and mixed metric dimension is provided in the end to decide which parameter is better to choose.

2. Mixed metric dimension results

As we discuss earlier, edge metric dimension of Möbius ladder network is three. The mixed metric dimension can be three or more. In this part of section we will prove that the mixed metric dimension of Möbius ladder network is four.

Theorem 2.1. Let $ML_m$ be a Möbius ladder network with $m\geq 3.$ Then

$dim_m(ML_m) = 4 .$

Proof. Let the mixed metric generator $Q_m = \{v_{1}, v_{2}, v_{\frac{m+4}{2}}, v_{\frac{3m+2}{2}}\},$ when $m$ is even and $Q_m = \{v_{1}, v_{2}, v_{\frac{m+3}{2}}, v_{m+1}\},$ when $m$ is odd. The labeling of vertices are defined in Figure 1.

Figure 1. Möbius Ladder Graph

$ML_m$ .

DownLoad: Full-Size Img PowerPoint

To prove that $dim_m(ML_m) = 4,$ we use the method of double inequality, for $dim_m(ML_m)\leq4,$ given below are the shortest distance between all edges and the mixed metric generator's vertices:

$d\left(v_{\epsilon}v_{m+\epsilon}, v_1\right) = \left\{ \begin{array}{ll} \epsilon-1 &\ \ {\rm{if}}\ \epsilon = 1, 2, \dots, \left\lfloor\frac{m+2}{2}\right\rfloor;\\ \left\lfloor\frac{m+3}{2}\right\rfloor-\epsilon+\left\lfloor\frac{m-1}{2}\right\rfloor &\ \ {\rm{if}}\ \epsilon = \left\lfloor\frac{m+3}{2}\right\rfloor, \dots, m. \end{array} \right.$

$d\left(v_{\epsilon}v_{\epsilon+1}, v_1\right) = \left\{ \begin{array}{ll} \epsilon-1 &\ \ {\rm{if}}\ \epsilon = 1, 2, \dots, \left\lfloor\frac{m+2}{2}\right\rfloor;\\ \left\lfloor\frac{m+4}{2}\right\rfloor-\epsilon+\left\lfloor\frac{m-1}{2}\right\rfloor &\ \ {\rm{if}}\ \epsilon = \left\lfloor\frac{m+4}{2}\right\rfloor, \dots, m. \end{array} \right.$

$d\left(v_{m+\epsilon}v_{m+\epsilon+1}, v_1\right) = \left\{ \begin{array}{ll} \epsilon &\ \ {\rm{if}}\ \epsilon = 1, 2, \dots, \left\lfloor\frac{m}{2}\right\rfloor;\\ \left\lfloor\frac{m-1}{2}\right\rfloor-\epsilon+\left\lfloor\frac{m+2}{2}\right\rfloor &\ \ {\rm{if}}\ \epsilon = \left\lfloor\frac{m+2}{2}\right\rfloor, \dots, m-1. \end{array} \right.$

$d\left(v_{2m}v_{1}, v_{1}\right) = 0.$

$d\left(v_{\epsilon}v_{m+\epsilon}, v_2\right) = \left\{ \begin{array}{ll} 1 &\ \ {\rm{if}}\ \epsilon = 1;\\ \epsilon-2 &\ \ {\rm{if}}\ \epsilon = 2, 3, \dots, \left\lfloor\frac{m+5}{2}\right\rfloor;\\ \left\lfloor\frac{m}{2}\right\rfloor-\epsilon+\left\lfloor\frac{m+7}{2}\right\rfloor &\ \ {\rm{if}}\ \epsilon = \left\lfloor\frac{m+7}{2}\right\rfloor, \dots, m. \end{array} \right.$

$d\left(v_{\epsilon}v_{\epsilon+1}, v_2\right) = \left\{ \begin{array}{ll} 0 &\ \ {\rm{if}}\ \epsilon = 1;\\ \epsilon-2 &\ \ {\rm{if}}\ \epsilon = 2, \dots, \left\lfloor\frac{m+4}{2}\right\rfloor;\\ \left\lfloor\frac{m+6}{2}\right\rfloor-\epsilon+\left\lfloor\frac{m-1}{2}\right\rfloor &\ \ {\rm{if}}\ \epsilon = \left\lfloor\frac{m+6}{2}\right\rfloor, \dots, m. \end{array} \right.$

$d\left(v_{m+\epsilon}v_{m+\epsilon+1}, v_2\right) = \left\{ \begin{array}{ll} 1 &\ \ {\rm{if}}\ \epsilon = 1;\\ \epsilon-1 &\ \ {\rm{if}}\ \epsilon = 2, \dots, \left\lfloor\frac{m+3}{2}\right\rfloor;\\ \left\lfloor\frac{m+5}{2}\right\rfloor-\epsilon+\left\lfloor\frac{m-1}{2}\right\rfloor &\ \ {\rm{if}}\ \epsilon = \left\lfloor\frac{m+5}{2}\right\rfloor, \dots, m-1. \end{array} \right.$

$d\left(v_{2m}v_{1}, v_{2}\right) = 2.$

$d\left(v_{\epsilon}v_{m+\epsilon}, v_{m+1}\right) = \left\{ \begin{array}{ll} \epsilon-1 &\ \ {\rm{if}}\ \epsilon = 1, 2, \dots, \left\lfloor\frac{m+2}{2}\right\rfloor;\\ \left\lfloor\frac{m+4}{2}\right\rfloor-\epsilon+\left\lfloor\frac{m-1}{2}\right\rfloor &\ \ {\rm{if}}\ \epsilon = \left\lfloor\frac{m+4}{2}\right\rfloor, \dots, m. \end{array} \right.$

$d\left(v_{\epsilon}v_{\epsilon+1}, v_{m+1}\right) = \left\{ \begin{array}{ll} \epsilon &\ \ {\rm{if}}\ \epsilon = 1, 2, \dots, \left\lfloor\frac{m}{2}\right\rfloor;\\ \left\lfloor\frac{m-1}{2}\right\rfloor-\epsilon+\left\lfloor\frac{m+2}{2}\right\rfloor &\ \ {\rm{if}}\ \epsilon = \left\lfloor\frac{m+2}{2}\right\rfloor, \dots, m. \end{array} \right.$

$d\left(v_{m+\epsilon}v_{m+\epsilon+1}, v_{m+1}\right) = \left\{ \begin{array}{ll} \epsilon-1 &\ \ {\rm{if}}\ \epsilon = 1, 2, \dots, \left\lfloor\frac{m+2}{2}\right\rfloor;\\ \left\lfloor\frac{m-1}{2}\right\rfloor-\epsilon+\left\lfloor\frac{m+4}{2}\right\rfloor &\ \ {\rm{if}}\ \epsilon = \left\lfloor\frac{m+4}{2}\right\rfloor, \dots, m-1. \end{array} \right.$

$d\left(v_{2m}v_{1}, v_{m+1}\right) = 1.$

$d\left(v_{\epsilon}v_{m+\epsilon}, v_{b}\right) = \left\{ \begin{array}{ll} \left\lfloor\frac{m-1}{2}\right\rfloor &\ \ {\rm{if}}\ \epsilon = 1;\\ \left| 2-\epsilon+\left\lfloor\frac{m}{2}\right\rfloor\right| &\ \ {\rm{if}}\ \epsilon = 2, 3, \dots, m. \end{array} \right.$

where $b = \left\lfloor\frac{m+4}{2}\right\rfloor = \frac{m+4}{2}\ (m = {\rm{even}}) = \frac{m+3}{2}\ (m = {\rm{odd}}),$

$d\left(v_{\epsilon}v_{\epsilon+1}, v_{b}\right) = \left\{ \begin{array}{ll} \left\lfloor\frac{m}{2}\right\rfloor-\epsilon+1 &\ \ {\rm{if}}\ \epsilon = 1, 2, \dots, \left\lfloor\frac{m+2}{2}\right\rfloor;\\ \epsilon-\left\lfloor\frac{m+4}{2}\right\rfloor &\ \ {\rm{if}}\ \epsilon = \left\lfloor\frac{m+4}{2}\right\rfloor, \dots, m. \end{array} \right.$

$d\left(v_{m+\epsilon}v_{m+\epsilon+1}, v_{b}\right) = \left\{ \begin{array}{ll} \left\lfloor\frac{m+1}{2}\right\rfloor &\ \ {\rm{if}}\ \epsilon = 1;\\ 2-\epsilon+\left\lfloor\frac{m-1}{2}\right\rfloor &\ \ {\rm{if}}\ \epsilon = 2, \dots, \left\lfloor\frac{m+2}{2}\right\rfloor;\\ \epsilon-\left\lfloor\frac{m+4}{2}\right\rfloor+1 &\ \ {\rm{if}}\ \epsilon = \left\lfloor\frac{m+4}{2}\right\rfloor, \dots, m-1. \end{array} \right.$

$d\left(v_{2m}v_{1}, v_{b}\right) = \left\lfloor\frac{m-1}{2}\right\rfloor.$

$d\left(v_{\epsilon}v_{m+\epsilon}, v_{c}\right) = \left|\frac{m}{2}-\epsilon+1\right|,$

where $c = \frac{3m+2}{2}$ ,

$d\left(v_{\epsilon}v_{\epsilon+1}, v_{c}\right) = \left\{ \begin{array}{ll} \frac{m}{2}-\epsilon+1 &\ \ {\rm{if}}\ \epsilon = 1, 2, \dots, \frac{m}{2};\\ \epsilon-\frac{m+2}{2}+1 &\ \ {\rm{if}}\ \epsilon = \frac{m+2}{2}, \dots, m. \end{array} \right.$

$d\left(v_{m+\epsilon}v_{m+\epsilon+1}, v_{c}\right) = \left\{ \begin{array}{ll} \frac{m-2}{2}-\epsilon+1 &\ \ {\rm{if}}\ \epsilon = 1, 2, \dots, \frac{m}{2};\\ \epsilon-\frac{m+2}{2} &\ \ {\rm{if}}\ \epsilon = \frac{m+2}{2}, \dots, m. \end{array} \right.$

$d\left(v_{2m}v_{1}, v_{c}\right) = \frac{m-2}{2}.$

It is easy to see the distances between all edges in relation to the chosen mixed metric generator $Q_m.$ Now, the vertices distances according to $Q_m$ are given as follows:

Distance of all vertices with respect to $v_1$ ;

$d(v_{\epsilon}, v_1) = \begin{cases}{} \epsilon-1, & \ {\rm{if}} \ \epsilon = 1, 2\dots, \left\lfloor\frac{m+2}{2}\right\rfloor;\\ m-\epsilon+2, & \ {\rm{if}} \ \epsilon = \left\lfloor\frac{m+2}{2}\right\rfloor+1, \dots, m+1;\\ \epsilon-m, & \ {\rm{if}} \ \epsilon = m+2, \dots, \left\lceil\frac{m+2}{2}\right\rceil;\\ 2m-\epsilon+1, & \ {\rm{if}} \ \epsilon = \left\lceil\frac{m+2}{2}\right\rceil+1, \dots, 2m. \end{cases}$

Distance of all vertices with respect to $v_2$ ;

$d(v_{\epsilon}, v_2) = \begin{cases}{} |\epsilon-2|, & \ {\rm{if}} \ \epsilon = 1, 2\dots, \left\lceil\frac{m+4}{2}\right\rceil;\\ m-\epsilon+3, & \ {\rm{if}} \ \epsilon = \left\lceil\frac{m+2}{2}\right\rceil+1, \dots, m+1;\\ \epsilon-m-1, & \ {\rm{if}} \ \epsilon = m+2, \dots, \left\lfloor\frac{3m}{2}\right\rfloor+1;\\ 2m-\epsilon+2, & \ {\rm{if}} \ \epsilon = \left\lfloor\frac{3m}{2}\right\rfloor+2, \dots, 2m. \end{cases}$

Distance of all vertices with respect to $v_{\frac{m+4}{2}}$ ;

$d(v_{\epsilon}, v_{\frac{m+4}{2}}) = \begin{cases}{} \frac{m}{2}, & \ {\rm{if}} \ \epsilon = 1;\\ \frac{m+4}{2}-\epsilon, & \ {\rm{if}} \ \epsilon = 2, 3, \dots, \frac{m+4}{2};\\ \epsilon-\frac{m+4}{2}, & \ {\rm{if}} \ \epsilon = \frac{m+6}{2}, \dots, m+1;\\ \frac{m}{2}, & \ {\rm{if}} \ \epsilon = m+2;\\ \frac{3m+6}{2}-\epsilon, & \ {\rm{if}} \ \epsilon = m+3, \dots, \frac{3m+4}{2};\\ 2+\epsilon-\frac{3m+6}{2}, & \ {\rm{if}} \ \epsilon = \frac{3m+6}{2}+2, \dots, 2m. \end{cases}$

Distance of all vertices with respect to $v_{\frac{3m+2}{2}}$ ;

$d(v_{\epsilon}, v_{\frac{3m+2}{2}}) = \begin{cases}{} \frac{m}{2}, & \ {\rm{if}} \ \epsilon = 1;\\ \frac{m+4}{2}-\epsilon, & \ {\rm{if}} \ \epsilon = 2, 3, \dots, \frac{m+2}{2};\\ \epsilon-\frac{m}{2}, & \ {\rm{if}} \ \epsilon = \frac{m+4}{2}, \dots, m;\\ \frac{m}{2}, & \ {\rm{if}} \ \epsilon = m+1;\\ \left|\frac{3m+2}{2}-\epsilon\right|, & \ {\rm{if}} \ \epsilon = m+2, \dots, 2m. \end{cases}$

Distance of all vertices with respect to $v_{\frac{m+3}{2}}$ ;

$d(v_{\epsilon}, v_{\frac{m+3}{2}}) = \begin{cases}{} \frac{m+3}{2}-\epsilon, & \ {\rm{if}} \ \epsilon = 1, 2, 3, \dots, \frac{m+3}{2};\\ \epsilon-\frac{m+3}{2}, & \ {\rm{if}} \ \epsilon = \frac{m+5}{2}, \dots, m+1;\\ \frac{3m+5}{2}-\epsilon, & \ {\rm{if}} \ \epsilon = m+2, \dots, \frac{3m+3}{2};\\ 2+\epsilon-\frac{3m+5}{2}, & \ {\rm{if}} \ \epsilon = \frac{3m+5}{2}+2, \dots, 2m. \end{cases}$

Distance of all vertices with respect to $v_{m+1}$ ;

$d(v_{\epsilon}, v_{m+1}) = \begin{cases}{} \epsilon, & \ {\rm{if}} \ \epsilon = 1, 2, \dots, \frac{m+1}{2};\\ m-\epsilon+1, & \ {\rm{if}} \ \epsilon = \frac{m+3}{2}, \dots, m+1;\\ \epsilon-m-1, & \ {\rm{if}} \ \epsilon = m+2, \dots, \frac{3m+2}{2};\\ 2m-\epsilon+2, & \ {\rm{if}} \ \epsilon = \frac{3m+4}{2}, \dots, 2m. \end{cases}$

It is clear to see the representations in term of distances of all edges and vertices with respect to mixed metric generator $Q_m$ are different, it proved that $dim_m(ML_m)\leq4.$ Now, the converse is directly deduced from Theorem 1.1, which implies that

$dim_m(ML_m) = 4.$

We know that edge metric dimension of hexagonal Möbius ladder network is three. By the relation in Theorem 1.1, mixed metric dimension can be three or more. In the following, we proved that the mixed metric dimension of the hexagonal Möbius ladder network is four.

Theorem 2.2. Let $HML_m$ be a hexagonal Möbius ladder network with $m\geq 2.$ Then

$dim_m(HML_m) = 4.$

Proof. Suppose that the mixed metric generator is $Q_m = \{v_{1}, v_{2}, v_{m+2}, v_{3m+2}\}.$ The labeling defined in Figure 2.

Figure 2. Hexagonal Möbius Ladder Graph

$HML_m$ .

DownLoad: Full-Size Img PowerPoint

To prove that $dim_m(HML_m) = 4,$ we will use the method of double inequality, for $dim_m(HML_m)\leq4,$ given below are the distances of all edges with respect to mixed metric generator:

$d\left(v_{\epsilon}v_{m+\epsilon}, v_1\right) = \left\{ \begin{array}{ll} \epsilon-1 &\ \ {\rm{if}}\ \epsilon = 1, 3, \dots, m+1\ {\rm{and}} \ m\ {\rm{even}};\\ 2m+1-\epsilon &\ \ {\rm{if}}\ \epsilon = m+3, m+5, \dots, 2m-1\ {\rm{and}} \ m\ {\rm{even}};\\ \epsilon-1 &\ \ {\rm{if}}\ \epsilon = 1, 3, \dots, m\ {\rm{and}} \ m\ {\rm{odd}};\\ 2m+1-\epsilon &\ \ {\rm{if}}\ \epsilon = m+2, m+4\dots, 2m-1\ {\rm{and}} \ m\ {\rm{odd}}. \end{array} \right.$

$d\left(v_{\epsilon}v_{\epsilon+1}, v_1\right) = \left\{ \begin{array}{ll} \epsilon-1 &\ \ {\rm{if}}\ \epsilon = 1, 2, \dots, m+1;\\ 2m+1-\epsilon &\ \ {\rm{if}}\ \epsilon = m+2, \dots, 2m. \end{array} \right.$

$d\left(v_{2m+\epsilon}v_{2m+\epsilon+1}, v_1\right) = \left\{ \begin{array}{ll} \epsilon &\ \ {\rm{if}}\ \epsilon = 1, 2, \dots, m;\\ 2m-\epsilon &\ \ {\rm{if}}\ \epsilon = m+1, \dots, 2m-1 . \end{array} \right.$

$d\left(v_{4m}v_{1}, v_1\right) = 0.$

$d\left(v_{\epsilon}v_{m+\epsilon}, v_2\right) = \left\{ \begin{array}{lll} 1 &\ \ {\rm{if}}\ \epsilon = 1;\\ \epsilon-2 &\ \ {\rm{if}}\ \epsilon = 3, 5, \dots, m+1\ {\rm{and}} \ m\ {\rm{even}};\\ 2m+3-\epsilon &\ \ {\rm{if}}\ \epsilon = m+3, m+5, \dots, 2m-1\ {\rm{and}} \ m\ {\rm{even}};\\ \epsilon-2 &\ \ {\rm{if}}\ \epsilon = 3, 5, \dots, m+2\ {\rm{and}} \ m\ {\rm{odd}};\\ 2m+3-\epsilon &\ \ {\rm{if}}\ \epsilon = m+4, m+6, \dots, 2m-1\ {\rm{and}} \ m\ {\rm{odd}}. \end{array} \right.$

$d\left(v_{\epsilon}v_{\epsilon+1}, v_2\right) = \left\{ \begin{array}{ll} 0 &\ \ {\rm{if}}\ \epsilon = 1;\\ \epsilon-2 &\ \ {\rm{if}}\ \epsilon = 2, \dots, m+2;\\ 2m+2-\epsilon &\ \ {\rm{if}}\ \epsilon = m+3, \dots, 2m. \end{array} \right.$

$d\left(v_{2m+\epsilon}v_{2m+\epsilon+1}, v_2\right) = \left\{ \begin{array}{ll} 2 &\ \ {\rm{if}}\ \epsilon = 1, 2;\\ \epsilon-1 &\ \ {\rm{if}}\ \epsilon = 3, \dots, m+1;\\ 2m+2-\epsilon &\ \ {\rm{if}}\ \epsilon = m+2, \dots, 2m-1. \end{array} \right.$

$d\left(v_{4m}v_{1}, v_2\right) = 2.$

$d\left(v_{\epsilon}v_{m+\epsilon}, v_{m+2}\right) = \left\{ \begin{array}{lll} m-1 &\ \ {\rm{if}}\ \epsilon = 1;\\ m+2-\epsilon &\ \ {\rm{if}}\ \epsilon = 3, 5, \dots, m+1\ {\rm{and}} \ m\ {\rm{even}};\\ \epsilon-m-2 &\ \ {\rm{if}}\ \epsilon = m+3, m+5, \dots, 2m-1\ {\rm{and}} \ m\ {\rm{even}};\\ m+2-\epsilon &\ \ {\rm{if}}\ \epsilon = 3, 5, \dots, m+2\ {\rm{and}} \ m\ {\rm{odd}};\\ \epsilon-m-2 &\ \ {\rm{if}}\ \epsilon = m+4, m+6\dots, 2m-1\ {\rm{and}} \ m\ {\rm{odd}}. \end{array} \right.$

$d\left(v_{\epsilon}v_{\epsilon+1}, v_{m+2}\right) = \left\{ \begin{array}{ll} m-\epsilon+1 &\ \ {\rm{if}}\ \epsilon = 1, 2, \dots, m+1;\\ \epsilon-m-2 &\ \ {\rm{if}}\ \epsilon = m+2, \dots, 2m. \end{array} \right.$

$d\left(v_{2m+\epsilon}v_{2m+\epsilon+1}, v_{m+2}\right) = \left\{ \begin{array}{lll} m &\ \ {\rm{if}}\ \epsilon = 1;\\ m+2-\epsilon &\ \ {\rm{if}}\ \epsilon = 2, \dots, m\ {\rm{and}} \ m\ {\rm{even}};\\ 2 &\ \ {\rm{if}}\ \epsilon = m+1, m+2\ {\rm{and}} \ m\ {\rm{even}};\\ \epsilon-m-1 &\ \ {\rm{if}}\ \epsilon = m+3, \dots, 2m-1\ {\rm{and}} \ m\ {\rm{even}};\\ m+2-\epsilon &\ \ {\rm{if}}\ \epsilon = 2, \dots, m+1\ {\rm{and}} \ m\ {\rm{odd}};\\ \epsilon-m-1 &\ \ {\rm{if}}\ \epsilon = m+2, \dots, 2m-1\ {\rm{and}} \ m\ {\rm{odd}}. \end{array} \right.$

$d\left(v_{4m}v_{1}, v_{m+2}\right) = \left\{ \begin{array}{lll} m &\ \ {\rm{when}}\ m = 2;\\ m-1 &\ \ {\rm{when}}\ m\geq3. \end{array} \right.$

$d\left(v_{\epsilon}v_{m+\epsilon}, v_{3m+2}\right) = \left\{ \begin{array}{lll} m-1 &\ \ {\rm{if}}\ \epsilon = 1;\\ m+2-\epsilon &\ \ {\rm{if}}\ \epsilon = 3, 5, \dots, m+1\ {\rm{and}} \ m\ {\rm{even}};\\ 2 &\ \ {\rm{if}}\ \epsilon = m+1, m+2, \ {\rm{and}} \ m\ {\rm{even}};\\ \epsilon-m-2 &\ \ {\rm{if}}\ \epsilon = m+3, m+5, \dots, 2m-1\ {\rm{and}} \ m\ {\rm{even}};\\ m+2-\epsilon &\ \ {\rm{if}}\ \epsilon = 3, 5, \dots, m+2\ {\rm{and}} \ m\ {\rm{odd}};\\ \epsilon-m-2 &\ \ {\rm{if}}\ \epsilon = m+4, m+6, \dots, 2m-1\ {\rm{and}} \ m\ {\rm{odd}}. \end{array} \right.$

$d\left(v_{\epsilon}v_{\epsilon+1}, v_{3m+2}\right) = \left\{ \begin{array}{lll} m &\ \ {\rm{if}}\ \epsilon = 1;\\ m+2-\epsilon &\ \ {\rm{if}}\ \epsilon = 2, \dots, m\ {\rm{and}} \ m\ {\rm{even}};\\ 2 &\ \ {\rm{if}}\ \epsilon = m+1, m+2\ {\rm{and}} \ m\ {\rm{even}};\\ \epsilon-m-1 &\ \ {\rm{if}}\ \epsilon = m+3, \dots, 2m\ {\rm{and}} \ m\ {\rm{even}};\\ m+2-\epsilon &\ \ {\rm{if}}\ \epsilon = 2, \dots, m+1\ {\rm{and}} \ m\ {\rm{odd}};\\ \epsilon-m-1 &\ \ {\rm{if}}\ \epsilon = m+2, \dots, 2m\ {\rm{and}} \ m\ {\rm{odd}}. \end{array} \right.$

$d\left(v_{2m+\epsilon}v_{2m+\epsilon+1}, v_{3m+2}\right) = \left\{ \begin{array}{ll} m+1-\epsilon &\ \ {\rm{if}}\ \epsilon = 1, \dots, m+1;\\ \epsilon-m-2 &\ \ {\rm{if}}\ \epsilon = m+2, \dots, 2m-1. \end{array} \right.$

$d\left(v_{4m}v_{1}, v_{3m+2}\right) = m-2.$

It is easy to verify that no two edges have the same representation with respect to the mixed metric generator. In order to complete the definition of mixed metric, we will check the vertices distance as well.

Distance of all vertices with respect to $v_1$ ;

$d(v_{\epsilon}, v_1) = \begin{cases}{} \epsilon-1, & \ {\rm{if}} \ \epsilon = 1, 2\dots, m+1;\\ 2m-\epsilon+2, & \ {\rm{if}} \ \epsilon = m+2, \dots, 2m+1;\\ \epsilon-2m, & \ {\rm{if}} \ \epsilon = 2m+2, \dots, 3m;\\ 4m-\epsilon+1, & \ {\rm{if}} \ \epsilon = 3m+1, \dots, 4m. \end{cases}$

Distance of all vertices with respect to $v_2$ ;

$d(v_{\epsilon}, v_2) = \begin{cases}{} |\epsilon-2|, & \ {\rm{if}} \ \epsilon = 1, 2\dots, m+2;\\ 2m-\epsilon+3, & \ {\rm{if}} \ \epsilon = m+3, \dots, 2m+1;\\ \epsilon-2m-1, & \ {\rm{if}} \ \epsilon = 2m+3, \dots, 3m+1;\\ \epsilon-2m+1, & \ {\rm{if}} \ \epsilon = 2m+2;\\ 4m-\epsilon+2, & \ {\rm{if}} \ \epsilon = 3m+2, \dots, 4m. \end{cases}$

Distance of all vertices with respect to $v_{m+2}$ ;

$d(v_{\epsilon}, v_{m+2}) = \begin{cases}{} m, & \ {\rm{if}} \ \epsilon = 1;\\ m-\epsilon+2, & \ {\rm{if}} \ \epsilon = 2, 3, \dots, m+2;\\ \epsilon-m-2, & \ {\rm{if}} \ \epsilon = m+3, \dots, 2m+1;\\ 3, & \ {\rm{if}} \ \epsilon = 2m+2\ {\rm{and}} \ m = 2;\\ m, & \ {\rm{if}} \ \epsilon = 2m+2\ {\rm{and}} \ m\geq3;\\ 3m+3-\epsilon, & \ {\rm{if}} \ \epsilon = 2m+3, \dots, 3m+1. \end{cases}$

$d(v_{\epsilon}, v_{m+2}) = \begin{cases}{} 3m+2-\epsilon+\left\lfloor\frac{m+6}{4}\right\rfloor, & \ {\rm{if}} \ \epsilon = 3m+2, \dots, 3m+2+\left\lfloor\frac{m+6}{4}\right\rfloor, \ {\rm{and}}\ m\ {\rm{even}};\\ \epsilon-3m-1-\left\lfloor\frac{m-1}{4}\right\rfloor, & \ {\rm{if}} \ \epsilon = 3m+3+\left\lfloor\frac{m-1}{4}\right\rfloor, \dots, 4m\ {\rm{and}}\ m\ {\rm{even}};\\ \epsilon-3m-1, & \ {\rm{if}} \ \epsilon = 3m+2, \dots, 4m\ {\rm{and}}\ m\ {\rm{odd}}. \end{cases}$

Distance of all vertices with respect to $v_{3m+2}$ ;

$d(v_{\epsilon}, v_{m+2}) = \begin{cases}{} m-2+\epsilon, & \ {\rm{if}} \ \epsilon = 1, 2;\\ m-\epsilon+3, & \ {\rm{if}} \ \epsilon = 3, 4, \dots, m+1\ {\rm{and}}\ m\ {\rm{even}};\\ 5-\epsilon+m, & \ {\rm{if}} \ \epsilon = m+2, m+3\ {\rm{and}}\ m\ {\rm{even}};\\ \epsilon-m-1, & \ {\rm{if}} \ \epsilon = m+4, \dots, 2m+1\ {\rm{and}}\ m\ {\rm{even}};\\ 3m+2-\epsilon, & \ {\rm{if}} \ \epsilon = 2m+2, \dots, 3m+2\ {\rm{and}}\ m\ {\rm{even}};\\ \epsilon-3m-2, & \ {\rm{if}} \ \epsilon = 3m+3, \dots, 4m\ {\rm{and}}\ m\ {\rm{even}};\\ m-\epsilon+3, & \ {\rm{if}} \ \epsilon = 3, 4, \dots, m+2\ {\rm{and}}\ m\ {\rm{odd}};\\ \epsilon-m-1, & \ {\rm{if}} \ \epsilon = m+3, \dots, 2m+1\ {\rm{and}}\ m\ {\rm{odd}};\\ 3m+2-\epsilon, & \ {\rm{if}} \ \epsilon = 2m+2, \dots, 3m+2\ {\rm{and}}\ m\ {\rm{odd}};\\ \epsilon-3m-2, & \ {\rm{if}} \ \epsilon = 3m+3, \dots, 4m\ {\rm{and}}\ m\ {\rm{odd}}. \end{cases}$

All the edges and vertices with respect to the selected mixed metric generator $Q_m$ have different representation, it is proved that $dim_m(HML_m)\leq4.$ Now, assume on the contrary that $dim_m(HML_m) = 3,$ which makes the minimum members in a mixed metric generator $Q_{m}^{'}$ are three, given below are few deliberation on the behalf of argument.

If all three vertices have a disparity of every arbitrary length, $Q_{m}^{'} = \{v_{i}, v_j, v_k\}$ with $1\leq$ $i, j, k$ $\leq\; 2m-1$ , then it denotes the same representation in the edges that are $d\left(v_1v_{2m+1}|Q_{m}^{'}\right) = d\left(v_1v_{4m}|Q_{m}^{'}\right)$ and including $i, j, k = 2m, 2m+1$ then the same representation in a vertex and an edge, $d\left(v_1|Q_{m}^{'}\right) = d\left(v_1v_{4m}|Q_{m}^{'}\right).$ When $2m+2\leq i, j, k\leq 4m-1$ , then it implies that the same representation in the edges which are $d\left(v_1v_{2m+1}|Q_{m}^{'}\right) = d\left(v_{2m}v_{2m+1}|Q_{m}^{'}\right).$ To generalize this concept, we choose the vertices $v_{i}, v_j, v_k\in V(HML_m),$ then it follows that the same representations in the edges which are either $d\left(v_{p}v_{p+1}|Q_{m}^{'}\right) = d\left(v_{2m+p}v_{2m+p+1}|Q_{m}^{'}\right)$ where $1\leq p\leq 2m$ or $d\left(v_{1}|Q_{m}^{'}\right) = d\left(v_{1}v_{2m+1}|Q_{m}^{'}\right)$ or $d\left(v_{2}|Q_{m}^{'}\right) = d\left(v_2v_{3}|Q_{m}^{'}\right)$ or $d\left(v_{3}|Q_{m}^{'}\right) = d\left(v_3v_{4}|Q_{m}^{'}\right)$ or $d\left(v_{i}|Q_{m}^{'}\right) = d\left(v_{i}v_{m+i}|Q_{m}^{'}\right)$ where $1\leq i\ ({\rm{odd}})\leq 2m-1$ . Thus, it is concluded that we can not take vertices in a mixed metric generator with cardinality three. Therefore, $dim_m(HML_m) = 4.$

In ^[11], it is discussed that the edge metric dimension of triangular ladder network is four. By the relation minimum mixed metric dimension can be four and it is proved in this part.

Theorem 2.3. Let $TL_m$ be a triangular ladder network with $m\geq 3.$ Then

$dim_m(TL_m) = 4.$

Proof. Suppose that the mixed metric generator $Q_m = \{v_{1}, v_{2}, v_{2m-1}, v_{2m}\}.$ In Figure 3 labeling is defined.

Figure 3. Triangular Ladder Graph

$TL_m$ .

DownLoad: Full-Size Img PowerPoint

To prove that $dim_m(TL_m) = 4,$ we first prove that $dim_m(TL_m)\leq4.$ The shortest distances of all the edges of our network according to the chosen mixed metric generator are given as follows:

$d(v_{\epsilon}v_{\epsilon+1}, v_1) = \begin{cases}{} \frac{\epsilon-1}{2}, & \ {\rm{if}} \ 1\leq{\epsilon}\ ( {\rm{odd}}) \leq 2m-1;\\ \frac{\epsilon+2}{2}, & \ {\rm{if}} \ 2\leq{\epsilon}\ ( {\rm{even}}) \leq 2m-2. \end{cases}$

$\ d(v_{\epsilon}v_{\epsilon+1}, v_1) = \begin{cases}{} \frac{\epsilon-1}{2}, & \ {\rm{if}} \ 1\leq{\epsilon}\ ( {\rm{odd}}) \leq 2m-3;\\ \frac{\epsilon}{2}, & \ {\rm{if}} \ 2\leq{\epsilon}\ ( {\rm{even}}) \leq 2m-2. \end{cases}$

$d(v_{\epsilon}v_{\epsilon+1}, v_2) = \begin{cases}{} \frac{\epsilon-1}{2}, & \ {\rm{if}} \ 1\leq{\epsilon}\ ( {\rm{odd}}) \leq 2m-1;\\ \frac{\epsilon-2}{2}, & \ {\rm{if}} \ 2\leq{\epsilon}\ ( {\rm{even}}) \leq 2m-2. \end{cases}$

$d(v_{\epsilon}v_{\epsilon+2}, v_2) = \begin{cases}{} 1, & \ {\rm{if}} \ \epsilon = 1;\\ \frac{\epsilon-1}{2}, & \ {\rm{if}} \ 1\leq{\epsilon}\ ( {\rm{odd}}) \leq 2m-3;\\ \frac{\epsilon-2}{2}, & \ {\rm{if}} \ 2\leq{\epsilon}\ ( {\rm{even}}) \leq 2m-2. \end{cases}$

$d(v_{\epsilon}v_{\epsilon+1}, v_{2m-1}) = \begin{cases}{} \frac{2m-\epsilon-1}{2}, & \ {\rm{if}} \ 1\leq{\epsilon}\ ( {\rm{odd}}) \leq 2m-1;\\ \frac{2m-\epsilon-2}{2}, & \ {\rm{if}} \ 2\leq{\epsilon}\ ( {\rm{even}}) \leq 2m-2.\\ \end{cases}$

$\ d(v_{\epsilon}v_{\epsilon+2}, v_{2m-1}) = \begin{cases}{} \frac{2m-\epsilon-1}{2}, & \ {\rm{if}} \ 1\leq{\epsilon}\ ( {\rm{odd}}) \leq 2m-3;\\ \frac{2m-\epsilon}{2}, & \ {\rm{if}} \ 2\leq{\epsilon}\ ( {\rm{even}}) \leq 2m-2. \end{cases}$

$d(v_{\epsilon}v_{\epsilon+1}, v_{2m}) = \begin{cases}{} \frac{2m-\epsilon-1}{2}, & \ {\rm{if}} \ 1\leq{\epsilon}\ ( {\rm{odd}}) \leq 2m-1;\\ \frac{2m-\epsilon+1}{2}, & \ {\rm{if}} \ 2\leq{\epsilon}\ ( {\rm{even}}) \leq 2m-2.\\ \end{cases}$

$\ d(v_{\epsilon}v_{\epsilon+2}, v_{2m}) = \begin{cases}{} \frac{2m-\epsilon+1}{2}, & \ {\rm{if}} \ 1\leq{\epsilon}\ ( {\rm{odd}}) \leq 2m-3;\\ \frac{2m-\epsilon-2}{2}, & \ {\rm{if}} \ 2\leq{\epsilon}\ ( {\rm{even}}) \leq 2m-2.\\ \end{cases}$

The representations of edges is unique with respect to the mixed metric generator. To complete the proof, we need to check the distances of vertices with respect to chosen mixed metric generator:

$d(v_{\epsilon}, v_{1}) = \left\lfloor\frac{\epsilon}{2}\right\rfloor,$

$d(v_{\epsilon}, v_{2}) = \begin{cases}{} \left\lfloor\frac{1}{\epsilon}\right\rfloor , & \ {\rm{if}} \ \epsilon = 1, 2;\\ \left\lfloor\frac{\epsilon-1}{2}\right\rfloor , & \ {\rm{if}} \ \epsilon = 3, 4, \dots, 2m. \end{cases}$

$d(v_{\epsilon}, v_{2m-1}) = \begin{cases}{} m-1-\left\lfloor\frac{\epsilon-1}{2}\right\rfloor, & \ {\rm{if}} \ \epsilon = 1, 2, \dots, 2m-1;\\ 1, & \ {\rm{if}} \ \epsilon = 2m. \end{cases}$

$d(v_{\epsilon}, v_{2m}) = m-\left\lfloor\frac{\epsilon}{2}\right\rfloor.$

The representations of all edges and vertices with respect to the mixed metric generator $Q_m$ are easily visible, it is proved that $dim_m(TL_m)\leq4.$ Now, for $dim_m(TL_m)\geq4$ , the proof can proceed similar to the proof of Theorem 2.1, as a result, we omit it.

Theorem 2.4. Let $TML_m$ be a triangular Möbius ladder network with $m\geq 5.$ Then

$dim_m(TML_m) = 5.$

Proof. Consider the mixed metric generator $Q_m = \{v_1, v_{2}, v_{4}, v_{5}, v_{6}\},$ when $m = 5,$ $Q_m = \{v_1, v_{3}, v_{4}, v_{m+1}, v_{m+2}\},$ when $m\geq6\ ({\rm{even}})$ and $Q_m = \{v_1, v_{4}, v_{5}, v_{m+3}, v_{m+4}\},$ when $m\geq7\ ({\rm{odd}}).$ The labeling of vertices is shown in Figure 4.

Figure 4. Triangonal Möbius Ladder Graph

$TML_m$ .

DownLoad: Full-Size Img PowerPoint

To prove that $dim_m(TML_m) = 5,$ we prove that $dim_m(TML_m)\leq5.$ The distances of all edges according to mixed metric generator are given below:

When $m = 5,$ the distance of the three vertices $v_1, v_4, v_5$ to all edges will be explained in later parts of proof.

$d(v_{\epsilon}v_{\epsilon+1}, v_{2}) = \begin{cases} \frac{\epsilon-1}{2}; & \ {\rm{if}} \ \epsilon = 1, 3, 5\\ 1; & \ {\rm{if}} \ \epsilon = 4, 6, 7\\ 0; & \ {\rm{if}} \ \epsilon = 2.\\ \end{cases}$

$d(v_{2m-2}v_2, v_{2}) = d(v_{2m-3}v_2, v_{2}) = 0.$

$d(v_{\epsilon}v_{\epsilon+2}, v_{2}) = \begin{cases} 1; & \ {\rm{if}} \ \epsilon = 1, 5\\ 2; & \ {\rm{if}} \ \epsilon = 3\\ \frac{\epsilon-2}{2}, & \ {\rm{if}} \ \epsilon = 2, 4, 6.\\ \end{cases}$

$d(v_{2m-1}v_1, v_{2}) = d(v_{2m-2}v_2, v_{6}) = d(v_{2m-1}v_1, v_{6}) = 1$

$d(v_{\epsilon}v_{\epsilon+1}, v_{6}) = \begin{cases} \left|\frac{5-\epsilon}{2}\right|, & \ {\rm{if}} \ 1\leq{\epsilon}\ ( {\rm{odd}}) \leq 2m-3;\\ \left|\frac{6-\epsilon}{2}\right|, & \ {\rm{if}} \ 2\leq{\epsilon}\ ( {\rm{even}}) \leq 2m-4. \end{cases}$

$d(v_{\epsilon}v_{\epsilon+2}, v_{6}) = \begin{cases} 2; & \ {\rm{if}}\ \epsilon = 1;\\ 1; & \ {\rm{if}}\ \epsilon = 2, 3, 5;\\ 0; & \ {\rm{if}} \ \epsilon = 4, 6.\\ \end{cases}$

$d(v_{2m-3}v_2, v_{6}) = 2.$

When $m\geq6 \ ({\rm{even}})$ , we obtain the following:

$d(v_{\epsilon}v_{\epsilon+1}, v_{1}) = \begin{cases} \frac{\epsilon-2}{2}; & \ {\rm{if}} \ \epsilon = 1, 3, \dots, m-1\\ \frac{2m-\epsilon-3}{2}; & \ {\rm{if}} \ \epsilon = m+1, m+3, \dots, 2m-3\\ \frac{\epsilon}{2}; & \ {\rm{if}} \ \epsilon = 2, 4, \dots, m\\ \frac{2m-\epsilon}{2}; & \ {\rm{if}} \ \epsilon = m+2, m+4, \dots, 2m-4.\\ \end{cases}$

$d(v_{2m-2}v_2, v_{1}) = d(v_{2m-3}v_2, v_{1}) = 1.$

$d(v_{\epsilon}v_{\epsilon+2}, v_{1}) = \begin{cases} \frac{\epsilon-2}{2}; & \ {\rm{if}} \ \epsilon = 1, 3, \dots, m-1\\ \frac{2m-\epsilon-1}{2}; & \ {\rm{if}} \ \epsilon = m+1, m+3, \dots, 2m-5\\ \frac{\epsilon}{2}; & \ {\rm{if}} \ \epsilon = 2, 4, \dots, m-2\\ \frac{2m-\epsilon-2}{2}; & \ {\rm{if}} \ \epsilon = m, m+2, \dots, 2m-4.\\ \end{cases}$

$d(v_{2m-1}v_{1}, v_{1}) = 0.$

$d(v_{\epsilon}v_{\epsilon+1}, v_{3}) = \begin{cases} 1; & \ {\rm{if}} \ \epsilon = 1\\ \frac{\epsilon-3}{2}; & \ {\rm{if}} \ \epsilon = 3, 5, \dots, m+3\\ \frac{2m+3-\epsilon}{2}; & \ {\rm{if}} \ \epsilon = m+5, m+7, \dots, 2m-3\\ \frac{\epsilon-2}{2}; & \ {\rm{if}} \ \epsilon = 2, 4, \dots, m+2\\ \frac{2m+2-\epsilon}{2}; & \ {\rm{if}} \ \epsilon = m+4, m+6, \dots, 2m-4.\\ \end{cases}$

$d(v_{2m-2}v_2, v_{3}) = d(v_{2m-3}v_2, v_{3}) = d(v_{2m-1}v_1, v_{3}) = 2.$

$d(v_{\epsilon}v_{\epsilon+2}, v_{3}) = \begin{cases} \epsilon-1; & \ {\rm{if}} \ \epsilon = 1, 2\\ \frac{\epsilon-3}{2}; & \ {\rm{if}} \ \epsilon = 3, 5, \dots, m+1\\ \frac{2m+1-\epsilon}{2}; & \ {\rm{if}} \ \epsilon = m+3, m+5, \dots, 2m-5\\ \frac{\epsilon-2}{2}; & \ {\rm{if}} \ \epsilon = 4, 6, \dots, m+2\\ \frac{2m+2-\epsilon}{2}; & \ {\rm{if}} \ \epsilon = m+4, m+6, \dots, 2m-4.\\ \end{cases}$

$d(v_{\epsilon}v_{\epsilon+1}, v_{4}) = \begin{cases} 1; & \ {\rm{if}} \ \epsilon = 1, 2\\ \frac{\epsilon-3}{2}; & \ {\rm{if}} \ \epsilon = 3, 5, \dots, m+3\\ \frac{2m-\epsilon+3}{2}; & \ {\rm{if}} \ \epsilon = m+5, m+7\dots, 2m-3\\ \frac{\epsilon-4}{2}; & \ {\rm{if}} \ \epsilon = 4, 6, \dots, m+2\\ \frac{2m-\epsilon+2}{2}; & \ {\rm{if}} \ \epsilon = m+4, m+6, \dots, 2m-4.\\ \end{cases}$

$d(v_{2m-2}v_2, v_{4}) = d(v_{2m-3}v_2, v_{4}) = d(v_{2m-1}v_1, v_{4}) = 2.$

$d(v_{\epsilon}v_{\epsilon+2}, v_{4}) = \begin{cases} 2-\epsilon; & \ {\rm{if}} \ \epsilon = 1, 2\\ \frac{\epsilon-1}{2}; & \ {\rm{if}} \ \epsilon = 3, 5, \dots, m+1\\ \frac{2m-\epsilon+1}{2}; & \ {\rm{if}} \ \epsilon = m+3, m+5, \dots, 2m-5\\ \frac{\epsilon-4}{2}; & \ {\rm{if}} \ \epsilon = 4, 6, \dots, m+2\\ \frac{2m-\epsilon+2}{2}; & \ {\rm{if}} \ \epsilon = m+4, m+6\dots, 2m-4.\\ \end{cases}$

$d(v_{\epsilon}v_{\epsilon+1}, v_{m+1}) = \begin{cases} \frac{m-2}{2}; & \ {\rm{if}} \ \epsilon = 1\\ \left|\frac{m-\epsilon+1}{2}\right|; & \ {\rm{if}} \ \epsilon = 3, 5, \dots, 2m-3\\ \left|\frac{m-\epsilon}{2}\right|; & \ {\rm{if}} \ \epsilon = 2, 4, \dots, 2m-4.\\ \end{cases}$

$d(v_{2m-2}v_2, v_{m+1}) = d(v_{2m-1}v_1, v_{m+1}) = \frac{m-2}{2}, \ d(v_{2m-3}v_2, v_{m+1}) = \frac{m-4}{2}.$

$d(v_{\epsilon}v_{\epsilon+2}, v_{m+1}) = \begin{cases} \frac{m-\epsilon-1}{2}; & \ {\rm{if}} \ \epsilon = 1, 3, \dots, m-1\\ \frac{\epsilon-m-1}{2}; & \ {\rm{if}} \ \epsilon = m+1, m+3, \dots, 2m-5\\ \frac{m-\epsilon-1}{2}; & \ {\rm{if}} \ \epsilon = 2, 4, \dots, m-2\\ 1; & \ {\rm{if}} \ \epsilon = m\\ \frac{\epsilon-m}{2}; & \ {\rm{if}} \ \epsilon = m+2, m+4, \dots, 2m-4.\\ \end{cases}$

$d(v_{\epsilon}v_{\epsilon+1}, v_{m+2}) = \begin{cases} \frac{m-2}{2}; & \ {\rm{if}} \ \epsilon = 1\\ \left|\frac{m-\epsilon+1}{2}\right|; & \ {\rm{if}} \ \epsilon = 3, 5, \dots, 2m-3\\ \left|\frac{m-\epsilon-2}{2}\right|; & \ {\rm{if}} \ \epsilon = 2, 4, \dots, 2m-4. \end{cases}$

$d(v_{2m-2}v_2, v_{m+2}) = d(v_{2m-3}v_2, v_{m+2}) = d(v_{2m-1}v_1, v_{m+2}) = \frac{m-4}{2}.$

$d(v_{\epsilon}v_{\epsilon+2}, v_{m+2}) = \begin{cases} \frac{m-\epsilon+1}{2}; & \ {\rm{if}} \ \epsilon = 1, 3, \dots, m-1\\ \frac{m-\epsilon+3}{2}; & \ {\rm{if}} \ \epsilon = m+1, m+3, \dots, 2m-5\\ \frac{m-\epsilon}{2}; & \ {\rm{if}} \ \epsilon = 2, 4, \dots, m\\ \frac{\epsilon-m-2}{2}; & \ {\rm{if}} \ \epsilon = m+2, m+4, \dots, 2m-4.\\ \end{cases}$

When $m\geq7 \ ({\rm{odd}})$ , we have

$d(v_{\epsilon}v_{\epsilon+1}, v_{1}) = \begin{cases} \frac{\epsilon-1}{2}; & \ {\rm{if}} \ \epsilon = 1, 3, \dots, m\\ \frac{2m-\epsilon-1}{2}; & \ {\rm{if}} \ \epsilon = m+2, m+4, \dots, 2m-3\\ \frac{\epsilon}{2}; & \ {\rm{if}} \ \epsilon = 2, 4, \dots, m-1\\ \frac{2m-\epsilon}{2}; & \ {\rm{if}} \ \epsilon = m+1, m+3, \dots, 2m-4.\\ \end{cases}$

$d(v_{2m-2}v_2, v_{1}) = d(v_{2m-3}v_2, v_{1}) = 1, \ d(v_{2m-1}v_1, v_{1}) = 0.$

$d(v_{\epsilon}v_{\epsilon+2}, v_{1}) = \begin{cases} \frac{\epsilon-1}{2}; & \ {\rm{if}} \ \epsilon = 1, 3, \dots, m\\ \frac{2m-\epsilon-1}{2}; & \ {\rm{if}} \ \epsilon = m+2, m+4, \dots, 2m-5\\ \frac{\epsilon}{2}; & \ {\rm{if}} \ \epsilon = 2, 4, \dots, m-1\\ \frac{2m-\epsilon-2}{2}; & \ {\rm{if}} \ \epsilon = m+1, m+3, \dots, 2m-4.\\ \end{cases}$

$d(v_{\epsilon}v_{\epsilon+1}, v_{4}) = \begin{cases} 1; & \ {\rm{if}} \ \epsilon = 1, 2\\ \frac{\epsilon-3}{2}; & \ {\rm{if}} \ \epsilon = 3, 5, \dots, m+2\\ \frac{2m-\epsilon+3}{2}; & \ {\rm{if}} \ \epsilon = m+4, m+6, \dots, 2m-3\\ \frac{\epsilon-4}{2}; & \ {\rm{if}} \ \epsilon = 4, 6, \dots, m+3\\ \frac{2m-\epsilon+4}{2}; & \ {\rm{if}} \ \epsilon = m+5, m+6, \dots, 2m-5.\\ \end{cases}$

$d(v_{2m-2}v_2, v_{4}) = d(v_{2m-3}v_2, v_{4}) = d(v_{2m-1}v_1, v_{4}) = 2.$

$d(v_{\epsilon}v_{\epsilon+2}, v_{4}) = \begin{cases} 2-\epsilon; & \ {\rm{if}} \ \epsilon = 1, 2\\ \frac{\epsilon-1}{2}; & \ {\rm{if}} \ \epsilon = 3, 5, \dots, m\\ \frac{2m-\epsilon+1}{2}; & \ {\rm{if}} \ \epsilon = m+2, m+4, \dots, 2m-5\\ \frac{\epsilon-4}{2}; & \ {\rm{if}} \ \epsilon = 4, 6, \dots, m+3\\ \frac{2m-\epsilon+2}{2}; & \ {\rm{if}} \ \epsilon = m+5, m+7, \dots, 2m-4.\\ \end{cases}$

$d(v_{\epsilon}v_{\epsilon+1}, v_{5}) = \begin{cases} \left|\frac{5-\epsilon}{2}\right|; & \ {\rm{if}} \ \epsilon = 1, 3, \dots, m+4\\ \frac{2m-\epsilon+5}{2}; & \ {\rm{if}} \ \epsilon = m+6, m+8, \dots, 2m-3\\ \frac{4-\epsilon}{2}; & \ {\rm{if}} \ \epsilon = 2, 4\\ \frac{\epsilon-4}{2}; & \ {\rm{if}} \ \epsilon = 6, 8, \dots, m+5\\ \frac{2m-\epsilon+3}{2}; & \ {\rm{if}} \ \epsilon = m+7, m+9, \dots, 2m-4.\\ \end{cases}$

$d(v_{2m-2}v_2, v_{5}) = d(v_{2m-3}v_2, v_{5}) = d(v_{2m-1}v_1, v_{5}) = 3.$

$d(v_{\epsilon}v_{\epsilon+2}, v_{5}) = \begin{cases} \frac{3-\epsilon}{2}; & \ {\rm{if}} \ \epsilon = 1, 3\\ \frac{\epsilon-5}{2}; & \ {\rm{if}} \ \epsilon = 5, 7, \dots, m+4\\ \frac{2m-\epsilon+3}{2}; & \ {\rm{if}} \ \epsilon = m+6, m+8, \dots, 2m-5\\ 1; & \ {\rm{if}} \ \epsilon = 2, 4\\ \frac{6-\epsilon}{2}; & \ {\rm{if}} \ \epsilon = 6, 8, \dots, m+2\\ \frac{2m-\epsilon+3}{2}; & \ {\rm{if}} \ \epsilon = m+4, m+6, \dots, 2m-4.\\ \end{cases}$

$d(v_{\epsilon}v_{\epsilon+1}, v_{m+3}) = \begin{cases} \frac{m-3}{2}; & \ {\rm{if}} \ \epsilon = 1\\ \left|\frac{m-\epsilon+2}{2}\right|; & \ {\rm{if}} \ \epsilon = 3, 5, \dots, 2m-3\\ \frac{m-1}{2}; & \ {\rm{if}} \ \epsilon = 2\\ \left|\frac{m-\epsilon+3}{2}\right|; & \ {\rm{if}} \ \epsilon = 4, 6\dots, 2m-4.\\ \end{cases}$

$d(v_{2m-2}v_2, v_{m+3}) = d(v_{2m-1}v_1, v_{m+3}) = \frac{m-5}{2}, \ d(v_{2m-3}v_2, v_{m+3}) = \frac{m-3}{2}.$

$d(v_{\epsilon}v_{\epsilon+2}, v_{m+3}) = \begin{cases} \frac{m-1}{2}; & \ {\rm{if}} \ \epsilon = 1\\ \frac{m-\epsilon+2}{2}; & \ {\rm{if}} \ \epsilon = 3, 5, \dots, m\\ 1; & \ {\rm{if}} \ \epsilon = m+2\\ \frac{\epsilon-m-2}{2}; & \ {\rm{if}} \ \epsilon = m+4, m+6, \dots, 2m-5\\ \frac{m-\epsilon+1}{2}; & \ {\rm{if}} \ \epsilon = 2, 4, \dots, m+1\\ \frac{\epsilon-m-3}{2}; & \ {\rm{if}} \ \epsilon = m+3, m+5, \dots, 2m-4.\\ \end{cases}$

$d(v_{\epsilon}v_{\epsilon+1}, v_{m+4}) = \begin{cases} \frac{m-5}{2}; & \ {\rm{if}} \ \epsilon = 1\\ \frac{m-1}{2}; & \ {\rm{if}} \ \epsilon = 3\\ \left|\frac{m-\epsilon+4}{2}\right|; & \ {\rm{if}} \ \epsilon = 5, 7, \dots, 2m-3\\ \frac{m-3}{2}; & \ {\rm{if}} \ \epsilon = 2\\ \left|\frac{m-\epsilon+3}{2}\right|; & \ {\rm{if}}\ \epsilon = 4, 6, \dots, 2m-4.\\ \end{cases}$

$d(v_{2m-2}v_2, v_{m+4}) = d(v_{2m-1}v_1, v_{m+4}) = \frac{m-5}{2}, \ d(v_{2m-3}v_2, v_{m+4}) = \frac{m-7}{2}.$

$d(v_{\epsilon}v_{\epsilon+2}, v_{m+4}) = \begin{cases} \frac{m-\epsilon-2}{2}; & \ {\rm{if}} \ \epsilon = 1, 3\\ \frac{m-\epsilon+4}{2}; & \ {\rm{if}} \ \epsilon = 5, 7, \dots, m+2\\ \frac{\epsilon-m-4}{2}; & \ {\rm{if}} \ \epsilon = m+4, m+6, \dots, 2m-5\\ \frac{m-3}{2}; & \ {\rm{if}} \ \epsilon = 2\\ \frac{m-\epsilon+3}{2}; & \ {\rm{if}} \ \epsilon = 4, 6, \dots, m+1\\ 1; & \ {\rm{if}} \ \epsilon = m+3\\ \frac{\epsilon-m-3}{2}; & \ {\rm{if}} \ \epsilon = m+5, m+7, \dots, 2m-4.\\ \end{cases}$

Now, the distances of vertices according to mixed metric generator are classified as follows:

When $m = 5,$ the distance of the vertices $v_1, v_4, v_5$ to all other vertices will be given in later parts of proof.

$d(v_{\epsilon}, v_{2}) = \begin{cases} 1; & \ {\rm{if}} \ \epsilon = 1, 3, 4, 7, 8\\ 2; & \ {\rm{if}} \ \epsilon = 5, 6\\ 0; & \ {\rm{if}} \ \epsilon = 2.\\ \end{cases}$

$d(v_{\epsilon}, v_{6}) = \begin{cases} 2; & \ {\rm{if}} \ \epsilon = 1, 2, 3\\ 1; & \ {\rm{if}} \ \epsilon = 4, 5, 7, 8\\ 0; & \ {\rm{if}} \ \epsilon = 6.\\ \end{cases}$

When $m\geq6 \ ({\rm{even}})$ , then

$d(v_{\epsilon}, v_{1}) = \begin{cases} \frac{\epsilon-1}{2}; & \ {\rm{if}} \ \epsilon = 1, 3, \dots, m+1\\ \frac{2m-\epsilon+1}{2}; & \ {\rm{if}} \ \epsilon = m+3, m+5, \dots, 2m-3\\ \frac{\epsilon}{2}; & \ {\rm{if}} \ \epsilon = 2, 4, \dots, m\\ \frac{2m-\epsilon}{2}; & \ {\rm{if}} \ \epsilon = m+2, m+4, \dots, 2m-2.\\ \end{cases}$

$d(v_{\epsilon}, v_{3}) = \begin{cases} 1; & \ {\rm{if}} \ \epsilon = 1, 2\\ \frac{\epsilon-3}{2}; & \ {\rm{if}} \ \epsilon = 3, 5, \dots, m+3\\ \frac{2m-\epsilon+3}{2}; & \ {\rm{if}} \ \epsilon = m+5, m+7, \dots, 2m-3\\ \frac{\epsilon-2}{2}; & \ {\rm{if}} \ \epsilon = 4, 6, \dots, m+2\\ \frac{2m-\epsilon+2}{2}; & \ {\rm{if}} \ \epsilon = m+4, m+6, \dots, 2m-2.\\ \end{cases}$

$d(v_{\epsilon}, v_{4}) = \begin{cases} \frac{5-\epsilon}{2}; & \ {\rm{if}} \ \epsilon = 1, 3\\ \frac{\epsilon-3}{2}; & \ {\rm{if}} \ \epsilon = 5, 7, \dots, m+1\\ \frac{2m-\epsilon+1}{2}; & \ {\rm{if}} \ \epsilon = m+3, m+5, \dots, 2m-3\\ \left|\frac{4-\epsilon}{2}\right|; & \ {\rm{if}} \ \epsilon = 2, 4, \dots, m+4\\ \frac{2m-\epsilon+4}{2}; & \ {\rm{if}} \ \epsilon = m+6, m+8, \dots, 2m-2.\\ \end{cases}$

$d(v_{\epsilon}, v_{m+1}) = \begin{cases} \left|\frac{m-\epsilon+1}{2}\right|; & \ {\rm{if}} \ \epsilon = 1, 3, \dots, 2m-3\\ \frac{m-\epsilon+2}{2}; & \ {\rm{if}} \ \epsilon = 2, 4, \dots, m\\ \frac{\epsilon-m}{2}; & \ {\rm{if}} \ \epsilon = m+2, m+4, \dots, 2m-2.\\ \end{cases}$

$d(v_{\epsilon}, v_{m+2}) = \begin{cases} \frac{m-2}{2}; & \ {\rm{if}} \ \epsilon = 1\\ \frac{m-\epsilon+3}{2}; & \ {\rm{if}} \ \epsilon = 3, 5, \dots, m+1\\ \frac{\epsilon-m}{2}; & \ {\rm{if}} \ \epsilon = m+2, m+4, \dots, 2m-3\\ \frac{m}{2}; & \ {\rm{if}} \ \epsilon = 2\\ \left|\frac{m-\epsilon+2}{2}\right|; & \ {\rm{if}} \ \epsilon = 4, 6, \dots, 2m-2.\\ \end{cases}$

When $m\geq7 \ ({\rm{odd}})$

$d(v_{\epsilon}, v_{1}) = \begin{cases} \frac{\epsilon-1}{2}; & \ {\rm{if}} \ \epsilon = 1, 3, \dots, m\\ \frac{2m-\epsilon+1}{2}; & \ {\rm{if}} \ \epsilon = m+2, m+4, \dots, 2m-3\\ \frac{\epsilon}{2}; & \ {\rm{if}} \ \epsilon = 2, 4, \dots, m-1\\ \frac{2m-\epsilon}{2}; & \ {\rm{if}} \ \epsilon = m+1, m+3, \dots, 2m-2.\\ \end{cases}$

$d(v_{\epsilon}, v_{4}) = \begin{cases} \frac{5-\epsilon}{2}; & \ {\rm{if}} \ \epsilon = 1, 3\\ \frac{\epsilon-3}{2}; & \ {\rm{if}} \ \epsilon = 5, 7, \dots, m+2\\ \frac{2m-\epsilon+1}{2}; & \ {\rm{if}} \ \epsilon = m+4, m+6, \dots, 2m-3\\ \left|\frac{4-\epsilon}{2}\right|; & \ {\rm{if}} \ \epsilon = 2, 4, \dots, m+3\\ \frac{2m-\epsilon+6}{2}; & \ {\rm{if}} \ \epsilon = m+5, m+7, \dots, 2m-2.\\ \end{cases}$

$d(v_{\epsilon}, v_{5}) = \begin{cases} \left|\frac{5-\epsilon}{2}\right|; & \ {\rm{if}} \ \epsilon = 1, 3, \dots, m+4\\ \frac{2m-\epsilon+7}{2}; & \ {\rm{if}} \ \epsilon = m+6, m+8, \dots, 2m-3\\ \frac{6-\epsilon}{2}; & \ {\rm{if}} \ \epsilon = 2, 4\\ \frac{\epsilon-4}{2}; & \ {\rm{if}} \ \epsilon = 6, 8, \dots, m+3\\ \frac{2m-\epsilon+4}{2}; & \ {\rm{if}} \ \epsilon = m+5, m+7, \dots, 2m-2.\\ \end{cases}$

$d(v_{\epsilon}, v_{m+3}) = \begin{cases} \frac{m+\epsilon-4}{2}; & \ {\rm{if}} \ \epsilon = 1, 3\\ \frac{m-\epsilon+4}{2}; & \ {\rm{if}} \ \epsilon = 5, 7, \dots, m+2\\ \frac{\epsilon-m-2}{2}; & \ {\rm{if}} \ \epsilon = m+4, m+6, \dots, 2m-3\\ \frac{m-3}{2}; & \ {\rm{if}} \ \epsilon = 2\\ \left|\frac{m-\epsilon+3}{2}\right|; & \ {\rm{if}} \ \epsilon = 4, 6, \dots, 2m-2.\\ \end{cases}$

$d(v_{\epsilon}, v_{m+4}) = \begin{cases} \frac{m-3}{2}; & \ {\rm{if}} \ \epsilon = 1, 3\\ \left|\frac{m-\epsilon+4}{2}\right|; & \ {\rm{if}} \ \epsilon = 5, 7, \dots, 2m-3\\ \frac{\epsilon+m-7}{2}; & \ {\rm{if}} \ \epsilon = 2, 4\\ \frac{m-\epsilon+5}{2}; & \ {\rm{if}} \ \epsilon = 6, 8, \dots, m+3\\ \frac{\epsilon-m-3}{2}; & \ {\rm{if}} \ \epsilon = m+5, m+7, \dots, 2m-2.\\ \end{cases}$

In the previously described distances, there are no two elements (vertices and edges) have same representations under the mixed metric generator $Q_m.$ It is shown that $dim_m(TML_m)\leq5.$ Now, a brief discussion for $dim_m(TML_m) = 4,$ which makes the minimum number of elements of the chosen mixed metric generator $Q_{m}^{'}$ equal to four.

If all four nodes have a gap of every arbitrary length, we can choose like, $Q_{m}^{'} = \{v_{i}, v_j, v_k, v_l\}$ with $1\leq i, j, k, l\leq 2m-2.$ Then it concludes to the identical representation in the edges which are $d\left(v_1v_{2}|Q_{m}^{'}\right) = d\left(v_1v_{2m-1}|Q_{m}^{'}\right)$ where indices are odd and if indices are even then the identical representation in edge either $d\left(v_1v_2|Q_{m}^{'}\right) = d\left(v_2v_{3}|Q_{m}^{'}\right)$ or $d\left(v_pv_{p+1}|Q_{m}^{'}\right) = d\left(v_qv_{q+1}|Q_{m}^{'}\right)$ where $1\leq p\ ({\rm{odd}})\leq 2m-3$ and $1\leq q\ ({\rm{even}})\leq 2m-4$ , to generalize this concept and taking vertices $v_{i}, v_j, v_k, v_l\in V(TML_m)$ then it implies to the identical representations in the edges which are either $d\left(v_pv_{p+1}|Q_{m}^{'}\right) = d\left(v_qv_{q+1}|Q_{m}^{'}\right)$ where $1\leq p\ ({\rm{odd}})\leq 2m-3$ and $1\leq q\ ({\rm{even}})\leq 2m-4$ or $d\left(v_{i}|Q_{m}^{'}\right) = d\left(v_pv_{p+2}|Q_{m}^{'}\right),$ where $i = {\rm{even}}$ and $2\leq p\ ({\rm{even}})\leq 2m-4$ or $d\left(v_{i}|Q_{m}^{'}\right) = d\left(v_pv_{p+2}|Q_{m}^{'}\right)$ where $i = {\rm{odd}}$ and $1\leq p\ ({\rm{odd}})\leq 2m-5,$ then it is concluded that we can not take vertices in mixed metric generator with cardinality four. Thus,

$dim_m(TML_m) = 5.$

The edge metric dimension of a ladder network is two. By the relation mixed metric dimension can be two or more. In the following result, we show that the mixed metric dimension of ladder network is indeed three.

Theorem 2.5. Let $L_m$ be a ladder network with $m\geq 2.$ Then

$dim_m(L_m) = 3.$

Proof. Suppose that the mixed metric generator is $Q_m = \{v_{1}, v_{2}, v_{2m-1}\}.$ The labeling of vertices is shown in Figure 5.

Figure 5. Ladder Graph

$L_m$ .

DownLoad: Full-Size Img PowerPoint

To prove that $dim_m(L_m) = 3,$ we use the method of the double inequality. First, we verify that $dim_m(L_m)\leq3$ by moving towards to show the distances of all edges according to mixed metric generator:

$d\left(v_{\epsilon}v_{\epsilon+1}|Q_m\right) = \left(\frac{\epsilon-1}{2}, \; \frac{\epsilon-1}{2}, \; m-1-\frac{\epsilon-1}{2}\right);\ \ \ 1\leq{\epsilon}\ ( {\rm{odd}})\leq 2m-1$

$d\left(v_{\epsilon}v_{\epsilon+2}|Q_m\right) = \left(\frac{\epsilon-1}{2}, \; \frac{\epsilon+1}{2}, \; m-2-\frac{\epsilon-1}{2}\right);\ \ \ 1\leq{\epsilon}\ ( {\rm{odd}})\leq 2m-3$

$d\left(v_{\epsilon}v_{\epsilon+1}|Q_m\right) = \left(\frac{\epsilon}{2}, \; \frac{\epsilon-2}{2}, \; m-\frac{\epsilon}{2}\right);\ \ \ 2\leq{\epsilon}\ ( {\rm{even}})\leq 2m-2$

Vertices representations according to selected mixed metric generator;

$d\left(v_{\epsilon}|Q_m\right) = \left\{ \begin{array}{ll} \left(\frac{\epsilon-1}{2}, \; \frac{\epsilon+1}{2}, \; m-\frac{\epsilon+1}{2}\right) &\ \ {\rm{if}}\ \epsilon = 1, 3, \dots, 2m-1;\\ \left(\frac{\epsilon}{2}, \; \frac{\epsilon-2}{2}, \; m-\frac{\epsilon-2}{2}\right) &\ \ {\rm{if}}\ \epsilon = 2, 4, \dots, 2m. \end{array} \right.$

It is easy to see that the distances between all edges and vertices based on the chosen mixed metric generator $Q_m$ are fulfilling the requirement of definition which means that $dim_m(L_m)\leq3.$ On the contrary, we choose the cardinality of mixed metric generator $Q_{m}^{'}$ to be two. Then the following discussion verifies the contradiction.

Now, it is possible that by taking $Q_{m}^{'}$ with cardinality $2,$ then either in two vertices or two edges have the same representations and it is also possible that an edge have the same distance according to $Q_{m}^{'}$ with a particular vertex. For example, if we take $Q_{m}^{'} = \{v_1, v_2\},$ then $d\left(v_1|Q_{m}^{'}\right) = d\left(v_1v_{3}|Q_{m}^{'}\right).$

Case 2.1. If we fix the first vertex $v_1$ and choose another vertex $v_i$ with $3\leq i\ ({\rm{odd}}) \leq 2m-3$ which means that $Q_{m}^{'} = \{v_1, v_{i}\},$ then the edges $v_{i}v_{i+1}$ & $v_{i}v_{i+2}$ have the same representation with respect to $Q_{m}^{'}.$ If $i = 2m-1,$ then the vertex $v_1$ and the edge $v_1v_{2}$ have the same representation with respect to $Q_{m}^{'}.$ Therefore, it is verified that the cardinality of mixed metric generator set cannot be equal to two.

Case 2.2. If two of vertices with moving indices with any kind of arbitrarily chosen gap size, we can choose $Q_{m}^{'} = \{v_{i}, v_j\}$ with $1\leq i\ ({\rm{odd}}) \leq 2m-1$ , then it follows that the identical representation in the edges which are $d\left(v_1v_{2}|Q_{m}^{'}\right) = d\left(v_2v_{4}|Q_{m}^{'}\right).$ it is verified that we can not take vertices in mixed metric generator with cardinality two.

Case 2.3. Now, taking even index vertices with fixed vertex $v_2$ can choose like, $Q_{m}^{'} = \{v_2, v_j\}$ with $2\leq i\ ({\rm{even}}) \leq 2m-4$ then it implies to the identical representation in the edges which are $d\left(v_{i-1}v_{i+1}|Q_{m}^{'}\right) = d\left(v_{i+1}v_{i+2}|Q_{m}^{'}\right)$ and when $j = 2m-2, 2m$ then the identical representation in a vertex with and edge, $d\left(v_2|Q_{m}^{'}\right) = d\left(v_1v_{3}|Q_{m}^{'}\right),$ it is verified that we can not take vertices in mixed metric generator with cardinality two.

Case 2.4. If both of two vertices from even and moving indices with any kind of arbitrarily chosen gap size can choose like, $Q_{m}^{'} = \{v_{i}, v_j\}$ with $2\leq i, j\leq 2m$ then it implies to the identical representation in the edges which are $d\left(v_1v_{2}|Q_{m}^{'}\right) = d\left(v_1v_{3}|Q_{m}^{'}\right)$ and it is verified that we can not take vertices in mixed metric generator with cardinality two.

Case 2.5. Take the vertices $v_{i}, v_j\in V(L_m)$ without choosing the size of gap, then there exist either two edges with distances or an edge with vertex i.e; $d\left(v_1v_{2}|Q_{m}^{'}\right) = d\left(v_1v_{3}|Q_{m}^{'}\right)$ or $d\left(v_1v_{3}|Q_{m}^{'}\right) = d\left(v_3v_{4}|Q_{m}^{'}\right)$ or $d\left(v_3v_{4}|Q_{m}^{'}\right) = d\left(v_{4}|Q_{m}^{'}\right)$ or $d\left(v_1v_{2}|Q_{m}^{'}\right) = d\left(v_2v_{4}|Q_{m}^{'}\right)$ or $d\left(v_3v_{5}|Q_{m}^{'}\right) = d\left(v_5v_{6}|Q_{m}^{'}\right)$ or $d\left(v_2|Q_{m}^{'}\right) = d\left(v_1v_{3}|Q_{m}^{'}\right)$ and finally verified that we can not take any type of vertices with any gap-size in the mixed metric generator with cardinality two, which implies to the contradiction that $dim_m(L_m)\neq2.$ Therefore, this proves the double inequality which is

$dim_m(L_m) = 3.$

3. Conclusions

Metric dimension or metric basis is a concept in which the whole vertex set of a structure is uniquely identified with a chosen subset named as locating set. Edge Metric dimension or edge metric basis is a concept in which the whole edge set of a structure is uniquely identified with a chosen subset from vertex set and it is named as edge locating set. Mixed-metric basis or mixed-metric dimension is a generalized version, in which the whole vertex and edge set of a structure is uniquely identified with a chosen subset from vertex set and it is named as edge mixed-metric locating set. In this comparative study, we discuss the relationships between metric, edge metric, and mixed metric dimension for different families of graphs. Precisely, the obtained results demonstrate the mixed metric dimension of some interesting graphs including Möbius ladder, hexagonal Möbius ladder, triangular Möbius ladder networks, in the given Table 1.

Table 1. Relation of metric parameters between graphs.

$G$	$dim$	$dim_{e}$	$dim_{m}$
$L_{m }$	$2$	$2$	$3$
$ML_{m }$	$3$	$4$	$4$
$HML_{m }$	$3$	$3$	$4$
$TL_{m }$	$3$	$4$	$4$
$TML_{m }$	$3$	$4$	$5$

| Show Table

DownLoad: CSV

Acknowledgments

M. Zayed extends thanks and appreciation to the Deanship of Scientific Research at King Khalid University, Saudi Arabia, for funding this work through research groups program under grant R.G.P.2/207/43.

Conflict of interest

The authors have no conflicts of interest to declare.

References

[1]	L. Debnath, Nonlinear partial differential equations for scientists and engineers, Massachusetts: Birkhäuser Boston, 2005. https://doi.org/10.1007/b138648
[2]	J. Yu, B. Ren, P. Liu, J. Zhou, CTE solvability, nonlocal symmetry, and interaction solutions of coupled integrable dispersion-less system, Complexity, 2022 (2022), 32211447. https://doi.org/10.1155/2022/3221447
[3]	K. Takasaki, Dispersionless Toda hierarchy and two-dimensional string theory, Commun. Math. Phys., 170 (1995), 101–116. https://doi.org/10.1007/BF02099441 doi: 10.1007/BF02099441
[4]	S. Aoyama, Y. Kodama, Topological conformal field theory with a rational W potential and the dispersionless KP hierarchy, Mod. Phys. Lett. A, 9 (1994), 2481–2492. https://doi.org/10.1142/S0217732394002355 doi: 10.1142/S0217732394002355
[5]	Z. Sagidullayeva, K. Yesmakhanova, R. Myrzakulov, Z. Myrzakulova, N. Serikbayev, G. Nugmanova, et al., Integrable generalized Heisenberg ferromagnet equations in 1+1 dimensions: reductions and gauge equivalence, arXiv: 2205.02073.
[6]	R. Myrzakulov, On some sigma models with potentials and the Klein-Gordon type equations, arXiv: hep-th/9812214.
[7]	K. Yesmakhanova, G. Nugmanova, G. Shaikhova, G. Bekova, R. Myrzakulov, Coupled dispersionless and generalized Heisenberg ferromagnet equations with self-consistent sources: geometry and equivalence, Int. J. Geom. Methods M., 17 (2020), 2050104. https://doi.org/10.1142/S0219887820501042 doi: 10.1142/S0219887820501042
[8]	M. Latha, C. Christal Vasanthi, An integrable model of (2+1)-dimensional Heisenberg ferromagnetic spin chain and soliton excitations, Phys. Scr., 89 (2014), 065204. https://doi.org/10.1088/0031-8949/89/6/065204
[9]	H. Triki, A. Wazwaz, New solitons and periodic wave solutions for the (2+1) dimensional Heisenberg ferromagnetic spin chain equation, J. Electromagnet. Wave., 30 (2016), 788–794. https://doi.org/10.1080/09205071.2016.1153986 doi: 10.1080/09205071.2016.1153986
[10]	M. Inc, A. Aliyu, A. Yusuf, D. Baleanu, Optical solitons and modulation instability analysis of an integrable model of (2+1)-Dimensional Heisenberg ferromagnetic spin chain equation, Micro Nanostructures, 112 (2017), 628–638. https://doi.org/10.1016/j.spmi.2017.10.018 doi: 10.1016/j.spmi.2017.10.018
[11]	S. Rayhanul Islam, M. Bashar, N. Muhammad, Immeasurable soliton solutions and enhanced (G'/G)-expansion method, Physics Open, 9 (2021), 100086. https://doi.org/10.1016/j.physo.2021.100086 doi: 10.1016/j.physo.2021.100086
[12]	B. Deng, H. Hao, Breathers, rogue waves and semi-rational solutions for a generalized Heisenberg ferromagnetic equation, Appl. Math. Lett., 140 (2023), 108550. https://doi.org/10.1016/j.aml.2022.108550
[13]	M. Daniel, L. Kavitha, R. Amuda, Soliton spin excitations in an anisotropic Heisenberg ferromagnet with octupole-dipole interaction, Phys. Rev. B, 59 (1999), 13774. https://doi.org/10.1103/PhysRevB.59.13774 doi: 10.1103/PhysRevB.59.13774
[14]	H. Triki, A. Wazwaz, New solitons and periodic wave solutions for the (2+1) dimensional Heisenberg ferromagnetic spin chain equation, J. Electromagnet. Wave., 30 (2016), 788–794. https://doi.org/10.1080/09205071.2016.1153986 doi: 10.1080/09205071.2016.1153986
[15]	M. Bashar, S. Rayhanul Islam, D. Kumar, Construction of traveling wave solutions of the (2+1)-dimensional Heisenberg ferromagnetic spin chain equation, Partial Differential Equations in Applied Mathematics, 4 (2021), 100040. https://doi.org/10.1016/j.padiff.2021.100040 doi: 10.1016/j.padiff.2021.100040
[16]	M. Bashar, S. Rayhanul Islam, Exact solutions to the (2+1)-Dimensional Heisenberg ferromagnetic spin chain equation by using modified simple equation and improve F-expansion methods, Physics Open, 5 (2020), 100027. https://doi.org/10.1016/j.physo.2020.100027
[17]	C. Christal Vasanthi, M. Latha, Heisenberg ferromagnetic spin chain with bilinear and biquadratic interactions in (2+1)-dimensions, Commun. Nonlinear Sci., 28 (2015), 109–122. https://doi.org/10.1016/j.cnsns.2015.04.012 doi: 10.1016/j.cnsns.2015.04.012
[18]	E. Zahran, A. Bekir, New unexpected variety of solitons arising from spatio-temporal dispersion (1+1) dimensional Ito-equation, Mod. Phys. Lett. B, 38 (2024), 2350258. https://doi.org/10.1142/S0217984923502585 doi: 10.1142/S0217984923502585
[19]	E. Zahran, A. Bekir, Optical soliton solutions to the perturbed Biswas-Milovic equation with Kudryashov's law of refractive index, Opt. Quant. Electron., 55 (2023), 1211. https://doi.org/10.1007/s11082-023-05453-w doi: 10.1007/s11082-023-05453-w
[20]	S. Kumar, R. Jiwari, R. Mittal, J. Awrejcewicz, Dark and bright soliton solutions and computational modeling of nonlinear regularized long wave model, Nonlinear Dyn., 104 (2021), 661–682. https://doi.org/10.1007/s11071-021-06291-9 doi: 10.1007/s11071-021-06291-9
[21]	E. Zahran, A. Bekir, New unexpected soliton solutions to the generalized (2+1) Schrödinger equation with its four mixing waves, Int. J. Mod. Phys. B, 36 (2022), 2250166. https://doi.org/10.1142/S0217979222501661 doi: 10.1142/S0217979222501661
[22]	M. Younis, T. Sulaiman, M. Bilal, S. Ur Rehman, U. Younas, Modulation instability analysis optical and other solutions to the modified nonlinear Schrödinger equation, Commun. Theor. Phys., 72 (2020), 065001. https://doi.org/10.1088/1572-9494/ab7ec8 doi: 10.1088/1572-9494/ab7ec8
[23]	E. Zahran, A. Bekir, R. Ibrahim, New optical soliton solutions of the popularized anti-cubic nonlinear Schrödinger equation versus its numerical treatment, Opt. Quant. Electron., 55 (2023), 377. https://doi.org/10.1007/s11082-023-04624-z doi: 10.1007/s11082-023-04624-z
[24]	E. Zahran, A. Bekir, M. Shehata, New diverse variety analytical optical soliton solutions for two various models that are emerged from the perturbed nonlinear Schrödinger equation, Opt. Quant. Electron., 55 (2023), 190. https://doi.org/10.1007/s11082-022-04423-y doi: 10.1007/s11082-022-04423-y
[25]	M. Ali Akbar, A. Wazwaz, F. Mahmud, D. Baleanu, R. Roy, H. Barman, et al., Dynamical behavior of solitons of the perturbed nonlinear Schrödinger equation and microtubules through the generalized Kudryashov scheme, Results Phys., 43 (2022), 106079. https://doi.org/10.1016/j.rinp.2022.106079
[26]	L. Ouahid, S. Owyed, M. Abdou, N. Alshehri, S. Elagan, New optical soliton solutions via generalized Kudryashov's scheme for Ginzburg-Landau equation in fractal order, Alex. Eng. J., 60 (2021), 5495–5510. https://doi.org/10.1016/j.aej.2021.04.030 doi: 10.1016/j.aej.2021.04.030
[27]	G. Genc, M. Ekici, A. Biswas, M. Belic, Cubic-quartic optical solitons with Kudryashov's law of refractive index by F-expansions schemes, Results Phys., 18 (2020), 103273. https://doi.org/10.1016/j.rinp.2020.103273 doi: 10.1016/j.rinp.2020.103273
[28]	D. Kumar, A. Seadawy, A. Joardar, Modified Kudryashov method via new exact solutions for some conformable fractional differential equations arising in mathematical biology, Chinese J. Phys., 56 (2018), 75–85. https://doi.org/10.1016/j.cjph.2017.11.020 doi: 10.1016/j.cjph.2017.11.020
[29]	C. Gomez S, H. Roshid, M. Inc, L. Akinyemi, H. Rezazadeh, On soliton solutions for perturbed Fokas-Lenells equation, Opt. Quant. Electron., 54 (2022), 370. https://doi.org/10.1007/s11082-022-03796-4 doi: 10.1007/s11082-022-03796-4
[30]	E. Zahran, A. Bekir, New variety diverse solitary wave solutions to the DNA Peyrard-Bishop model, Mod. Phys. Lett. B, 37 (2023), 2350027. https://doi.org/10.1142/S0217984923500276 doi: 10.1142/S0217984923500276
[31]	E. Zahran, A. Bekir, New solitary solutions to the nonlinear Schrödinger equation under the few-cycle pulse propagation property, Opt. Quant. Electron., 55 (2023), 696. https://doi.org/10.1007/s11082-023-04916-4 doi: 10.1007/s11082-023-04916-4
[32]	E. Zahran, A. Bekir, New diverse soliton solutions for the coupled Konno-Oono equations, Opt. Quant. Electron., 55 (2023), 112. https://doi.org/10.1007/s11082-022-04376-2 doi: 10.1007/s11082-022-04376-2
[33]	E. Zahran, H. Ahmad, T. Saeed, T. Botmart, New diverse variety for the exact solutions to Keller-Segel-Fisher system, Results Phys., 35 (2022), 105320. https://doi.org/10.1016/j.rinp.2022.105320 doi: 10.1016/j.rinp.2022.105320
[34]	A. Hyder, M. Barakat, General improved Kudryashov method for exact solutions of nonlinear evolution equations in mathematical physics, Phys. Scr., 95 (2020), 045212. https://doi.org/10.1088/1402-4896/ab6526 doi: 10.1088/1402-4896/ab6526

This article has been cited by:

1.	Basma Mohamed, Linda Mohaisen, Mohamed Amin, Computing Connected Resolvability of Graphs Using Binary Enhanced Harris Hawks Optimization, 2023, 36, 1079-8587, 2349, 10.32604/iasc.2023.032930
2.	Muhammad Imran, Ricai Luo, Muhammad Kamran Jamil, Muhammad Azeem, Khawaja Muhammad Fahd, Geometric perspective to Degree–Based topological indices of supramolecular chain, 2022, 16, 25901230, 100716, 10.1016/j.rineng.2022.100716
3.	Shahid Zaman, Mahnoor Mushtaq, Muhammad Danish, Parvez Ali, Sadaf Rasheed, Topological Characterization of Some New Anti-Viral Drugs for Cancer Treatment, 2024, 2191-1630, 10.1007/s12668-024-01500-2
4.	Wenjing Xiong, Xingchao Deng, A note on nonlocal metric dimension of some networks, 2024, 1744-5760, 1, 10.1080/17445760.2024.2429106
5.	Manonmani R., Selvarani P., Multilinear regression analysis of antiviral medications with topological indices, 2025, 10933263, 109104, 10.1016/j.jmgm.2025.109104

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)