Metric-Based Fractional Dimension of Rotationally-Symmetric Line Networks

Abstract

The parameter of distance plays an important role in studying the properties symmetric networks such as connectedness, diameter, vertex centrality and complexity. Particularly different metric-based fractional models are used in diverse fields of computer science such as integer programming, pattern recognition, and in robot navigation. In this manuscript, we have computed all the local resolving neighborhood sets and established sharp bounds of a metric-based fractional dimension called by the local fractional metric dimension of the rotationally symmetric line networks of wheel and prism networks. Furthermore, the bounded and unboundedness of these networks is also checked under local fractional metric dimension when the order of these networks approaches to infinity. The lower and upper bounds of local fractional metric dimension of all the rotationally symmetric line networks is also analyzed by using $3D$ shapes.

1. Introduction

The fundamental concepts of the metric dimension ($MD$) of connected networks were first revealed by Slater in 1975 [1] and the notion of $MD$ was initiated by Melter and Hararay in 1976 [2]. Robot navigation in a network space was studied by Khuller et al. with the help of $MD$ [3]. Gerey and Johnson proved that computing $MD$ for any connected network is an NP-complete problem in general [4]. Melter and Tomescu studied metric basis in digital geometry and they also computed the MD of grid-related networks [5]. $MD$ has applications in the processing of maps, pattern reorganization, robot navigation [6], network discovery and verification [7], hierarchical lattice [8], pharmaceutical chemistry, and in integer programming [9]. Since then, various types of $MD$ such as edge $MD$ [10], mixed $MD$ [11], K $MD$ [12], partition dimension [13] have been discovered. The general definition of cone metric spaces in the context of neutrosophic cone metric space theory was developed by Al-Omeri et al. and they have developed some fundamental results as well [14]; to study the common fixed point theorems in neutrosophic cone metric space, see [15]. The most recent development in this field of $MD$ has been made by Bokhary et al., and they have computed the $MD$ of subdivision of circulant networks [16].

The notion of fractional metric dimension ($FMD$) has been introduced by Currie and Ollermann, they proposed that the finding of the $FMD$ of a network is formulated as a certain integer programming problem [17] and the idea of $FMD$ in the field of networking theory was introduced by S. Arumugam and V. Mathew. They have developed different techniques to find the $FMD$ of diversely connected networks and they have also obtained the exact value of $FMD$ of famous networks such as Petersen, cycle, friendship, hypercubes, wheel, and grid networks [18]. Feng et al. determined the FMD of distance regular networks and FMD of Hamming and Johnson networks. Moreover, they proposed an inequality for the $MD$ and $FMD$ [19] and $FMD$ of trees, and unicyclic networks were obtained by Krismanto et al. [20]. Zafar et al. obtained the exact value of the $FMD$ of prism and path-related networks [21].

The latest invariant of $FMD$ called local fractional metric dimension ($LFMD$) was introduced by Asiyah et al., and they computed the exact value of the $LFMD$ of the corona product of different types of networks [22]. Liu et al. derived some significant results on the upper bounds of $LFMD$ of rotationally symmetric and planner networks [23] and Ali et al. recently extended the work of Liu et al. and also computed upper bounds of the $LFMD$ of some rotationally symmetric planner networks [24]. Javaid et al. established the bounds of the $LFMD$ of all the networks and they also obtained the exact value of the $LFMD$ of path, cycle, bipartite, and complete networks [25]. The lower bound of the $LFMD$ is improved by Javaid et al. and they also established the bounds of the $LFMD$ of antiprism and sun flower networks [26]. Since discovering the bounds of the $LFMD$ of generalized sunlet [27] and convex polytopes, [28,29], Sierpinski networks have been established [30].

Now, we are presenting some applications of $MD$ in the field of chemical graph theory; the chemical graph theory applies in chemistry and focuses on the molecular topology. After converting a chemical structure into a specific network, a comprehensive structural analysis can be performed. Some of the chemical compounds are considered as functional groups, where atoms represented by nodes and bonds among them represented by edges. By using the idea of characteristic polynomials the different common substructures are characterized and the certain resolving sets are used to find the specific position when two chemical structures have the same functional group. This study has been used in pharmaceutical activities and in drug discovery [9,31].

This article is an extension of work done by Ali et al. [24], as they have established upper bounds of $LFMD$s for certain rotationally symmetric networks. In this manuscript, our aim is to compute both the upper and lower bounds of $LFMD$s of rotationally symmetric line networks of wheels and prism networks. The detail of line networks prism and wheel network is given from Figure 1 and Figure 2, $3D$ representation of all the obtained results is given from Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9. The boundedness and unboundedness of all these networks is also obtained. Section 2 contains preliminary concepts, Section 3 deals with the main results, and Section 4 represents the conclusion of the manuscript.

2. Preliminaries

Let $\mathbb{B}$ be a connected network with vertex set $V=V\left(\mathbb{B}\right)$ and edge set $E=E\left(\mathbb{B}\right).$ A walk between two vertices $u_i$ and $u_j$ is the sequence of edges and vertices. A path between two vertices $u_i$ and $u_j$ is a walk in which neither vertex nor edge is repeated. The distance between any two vertices $u_i$ and $u_j$ ($d(u_i,u_j)$) is the length of the shortest path connecting them. For further study about the preliminary concepts of networking theory see [32]. A vertex $u\in V\left(\mathbb{B}\right)$ resolves a pair $(v,w)$ if the distance from u to v is not equal to the distance from u to w$\left(d\right(u,v)\ne d(u,w\left)\right)$. Let $L=\{u_1,u_2,u_3,\dots ,u_t\}\subseteq V\left(\mathbb{B}\right)$, then t tuple representation of v with respect to L is $d(u|L)=(d(v,u_1),d(v,u_2),d(v,u_3),\dots ,d(v,u_t))$. If distinct elements of $\mathbb{B}$ have a unique representation with respect to L, then L becomes a resolving set. The minimum cardinality of a resolving set is called $MD$ of $\mathbb{B}$, thus $MD$ of $\mathbb{B}$ is defined as follows:

$$dim\left(\mathbb{B}\right)=min\{|L|:Lisaresolvingsetof\mathbb{B}\}.$$

In a connected network $\mathbb{B}$ for $uv\in E\left(\mathbb{B}\right)$ a vertex $x\in V\left(\mathbb{B}\right)$ is said to resolve adjacent pairs of vertices as $L_r\left(uv\right)=\{x\in V\left(\mathbb{B}\right):d(x,u)\ne d(x,u)\}$ and it is called a local resolving neighborhood $\left(LRN\right)$ set of an edge $uv\in E\left(\mathbb{B}\right)$. A real-valued function $\lambda :V\left(\mathbb{B}\right)\to [0,1]$ is called a local resolving function $\left(LRF\right)$ of $\mathbb{B}$ if $\lambda \left(L_r\left(uv\right)\right)\ge 1$ for an edge $uv\in E\left(\mathbb{B}\right)$, where $\lambda \left(L_r\left(uv\right)\right)={\displaystyle \sum _{x\in L_r\left(uv\right)}}\lambda \left(x\right).$ An $LRF$ $\lambda$ is called a minimal $LRF$ if there exists another $LRF$ ${\lambda }^{\prime }:V\left(\mathbb{B}\right)\to [0,1]$ such that $|{\lambda }^{\prime }|<|\lambda |$ and ${\lambda }^{\prime }\left(x\right)\ne \lambda \left(x\right)$ for at least $x\in V\left(\mathbb{B}\right)$ that is not $LRF$ of $\mathbb{B}$. Thus, the $LFMD$ of $\mathbb{B}$ is defined as follows:

$$Ldi{m}_F\left(\mathbb{B}\right)=min\{|\lambda |:\lambda isaminimallocalresolvingfunctionof\mathbb{B}\}.$$

A line network $L\left(\mathbb{B}\right)$ of $\mathbb{B}$ is a network whose vertices are the edges of $\mathbb{B}$, and two vertices $u,v\in L\left(\mathbb{B}\right)$ are connected $iff$ they have a common end vertex in $\mathbb{B}.$ For more results about line networks and their $MD$, we refer [33,34].

3. Main Results

In this section, we have computed the $LRN$ sets of rotationally symmetric line networks of prism and wheel networks and the bounds of the $LFMD$s of these networks are also established. Furthermore, all the theorems are divided into two cases, the case 1 is particular and case 2 is general.

3.1. LRN Sets and LFMD of Line Network of Wheel Network

In this subsection, our aim to compute the $LRN$ sets and the $LFMD$s of the line network of wheel networks. The network is defined as follows:

Let $L{W}_t$ be a line network of a wheel network with a vertex set $V\left(L{W}_t\right)=\{a_i,b_i:1\le i\le t\}$ and edge set $E\left(L{W}_t\right)=\{b_i{b}_{i+1},b_i{a}_{i+1},a_i{b}_i,a_i{a}_{i+2},a_i{a}_{i+3},a_i{a}_{i+4},\dots ,a_i{a}_{i+t}:1\le i\le t\}$ with order $2t$ and size $\frac{t^2+5t}{2}$. For more information about $L{W}_t$, see Figure 1.

Lemma 1.

Let $L{W}_t$ be the line network of wheel network, where $t\cong 1\left(mod2\right).$ Then

(a)

$|L_r\left(a_i{a}_{i+1}\right)|=4$ and $|{\displaystyle \bigcup _{i=1}^t}{L}_r\left(a_i{a}_{i+1}\right)|=|V\left(L{W}_t\right)|.$

(b)

$|(L_r\left(a_i{a}_{i+1}\right)|\le |L_r\left(x\right)|$ and $|L_r\left(x\right)\cap {\displaystyle \bigcup _{i=1}^t}{L}_r\left(a_i{a}_{i+1}\right)|\ge |L_r\left(a_i{a}_{i+1}\right)|\forall x\in E\left(L{W}_t\right).$

Proof. 

Consider $a_i$ inner and $b_i$ are outer vertices of $L{W}_t$, where $1\le i\le t$ and $t+1\cong 1\left(modt\right).$

(a)

$L_r\left(a_i{a}_{i+1}\right)=\{a_i,a_{i+1},b_{i+1},b_{t+i-1}\}$ therefore, $|L_r\left(a_i{a}_{i+1}\right)|=4$ also $|{\displaystyle \bigcup _{i=1}^t}LR\left(a_i{a}_{i+1}\right)|=2t=|V\left(L{W}_t\right)|.$

(b)

The LRN sets other than $L_r\left(a_i{a}_{i+1}\right)$ are $L_r\left(b_i{b}_{i+1}\right)=V\left(L{W}_t\right)-\{b_{\frac{t+2i+1}{2}},a_{i+1},b_{\frac{t+2i+3}{2}},$ $b_{\frac{t+2i+5}{2}},\dots \dots ,b_{t+i-2},b_{t+i-1}\},$ $L_r\left(a_i{b}_i\right)=V\left(L{W}_t\right)-\{a_{i+1},b_{i+2},b_{i-1}\}.$ Since ${\displaystyle \bigcup _{i=1}^t}{L}_r\left(a_i{a}_{i+1}\right)$ $=V\left(L{W}_t\right)$, $|L_r\left(x\right)\cap {\displaystyle \bigcup _{i=1}^t}{L}_r\left(a_i{a}_{i+1}\right)|\ge |L_r\left(a_i{a}_{i+1}\right)|.$ The comparison among the cardinalities of all the $LRN$ sets of $L{W}_t$ is given in

Table 1. Comparison between the cardinalities of LRN sets of $L{W}_t$.

LRN Set	Cardinality
$L_r\left(a_i{b}_i\right)$	$2t-3>4$
$L_r\left(b_i{b}_{i+1}\right)$	$2t-4>4$

.

□

It is clear from Table 1 that $|L_r\left(a_i{a}_{i+1}\right)|<|L_r\left(x\right)|$, where $L_r\left(x\right)$ are the other $LRN$ sets of $L{W}_t$.

Theorem 1.

Let $L{W}_3$ be a line network of generalized wheel network, then

$$Ldi{m}_F\left(L{W}_3\right)=\frac{3}{2}.$$

Proof. 

The $LRN$ sets of $L{W}_3$ are as follows:

$L_r\left(b_1{b}_2\right)=\{b_1,b_2,a_1,a_3\},$

$L_r\left(b_2{b}_3\right)=\{b_2,b_3,a_2,a_1\},$

$L_r\left(b_3{b}_1\right)=\{b_3,b_1,a_3,a_2\},$

$L_r\left(a_1{b}_1\right)=\{a_1,a_3,b_1,b_2\},$

$L_r\left(a_2{b}_2\right)=\{a_2,a_1,b_2,b_3\},$

$L_r\left(a_3{b}_3\right)=\{a_3,a_2,b_3,b_1\},$

$L_r\left(a_1{a}_2\right)=\{a_1,a_2,b_2,b_3\},$

$L_r\left(a_2{a}_3\right)=\{a_2,a_3,b_3,b_1\},$

$L_r\left(a_3{a}_1\right)=\{a_3,a_1,b_1,b_3\}.$

From above, $LRN$ sets the cardinality of all the $LRN$ sets as 4, therefore, we define a constant mapping $\lambda \left(V\left(L{W}_3\right)\right)\to [0,1]$ as $\frac{1}{4}$ to each $v\in V\left(L{W}_3\right)$, hence

$$Ldi{m}_F\left(L{W}_3\right)={\displaystyle \sum _{i=1}^6}\frac{1}{4}=\frac{3}{2}.$$

□

Theorem 2.

Let $L{W}_t$ be a line network of a wheel network, where $t\cong 1\left(mod2\right).$ Then

$$\frac{2t}{2t-3}\le Ldi{m}_F\left(L{W}_t\right)\le \frac{t}{2}.$$

Proof. 

To prove the theorem, we have divided it in two cases:

Case 1:

For $t=5$, we have following $LRN$ sets:

$L_r\left(b_1{b}_2\right)=V\left(L{W}_5\right)-\{b_4,a_2,a_4,a_5\},$

$L_r\left(b_2{b}_3\right)=V\left(L{W}_5\right)-\{b_5,a_3,a_5,a_1\},$

$L_r\left(b_3{b}_4\right)=V\left(L{W}_5\right)-\{b_1,a_4,a_1,a_2\},$

$L_r\left(b_4{b}_5\right)=V\left(L{W}_5\right)-\{b_2,a_5,a_2,a_3\},$

$L_r\left(b_5{b}_1\right)=V\left(L{W}_5\right)-\{b_3,a_1,a_3,a_4\},$

$L_r\left(a_1{b}_1\right)=V\left(L{W}_5\right)-\{b_3,b_4,a_2\},$

$L_r\left(a_2{b}_2\right)=V\left(L{W}_5\right)-\{b_4,b_5,a_3\},$

$L_r\left(a_3{b}_3\right)=V\left(L{W}_5\right)-\{b_5,b_1,a_4\},$

$L_r\left(a_4{b}_4\right)=V\left(L{W}_5\right)-\{b_1,b_2,a_5\},$

$L_r\left(a_5{b}_5\right)=V\left(L{W}_5\right)-\{b_2,b_3,a_1\},$

$L_r\left(a_1{a}_2\right)=\{a_1,a_2,b_2,b_5\},$

$L_r\left(a_2{a}_3\right)=\{a_2,a_3,b_3,b_1\},$

$L_r\left(a_3{a}_4\right)=\{a_3,a_4,b_4,b_2\},$

$L_r\left(a_4{a}_5\right)=\{a_4,a_5,b_5,b_3\},$

$L_r\left(a_5{a}_1\right)=\{a_5,a_1,b_1,b_2\}.$

From above, $LRN$ sets the minimum cardinality of $LRN$ set $L_r\left(a_i{a}_{i+1}\right)$ as 4, where $1\le i\le 5$; therefore, we define a minimal $LRF$ $\lambda \left(V\left(L{W}_5\right)\right)\to [0,1]$ as $\frac{1}{4}$ to each $v\in V\left(L{W}_5\right)$, hence $Ldi{m}_F\left(L{W}_5\right)\le {\displaystyle \sum _{i=1}^{10}}\frac{1}{4}=\frac{5}{2}$. The maximum cardinality of $\left(LRN\right)$ set $L_r\left(a_i{b}_i\right)$ is 7; therefore, we define a maximal $LRF$ ${\lambda }^{\prime }\left(V\left(L{W}_5\right)\right)\to [0,1]$ as $\frac{1}{7}$ to each $v\in V\left(L{W}_5\right)$, hence $Ldi{m}_F\left(L{W}_5\right)\ge {\displaystyle \sum _{i=1}^{10}}\frac{1}{7}=\frac{10}{7}.$

$$\frac{10}{7}\le Ldi{m}_F\left(L{W}_5\right)\le \frac{5}{2}.$$

Case 2:

For $t\ge 7$, in the view of Lemma 1 the cardinality of $LRN$ set $L_r\left(a_i{a}_{i+1}\right)$ is 4 and $|L_r\left(a_i{a}_{i+1}\right)|<|L_r\left(x\right)|,$ where $L_r\left(x\right)$ are other LRN sets of $L{W}_t$. Therefore, we define a minimal, $LRF$ $\lambda \left(V\left(L{W}_t\right)\right)\to [0,1]$ as $\frac{1}{4}$ to each $v\in V\left(L{W}_t\right)$, hence $Ldi{m}_F\left(L{W}_t\right)\le {\displaystyle \sum _{i=1}^{2t}}\frac{1}{4}=\frac{t}{2}$. In the same context, by Lemma 1, the maximum cardinality of $LRN$ set $L_r\left(a_i{b}_i\right)$ is $2t-3$ and $|L_r\left(a_i{b}_i\right)|>|L_r\left(x\right)|,$ where $L_r\left(x\right)$ are other $LRN$ sets of $L{W}_t$. Therefore, we define a maximal $LRF$ ${\lambda }^{\prime }\left(V\left(L{W}_t\right)\right)\to [0,1]$ as $\frac{1}{2t-3}$ to each $v\in V\left(L{W}_t\right)$, hence $Ldi{m}_F\left(L{W}_t\right)$ $\ge {\displaystyle \sum _{i=1}^{2t}}\frac{1}{2t-3}=\frac{2t}{2t-3}.$

$$\frac{2t}{2t-3}⩽Ldi{m}_F\left(L{W}_t\right)⩽\frac{t}{2}$$

□

Lemma 2.

Let $L{W}_t$ be the line network of wheel network then, where $t\cong 0\left(mod2\right).$ Then

(a)

$|L_r\left(a_i{a}_{i+1}\right)|=4$ and ${\displaystyle \bigcup _{i=1}^t}{L}_r\left(a_i{a}_{i+1}\right)=V\left(L{W}_t\right);$

(b)

$|L_r\left(a_i{a}_{i+1}\right)|\le |L_r\left(x\right)|$ and $|L_r\left(x\right)\cap {\displaystyle \bigcup _{i=1}^t}{L}_r\left(a_i{a}_{i+1}\right)|\ge |L_r\left(a_i{a}_{i+1}\right)|,\forall x\in E\left(L{W}_t\right).$

Proof. 

Consider $a_i$ inner and $b_i$ are outer vertices of $L{W}_t$, where $1\le i\le t$ and $t+1\cong 1\left(modt\right)$.

(a)

$L_r\left(a_i{a}_{i+1}\right)=\{a_i,a_{i+1},b_{i+1},b_{t+i-1}\}$, therefore, $|L_r\left(a_i{a}_{i+1}\right)|=4$ also $|{\displaystyle \bigcup _{i=1}^t}LR\left(a_i{a}_{i+1}\right)|=2t$.

(b)

The $LRN$ sets other than $L_r\left(a_i{a}_{i+1}\right)$ are $L_r\left(b_i{b}_{i+1}\right)=\{b_i,b_{i+1},b_{i+2},\dots \dots \dots ,b_{t+i-1}\}\cup \{a_i,a_{i+2}\}$ and $L_r\left(a_i{b}_i\right)=V\left(L{W}_t\right)-\{a_{i+1},b_{i+2},b_{t+i-2},b_{t+i-1}\}.$ Since ${\displaystyle \bigcup _{i=1}^t}{L}_r\left(a_i{a}_{i+1}\right)=V\left(L{W}_t\right)$, therefore $|L_r\left(x\right)\cap {\displaystyle \bigcup _{i=1}^t}{L}_r\left(a_i{a}_{i+1}\right)|\ge |L_r\left(a_i{a}_{i+1}\right)|.$ The comparison among the cardinalities of all the $LRN$ sets is given in

Table 2. Comparison between the cardinalities of $LRN$ sets of $L{W}_t$.

LRN Set	Cardinality
$L_r\left(a_i{b}_i\right)$	$2t-4>4$
$L_r\left(b_i{b}_{i+1}\right)$	$t+2>4$

.

□

It is clear from Table 2 that $|L_r\left(a_i{a}_{i+1}\right)|<|L_r\left(x\right)|$, where $L_r\left(x\right)$ are the other $LRN$ sets of $L{W}_t$.

Theorem 3.

Let $L{W}_4$ be the line network of generalized wheel network then

$$\frac{4}{3}\le Ldi{m}_F\left(L{W}_4\right)\le 2.$$

Proof. 

The $LRN$ sets of $L{W}_4$ are given as follows:

$L_r\left(b_1{b}_2\right)=V\left(L{W}_4\right)-\{a_2,a_4\},$

$L_r\left(b_2{b}_3\right)=V\left(L{W}_4\right)-\{a_3,a_1\},$

$L_r\left(b_3{b}_4\right)=V\left(L{W}_4\right)-\{a_4,a_2\},$

$L_r\left(b_4{b}_1\right)=V\left(L{W}_4\right)-\{a_1,a_3\},$

$L_r\left(a_1{b}_1\right)=V\left(L{W}_4\right)-\{a_2,b_3,b_4\},$

$L_r\left(a_2{b}_2\right)=V\left(L{W}_4\right)-\{a_3,b_4,b_1\},$

$L_r\left(a_3{b}_3\right)=V\left(L{W}_4\right)-\{a_4,b_1,b_2\},$

$L_r\left(a_4{b}_4\right)=V\left(L{W}_4\right)-\{a_1,b_2,b_3\},$

$L_r\left(a_1{a}_2\right)=\{a_1,a_2,b_2,b_4\},$

$L_r\left(a_2{a}_3\right)=\{a_2,a_3,b_3,b_1\},$

$L_r\left(a_3{a}_4\right)=\{a_3,a_1,b_1,b_2\},$

$L_r\left(a_4{a}_1\right)=\{a_4,a_4,b_2,b_3\}.$

From the above, $LRN$ sets the minimum cardinality of $LRN$ set $L_r\left(a_i{a}_{i+1}\right)$ as 4, where $1\le i\le 4$; therefore, we define a minimal $LRF$ $\lambda \left(V\left(L{W}_4\right)\right)\to [0,1]$ as $\frac{1}{4}$ to each $v\in V\left(L{W}_4\right)$, hence $Ldi{m}_F\left(L{W}_4\right)\le {\displaystyle \sum _{i=1}^8}\frac{1}{4}=2$. The maximum cardinality of $L_r\left(b_i{b}_{i+1}\right)$ is 6; therefore, we define a maximal $LRF$ $\lambda \left(V\left(L{W}_4\right)\right)\to [0,1]$ as $\frac{1}{6}$ to each $v\in V\left(L{W}_4\right)$, hence $Ldi{m}_F\left(L{W}_4\right)\ge {\displaystyle \sum _{i=1}^8}\frac{1}{6}=\frac{4}{3}$.

$$\frac{4}{3}\le Ldi{m}_F\left(L{W}_4\right)\le 2.$$

□

Theorem 4.

Let $L{W}_t$ be a line network of a generalized wheel network, where $t\cong 0\left(mod2\right)$, then

$$\frac{t}{t-2}\le Ldi{m}_F\left(L{W}_t\right)\le \frac{t}{2}.$$

Proof. 

To prove the theorem, we have divided it in two cases:

Case 1:

For $t=6$, we have the following possible $LRN$ sets:

$L_r\left(b_1{b}_2\right)=V\left(L{W}_6\right)-\{a_2,a_4,a_5,a_6\},$

$L_r\left(b_2{b}_3\right)=V\left(L{W}_6\right)-\{a_3,a_5,a_6,a_1\},$

$L_r\left(b_3{b}_4\right)=V\left(L{W}_6\right)-\{a_4,a_6,a_1,a_2\},$

$L_r\left(b_4{b}_5\right)=V\left(L{W}_6\right)-\{a_5,a_1,a_2,a_3\},$

$L_r\left(b_5{b}_6\right)=V\left(L{W}_6\right)-\{a_6,a_2,a_3,a_4\},$

$L_r\left(b_6{b}_1\right)=V\left(L{W}_6\right)-\{a_1,a_3,a_4,a_5\},$

$L_r\left(a_1{b}_1\right)=V\left(L{W}_6\right)-\{a_2,b_3,b_5,b_6\},$

$L_r\left(a_2{b}_2\right)=V\left(L{W}_6\right)-\{a_3,b_4,b_6,b_1\},$

$L_r\left(a_3{b}_3\right)=V\left(L{W}_6\right)-\{a_4,b_5,b_1,b_2\},$

$L_r\left(a_4{b}_4\right)=V\left(L{W}_6\right)-\{a_5,b_6,b_2,b_3\},$

$L_r\left(a_5{b}_5\right)=V\left(L{W}_6\right)-\{a_6,b_1,b_3,b_4\},$

$L_r\left(a_6{b}_6\right)=V\left(L{W}_6\right)-\{a_1,b_2,b_4,b_5\},$

$L_r\left(a_1{a}_2\right)=\{a_1,a_2,b_2,b_6\},$

$L_r\left(a_2{a}_3\right)=\{a_2,a_3,b_3,b_1\},$

$L_r\left(a_3{a}_4\right)=\{a_3,a_4,b_4,b_2\},$

$L_r\left(a_4{a}_5\right)=\{a_4,a_5,b_5,b_3\},$

$L_r\left(a_5{a}_6\right)=\{a_5,a_6,b_6,b_4\},$

$L_r\left(a_6{a}_1\right)=\{a_6,a_3,b_1,b_5\}.$

From above, $LRN$ sets the cardinality of $LRN$ set $L_r\left(a_i{a}_{i+1}\right)$ as 4, where $1\le i\le 6$; therefore, we define a minimal $LRF$ $\lambda \left(V\left(L{W}_6\right)\right)\to [0,1]$ as $\frac{1}{4}$ to each $v\in V\left(L{W}_6\right)$, hence $Ldi{m}_F\left(L{W}_6\right)\le {\displaystyle \sum _{i=1}^{12}}\frac{1}{4}=3$. The maximum cardinality of $LRN$ set $L_r\left(b_i{b}_{i+1}\right)$ is 8; therefore, we define a maximal $LRF$ ${\lambda }^{\prime }\left(V\left(L{W}_6\right)\right)\to [0,1]$ as $\frac{1}{8}$ to each $v\in V\left(L{W}_6\right)$, hence $Ldi{m}_F\left(L{W}_6\right)\ge {\displaystyle \sum _{i=1}^{12}}\frac{1}{8}=\frac{3}{2}.$

$$\frac{3}{2}\le Ldi{m}_F\left(L{W}_6\right)\le 3.$$

Case 2:

For $t\ge 8$, in the view of Lemma 2, the cardinality of $LRN$ set $L_r\left(a_i{a}_{i+1}\right)$ is 4 and $|L_r\left(a_i{a}_{i+1}\right)|<|L_r\left(x\right)|,$ where $L_r\left(x\right)$ are other $LRN$ sets of $L{W}_t$. Therefore, we define a minimal $LRF$ $\lambda \left(V\left(L{W}_t\right)\right)\to [0,1]$ as $\frac{1}{4}$ to each $v\in V\left(L{W}_t\right)$, hence $Ldi{m}_F\left(L{W}_t\right)\le {\displaystyle \sum _{i=1}^{2t}}\frac{1}{4}=\frac{t}{2}$. In the same context, by Lemma 2, the maximum cardinality of $LRN$ set $L_r\left(a_i{b}_i\right)$ is $2t-4$ and $|L_r\left(a_i{b}_i\right)|>|L_r\left(x\right)|,$ where $L_r\left(x\right)$ are the other $LRN$ sets of $L{W}_t$. Therefore, we define a maximal $LRF$ ${\lambda }^{\prime }\left(V\left(L{W}_t\right)\right)\to [0,1]$ as $\frac{1}{2t-4}$ to each $v\in V\left(L{W}_t\right)$, hence $Ldi{m}_F\left(L{W}_t\right)\ge {\displaystyle \sum _{i=1}^{2t}}\frac{1}{2t-4}=\frac{t}{t-2}$.

$$\frac{t}{t-2}\le Ldi{m}_F\left(L{W}_t\right)\le \frac{t}{2}.$$

□

3.2. Line Network of Prism Network $L{D}_t$

In this subsection, our aim is to compute $LRN$ sets and $LFMD$ of the line network of prism network. The line network of prism network is defined as follows:

Let $L{D}_t$ be the line network of prism network with vertex set $V\left(L{D}_t\right)=\{a_i,b_i,c_i:1\le i\le t\}$ and edge set $E\left(L{D}_t\right)=\{a_i{a}_{i+1},a_i{b}_i,b_i{a}_{i+1},c_i{b}_i,c_i{c}_{i+1}:1\le i\le t\}$ with order $3t$ and size $6t$. For more information see Figure 2.

Lemma 3.

Let $L{D}_t$ be the line network of prism network, where $t\cong 1\left(mod2\right).$ Then

(a)

$|L_r\left(b_i{a}_{i+1}\right)|=t+3$ and $|{\displaystyle \bigcup _{i=1}^t}{L}_r\left(b_i{a}_{i+1}\right)|=3t.$

(b)

$|L_r\left(b_i{a}_{i+1}\right)|\le |L_r\left(x\right)|$ and $|L_r\left(x\right)\cap {\displaystyle \bigcup _{i=1}^t}{L}_r\left(b_i{a}_{i+1}\right)|\ge |L_r\left(b_i{a}_{i+1}\right)|,\forall x\in E\left(L{D}_t\right).$

Proof. 

Consider $a_i$ inner, $b_i$ middle, and $c_i$ are outer vertices of $L{D}_t$, where $1\le i\le t$ and $t+1\cong 1\left(modt\right).$

(a)

$L_r\left(b_i{a}_{i+1}\right)=V\left(L{D}_t\right)-\{a_i,a_{\frac{t+2i+3}{2}},a_{\frac{t+2i+5}{2}},\dots ,a_{t+i-1},b_{\frac{t+2i+1}{2}},b_{\frac{t+2i+3}{2}},\dots ,b_{t+i-1},$

$c_{i+2},c_{i+3},\dots ,c_{t+i-1}\}$; therefore, $|L_r\left(b_i{a}_{i+1}\right)|=t+3$ also $|{\displaystyle \bigcup _{i=1}^t}{L}_r\left(b_i{a}_{i+1}\right)|=3t.$

(b)

The $LRN$ sets other then $L_r\left(b_i{a}_{i+1}\right)$ are $L_r\left(a_i{b}_i\right)=V\left(L{D}_t\right)-\{a_{i+1},a_{i+2},a_{i+3},\dots ,a_{\frac{t+2i-1}{2}}\}$ $\cup \{b_{i+1},b_{i+2},b_{i+3},\dots ,b_{\frac{t+2i-1}{2}}\}\cup \{c_{i+2},c_{i+3},\dots ,c_{t+i-1}\}$, $L_r\left(a_i{a}_{i+1}\right)=V\left(L{D}_t\right)-\{a_{\frac{t+2i+1}{2}}\}$ $\cup \{b_i\}\cup \{c_i,c_{i+1}\}$, $L_r\left(b_i{c}_i\right)=V\left(L{D}_t\right)-\{a_{\frac{t+2i+1}{2}},a_{\frac{t+2i+3}{2}},a_{\frac{t+2i+5}{2}},\dots ,a_{t+i-1}\}$ $\cup \{b_{i+2},b_{i+3},b_{i+4},\dots ,b_{\frac{t+2i}{2}}\}$ $\cup \{c_{i+2},c_{i+3},c_{i+4},\dots ,c_{\frac{t+2i}{2}}\}$, $L_r\left(c_i{c}_{i+1}\right)=V\left(L{D}_t\right)$ $-\{a_i,a_{i+1},a_{\frac{t+2i+1}{2}}\}\cup \{b_i\}\cup \{c_{\frac{t+2i+1}{2}}\}$ and $L_r\left(b_i{c}_{i+1}\right)=V\left(L{D}_t\right)-\{a_{i+2},a_{i+4},a_{\frac{t+2i+1}{2}}\}\cup$ $\{b_{\frac{t+2i+1}{2}},b_{\frac{t+2i+3}{2}},b_{\frac{t+2i+5}{2}},\dots ,b_{t+i-1}\}$ $\cup \{c_i,c_{\frac{t+2i+1}{2}},$ $c_{\frac{t+2i+3}{2}},\dots ,c_{t+i-1}\}.$ Since ${\displaystyle \bigcup _{i=1}^t}{L}_r\left(b_i{a}_{i+1}\right)$ $=V\left(L{D}_t\right)$, $|L_r\left(x\right)\cap {\displaystyle \bigcup _{i=1}^t}{L}_r$ $\left(b_i{a}_{i+1}\right)|\ge |L_r\left(b_i{a}_{i+1}\right)|$. The comparison among the cardinalities of all the $LRN$ sets is given in

Table 3. Comparison between the cardinalities of $LRN$ sets of $L{D}_t$.

LRN Set	Cardinality
$L_r\left(a_i{a}_{i+1}\right)$	$3t-5>t+3$
$L_r\left(a_i{b}_i\right)$	$\frac{3t+3}{2}>t+3$
$L_r\left(b_i{c}_i\right)$	$t+7>t+3$
$L_r\left(c_i{c}_{i+1}\right)$	$\frac{5t-1}{2}>t+3$
$L_r\left(b_i{c}_{i+1}\right)$	$\frac{3t+3}{2}>t+3$

.

 □

It is clear from Table 3 that $|L_r\left(b_i{a}_{i+1}\right)|<|L_r\left(x\right)|$, where $L_r\left(x\right)$ are the other $LRN$ sets of $L{D}_t$.

Theorem 5.

Let $L{D}_t$ be a line network of prism network then

$$Ldi{m}_F\left(L{D}_5\right)=\frac{3}{2}.$$

Proof. 

The $LRN$ sets of $L{D}_5$ are given by:

$L_r\left(a_1{a}_2\right)=V\left(L{D}_5\right)-\{a_4,b_1,c_1,c_2,c_4\}$,

$L_r\left(a_2{a}_3\right)=V\left(L{D}_5\right)-\{a_5,b_2,c_2,c_3,c_5\}$,

$L_r\left(a_3{a}_4\right)=V\left(L{D}_5\right)-\{a_1,b_3,c_3,c_4,c_1\}$,

$L_r\left(a_4{a}_5\right)=V\left(L{D}_5\right)-\{a_2,b_4,c_4,c_5,c_2\}$,

$L_r\left(a_5{a}_1\right)=V\left(L{D}_5\right)-\{a_3,b_5,c_5,c_1,c_3\}$,

$L_r\left(a_1{b}_1\right)=V\left(L{D}_5\right)-\{a_2,a_3,b_2,c_4,c_5\}$,

$L_r\left(a_2{b}_2\right)=V\left(L{D}_5\right)-\{a_3,a_4,b_3,c_5,c_1\}$,

$L_r\left(a_3{b}_3\right)=V\left(L{D}_5\right)-\{a_4,a_5,b_4,c_1,c_2\}$,

$L_r\left(a_4{b}_4\right)=V\left(L{D}_5\right)-\{a_5,a_1,b_5,c_2,c_3\}$,

$L_r\left(a_5{b}_5\right)=V\left(L{D}_5\right)-\{a_1,a_2,b_1,c_3,c_4\}$,

$L_r\left(b_1{a}_2\right)=V\left(L{D}_5\right)-\{a_1,a_5,b_4,b_5,c_4\}$,

$L_r\left(b_2{a}_3\right)=V\left(L{D}_5\right)-\{a_2,a_1,b_5,b_1,c_5\}$,

$L_r\left(b_3{a}_4\right)=V\left(L{D}_5\right)-\{a_3,a_2,b_1,b_2,c_1\}$,

$L_r\left(b_4{a}_5\right)=V\left(L{D}_5\right)-\{a_4,a_3,b_2,b_3,c_2\}$,

$L_r\left(b_5{a}_1\right)=V\left(L{D}_5\right)-\{a_5,a_4,b_3,b_4,c_3\}$,

$L_r\left(b_1{c}_1\right)=V\left(L{D}_5\right)-\{a_4,a_5,b_4,b_5,c_3\}$,

$L_r\left(b_2{c}_2\right)=V\left(L{D}_5\right)-\{a_5,a_1,b_5,b_1,c_4\}$,

$L_r\left(b_3{c}_3\right)=V\left(L{D}_5\right)-\{a_1,a_5,b_4,b_5,c_5\}$,

$L_r\left(b_4{c}_4\right)=V\left(L{D}_5\right)-\{a_2,a_1,b_5,b_1,c_1\}$,

$L_r\left(b_5{c}_5\right)=V\left(L{D}_5\right)-\{a_3,a_2,b_1,b_2,c_2\}$,

$L_r\left(b_1{c}_2\right)=V\left(L{D}_5\right)-\{a_3,a_4,b_4,b_5,c_1\}$,

$L_r\left(b_2{c}_3\right)=V\left(L{D}_5\right)-\{a_4,a_5,b_5,b_1,c_2\}$,

$L_r\left(b_3{c}_4\right)=V\left(L{D}_5\right)-\{a_5,a_1,b_1,b_2,c_3\}$,

$L_r\left(b_4{c}_5\right)=V\left(L{D}_5\right)-\{a_1,a_2,b_2,b_3,c_4\}$,

$L_r\left(b_5{c}_1\right)=V\left(L{D}_5\right)-\{a_2,a_3,b_3,b_4,c_5\}$,

$L_r\left(c_1{c}_2\right)=V\left(L{D}_5\right)-\{a_1,a_2,a_4,b_1,c_4\}$,

$L_r\left(c_2{c}_3\right)=V\left(L{D}_5\right)-\{a_2,a_3,a_5,b_2,c_5\}$,

$L_r\left(c_3{c}_4\right)=V\left(L{D}_5\right)-\{a_3,a_4,a_1,b_3,c_1\}$,

$L_r\left(c_4{c}_5\right)=V\left(L{D}_5\right)-\{a_4,a_5,a_2,b_4,c_2\}$,

$L_r\left(c_5{c}_1\right)=V\left(L{D}_5\right)-\{a_5,a_1,a_3,b_5,c_3\}$.

Since the cardinality of each $LRN$ set of $L{D}_5$ is 10, we define a constant $LRF$ $\lambda \left(V\left(L{D}_5\right)\right)$ $\to [0,1]$ as $\frac{1}{10}$ to each $v\in V\left(L{D}_5\right)$, hence

$$Ldi{m}_F\left(L{D}_5\right)={\displaystyle \sum _{i=1}^{15}}\frac{1}{10}=\frac{3}{2}.$$

□

Theorem 6.

Let $L{D}_t$ be a line network of prism network, where $t\cong 1\left(mod2\right)$, then

$$\frac{3t}{3t-5}\le Ldi{m}_F\left(L{D}_t\right)\le \frac{3t}{t+3}.$$

Proof. 

To prove the theorem, we have divided it in two cases:

Case 1:

The $LRN$ sets of $L{D}_7$ are given by:

$L_r\left(b_1{c}_1\right)=V\left(L{D}_7\right)-\{a_5,a_6,a_7,b_5,b_6,b_7,c_2,c_3,c_4\}$,

$L_r\left(b_2{c}_2\right)=V\left(L{D}_7\right)-\{a_6,a_7,a_1,b_6,b_7,b_1,c_3,c_4,c_5\}$,

$L_r\left(b_3{c}_3\right)=V\left(L{D}_7\right)-\{a_7,a_1,a_2,b_7,b_1,b_2,c_4,c_5,c_6\}$,

$L_r\left(b_4{c}_4\right)=V\left(L{D}_7\right)-\{a_1,a_2,a_3,b_1,b_2,b_3,c_5,c_6,c_7\}$,

$L_r\left(b_5{c}_5\right)=V\left(L{D}_7\right)-\{a_2,a_3,a_4,b_2,b_3,b_4,c_6,c_7,c_1\}$,

$L_r\left(b_6{c}_6\right)=V\left(L{D}_7\right)-\{a_3,a_4,a_5,b_3,b_4,b_5,c_7,c_1,c_2\}$,

$L_r\left(b_7{c}_7\right)=V\left(L{D}_7\right)-\{a_4,a_5,a_6,b_4,b_5,b_6,c_1,c_2,c_3\}$,

$L_r\left(a_1{b}_1\right)=V\left(L{D}_7\right)-\{a_2,a_3,a_4,b_2,b_3,b_4,c_5,c_6,c_7\}$,

$L_r\left(a_2{b}_2\right)=V\left(L{D}_7\right)-\{a_3,a_4,a_5,b_3,b_4,b_5,c_6,c_7,c_1\}$,

$L_r\left(a_3{b}_3\right)=V\left(L{D}_7\right)-\{a_4,a_5,a_6,b_4,b_5,b_6,c_7,c_1,c_2\}$,

$L_r\left(a_4{b}_4\right)=V\left(L{D}_7\right)-\{a_5,a_6,a_7,b_5,b_6,b_7,c_1,c_2,c_3\}$,

$L_r\left(a_5{b}_5\right)=V\left(L{D}_7\right)-\{a_6,a_7,a_1,b_6,b_7,b_1,c_2,c_3,c_4\}$,

$L_r\left(a_6{b}_6\right)=V\left(L{D}_7\right)-\{a_7,a_1,a_2,b_7,b_1,b_2,c_3,c_4,c_5\}$,

$L_r\left(a_7{b}_7\right)=V\left(L{D}_7\right)-\{a_1,a_2,a_3,b_1,b_2,b_3,c_4,c_5,c_6\}$,

$L_r\left(b_1{a}_2\right)=V\left(L{D}_7\right)-\{a_1,a_6,a_7,b_5,b_6,b_7,c_3,c_4,c_5\}$,

$L_r\left(b_2{a}_3\right)=V\left(L{D}_7\right)-\{a_2,a_7,a_1,b_6,b_7,b_1,c_4,c_5,c_6\}$,

$L_r\left(b_3{a}_4\right)=V\left(L{D}_7\right)-\{a_3,a_1,a_2,b_7,b_1,b_2,c_5,c_6,c_7\}$,

$L_r\left(b_4{a}_5\right)=V\left(L{D}_7\right)-\{a_4,a_2,a_3,b_1,b_2,b_3,c_6,c_7,c_1\}$,

$L_r\left(b_5{a}_6\right)=V\left(L{D}_7\right)-\{a_5,a_3,a_4,b_2,b_3,b_4,c_7,c_1,c_2\}$,

$L_r\left(b_6{a}_7\right)=V\left(L{D}_7\right)-\{a_6,a_4,a_5,b_3,b_4,b_5,c_1,c_2,c_3\}$,

$L_r\left(a_1{a}_2\right)=V\left(L{D}_7\right)-\{a_5,b_1,c_1,c_2,c_5\}$,

$L_r\left(a_2{a}_3\right)=V\left(L{D}_7\right)-\{a_6,b_2,c_2,c_3,c_6\}$,

$L_r\left(a_3{a}_4\right)=V\left(L{D}_7\right)-\{a_7,b_3,c_3,c_4,c_7\}$,

$L_r\left(a_4{a}_5\right)=V\left(L{D}_7\right)-\{a_1,b_4,c_4,c_5,c_1\}$,

$L_r\left(a_5{a}_5\right)=V\left(L{D}_7\right)-\{a_2,b_5,c_5,c_6,c_2\}$,

$L_r\left(a_6{a}_7\right)=V\left(L{D}_7\right)-\{a_3,b_6,c_6,c_7,c_3\}$,

$L_r\left(a_7{a}_1\right)=V\left(L{D}_7\right)-\{a_4,b_7,c_7,c_1,c_4\}$,

$L_r\left(c_1{c}_2\right)=V\left(L{D}_7\right)-\{a_1,a_2,a_5,b_1,c_5\}$,

$L_r\left(c_2{c}_3\right)=V\left(L{D}_7\right)-\{a_2,a_3,a_6,b_2,c_6\}$,

$L_r\left(c_3{c}_4\right)=V\left(L{D}_7\right)-\{a_3,a_4,a_7,b_3,c_7\}$,

$L_r\left(c_4{c}_5\right)=V\left(L{D}_7\right)-\{a_4,a_5,a_1,b_4,c_1\}$,

$L_r\left(c_5{c}_6\right)=V\left(L{D}_7\right)-\{a_5,a_6,a_2,b_5,c_2\}$,

$L_r\left(c_6{c}_7\right)=V\left(L{D}_7\right)-\{a_6,a_7,a_3,b_6,c_3\}$,

$L_r\left(c_7{c}_1\right)=V\left(L{D}_7\right)-\{a_7,a_1,a_4,b_7,c_4\}$.

From the above $LRN$ sets, the $LRN$ sets having the minimum cardinalities are $L_r\left(b_i{c}_i\right)$, $L_r\left(a_i{b}_i\right)$ and $L_r\left(b_i{a}_{i+1}\right)$ and the cardinality of each of them is 12, where $1\le i\le 7$ therefore, we define a minimal $LRF$ $\lambda \left(V\left(L{D}_7\right)\right)\to [0,1]$ as $\frac{1}{12}$ to each $v\in V\left(L{W}_7\right)$, hence $Ldi{m}_F\left(L{D}_7\right)\le {\displaystyle \sum _{i=1}^{21}}\frac{1}{12}=\frac{7}{4}$. The $LRN$ sets having maximum cardinality are $L_r\left(a_i{a}_{i+1}\right)$, $L_r\left(c_i{c}_{i+1}\right)$, where $1\le i\le 7$ and cardinality of each of them is 17; therefore, we define a maximal $LRF$ ${\lambda }^{\prime }\left(V\left(L{D}_7\right)\right)\to [0,1]$ as $\frac{1}{17}$ to each $v\in V\left(L{D}_7\right)$, hence $Ldi{m}_F\left(L{D}_7\right)\ge {\displaystyle \sum _{i=1}^{21}}\frac{1}{17}=\frac{21}{17}.$ The bounds of $LFMD$ of $L{D}_7$ are given as follows:

$$\frac{21}{17}\le Ldi{m}_F\left(L{D}_7\right)\le \frac{7}{4}.$$

Case 2:

For $t\ge 7$, in the view of Lemma 3, the cardinality of $LRN$ set $L_r\left(b_i{a}_{i+1}\right)$ is $t+3$ and $|L_r\left(b_i{a}_{i+1}\right)|<|L_r\left(x\right)|,$ where $L_r\left(x\right)$ are other $LRN$ sets of $L{D}_t$, where $1\le i\le t.$ Therefore, we define a minimal $LRF$ $\lambda (V\left(L{D}_t\right)\to [0,1]$ as $\frac{1}{t+3}$ to each $v\in V\left(L{D}_t\right)$, hence $Ldi{m}_F\left(L{D}_t\right)\le {\displaystyle \sum _{i=1}^{3t}}\frac{1}{t+3}=\frac{3t}{t+3}$. In the same context by Lemma 3 the maximum cardinality of $LRN$ set $L_r\left(a_i{a}_{i+1}\right)$ is $3t-5$ and $|L_r\left(a_i{a}_{i+1}\right)|>|L_r\left(x\right)|,$ where $L_r\left(x\right)$ are other $LRN$ sets of $L{W}_t$, where $1\le i\le t.$ Therefore, we define a maximal $LRF$ ${\lambda }^{\prime }\left(V\left(L{W}_t\right)\right)\to [0,1]$ as $\frac{1}{3t-5}$ to each $v\in V\left(L{D}_t\right)$, hence $Ldi{m}_F\left(L{D}_t\right)\ge {\displaystyle \sum _{i=1}^{3t}}\frac{1}{3t-5}=\frac{3t}{3t-5}$. The bounds of $LFMD$ of $L{D}_t$ are given as follows:

$$\frac{3t}{3t-5}\le Ldi{m}_F\left(L{D}_t\right)\le \frac{3t}{t+3}.$$

□

Lemma 4.

Let $L{D}_t$ be the line network of prism network then, where $t\cong 0\left(mod2\right)$ then

(a)

$|L_r\left(b_i{a}_{i+1}\right)|=\frac{3t+4}{2}$ and $|{\displaystyle \bigcup _{i=1}^t}{L}_r\left(b_i{a}_{i+1}\right)|=3t;$

(b)

$|L_r\left(b_i{a}_{i+1}\right)|\le |L_r\left(x\right)|$ and $|L_r\left(x\right)\cap {\displaystyle \bigcup _{i=1}^t}{L}_r\left(b_i{a}_{i+1}\right)|\ge |L_r\left(b_i{a}_{i+1}\right)|,\forall x\in E\left(L{D}_t\right).$

Proof. 

Consider $a_i$ inner, $b_i$ middle, and $c_i$ are outer vertices of $L{D}_t$, where $1\le i\le t$ and $t+1\cong 1\left(modt\right).$

(a)

$L_r\left(a_i{b}_i\right)=V\left(L{D}_t\right)-\{a_{i+1},a_{i+2},a_{i+3},\dots ,a_{\frac{n+2i}{2}}\}\cup \{b_{i+1},b_{i+2},\dots ,b_{\frac{t+2i-2}{2}},b_{\frac{t+2i+2}{2}}\}$

$\cup \{c_{\frac{t+2i+2}{2}},c_{\frac{t+2i+4}{2}},c_{\frac{t+2i+6}{2}},\dots ,c_{t+i-1}\}$, $L_r\left(b_i{a}_{i+1}\right)=V\left(L{D}_t\right)-\{a_{i+1},a_{i+2},a_{i+3},\dots ,a_{\frac{t+2i}{2}}\}$

$\cup \{b_{i+1},b_{i+2},\dots ,b_{\frac{t+2i-2}{2}},b_{\frac{t+2i+2}{2}}\}\cup \{c_{\frac{t+2i+2}{2}},c_{\frac{t+2i+4}{2}},c_{\frac{t+2i+6}{2}},\dots ,c_{t+i-1}\}$, $L_r\left(b_i{c}_i\right)=V\left(L{D}_t\right)$$-\{a_{\frac{t+2i+2}{2}},a_{\frac{t+2i+4}{2}},a_{\frac{t+2i+6}{2}},\dots ,a_{t+i-1}\}\cup \{b_{i+2},b_{i+3},b_{i+4},\dots ,b_{\frac{t+2i-2}{2}}\}\cup \{c_{i+1},c_{i+2},c_{i+3},\dots ,$$c_{\frac{t+2i}{2}}\}$. Therefore, $|L_r\left(b_i{a}_{i+1}\right)|=|L_r\left(a_i{b}_i\right)|=|L_r\left(b_i{c}_i\right)|=\frac{3t+4}{2}$ also $|{\displaystyle \bigcup _{i=1}^t}LR\left(b_i{a}_{i+1}\right)|=3t;$

(b)

$L_r\left(a_i{a}_{i+1}\right)=V\left(L{D}_t\right)-\{b_i,c_i,c_{i+1}\}$ and $L_r\left(c_i{c}_{i+1}\right)=V\left(L{D}_t\right)-\{a_i,a_{i+1},b_i\}.$ The comparison among the cardinalities of all the $LRN$ sets is given in

Table 4. Comparison among the cardinalities of $LRN$ sets of $L{D}_t$.

LRN Set	Cardinality
$L_r\left(a_i{a}_{i+1}\right)$	$3t-3>\frac{3t+4}{2}$
$L_r\left(c_i{c}_{i+1}\right)$	$3t-3>\frac{3t+4}{2}$

.

□

Theorem 7.

Let $L{D}_4$ be a line network of prism network, then

$$Ldi{m}_F\left(L{D}_4\right)=\frac{3}{2}.$$

Proof. 

For $t=4$, we have following LRN sets:

$L_r\left(a_1{a}_2\right)=V\left(L{D}_4\right)-\{b_1,b_3,c_1,c_2\}$,

$L_r\left(a_2{a}_3\right)=V\left(L{D}_4\right)-\{b_2,b_4,c_2,c_3\}$,

$L_r\left(a_3{a}_4\right)=V\left(L{D}_4\right)-\{b_3,b_5,c_3,c_4\}$,

$L_r\left(a_4{a}_1\right)=V\left(L{D}_4\right)-\{b_4,b_6,c_4,c_1\}$,

$L_r\left(a_1{b}_1\right)=V\left(L{D}_4\right)-\{a_2,a_3,b_2,c_4\}$,

$L_r\left(a_2{b}_2\right)=V\left(L{D}_4\right)-\{a_3,a_4,b_3,c_1\}$,

$L_r\left(a_3{b}_3\right)=V\left(L{D}_4\right)-\{a_4,a_1,b_4,c_2\}$,

$L_r\left(a_4{b}_4\right)=V\left(L{D}_4\right)-\{a_1,a_2,b_1,c_3\}$,

$L_r\left(b_1{a}_2\right)=V\left(L{D}_4\right)-\{a_1,a_4,b_3,c_3\}$,

$L_r\left(b_2{a}_3\right)=V\left(L{D}_4\right)-\{a_2,a_1,b_4,c_4\}$,

$L_r\left(b_3{a}_4\right)=V\left(L{D}_4\right)-\{a_3,a_2,b_1,c_1\}$,

$L_r\left(b_4{a}_1\right)=V\left(L{D}_4\right)-\{a_4,a_3,b_2,c_2\}$,

$L_r\left(b_1{c}_1\right)=V\left(L{D}_4\right)-\{a_4,b_2,c_2,c_3\}$,

$L_r\left(b_2{c}_2\right)=V\left(L{D}_4\right)-\{a_1,b_3,c_3,c_4\}$,

$L_r\left(b_2{c}_3\right)=V\left(L{D}_4\right)-\{a_2,b_4,c_4,c_1\}$,

$L_r\left(b_1{c}_1\right)=V\left(L{D}_4\right)-\{a_4,b_2,c_2,c_3\}$,

$L_r\left(b_2{c}_2\right)=V\left(L{D}_4\right)-\{a_1,b_3,c_3,c_4\}$,

$L_r\left(b_3{c}_3\right)=V\left(L{D}_4\right)-\{a_2,b_4,c_4,c_1\}$,

$L_r\left(b_4{c}_4\right)=V\left(L{D}_4\right)-\{a_3,b_1,c_1,c_2\}$,

$L_r\left(b_1{c}_2\right)=V\left(L{D}_4\right)-\{a_3,b_4,c_1,c_4\}$,

$L_r\left(b_2{c}_3\right)=V\left(L{D}_4\right)-\{a_4,b_1,c_2,c_1\}$,

$L_r\left(b_2{c}_3\right)=V\left(L{D}_4\right)-\{a_1,b_2,c_3,c_2\}$,

$L_r\left(b_3{c}_4\right)=V\left(L{D}_4\right)-\{a_2,b_3,c_4,c_3\}$,

$L_r\left(c_1{a}_2\right)=V\left(L{D}_4\right)-\{a_1,a_2,b_1,b_3\}$,

$L_r\left(c_2{a}_3\right)=V\left(L{D}_4\right)-\{a_2,a_3,b_2,b_4\}$,

$L_r\left(c_3{a}_4\right)=V\left(L{D}_4\right)-\{a_3,a_4,b_3,b_1\}$,

$L_r\left(c_4{a}_1\right)=V\left(L{D}_4\right)-\{a_4,a_1,b_4,b_2\}$.

Since the cardinality of each $LRN$ set of $L{D}_4$ is 10, therefore, we define a constant $LRF$ $\lambda \left(V\left(L{D}_4\right)\right)\to [0,1]$ as $\frac{1}{8}$ to each $v\in V\left(L{D}_4\right)$, hence

$$Ldi{m}_F\left(L{D}_4\right)={\displaystyle \sum _{i=1}^{12}}\frac{1}{8}=\frac{3}{2}.$$

□

Theorem 8.

Let $L{D}_t$ be a line network of prism network, where $t\cong 0\left(mod2\right).$ Then

$$\frac{t}{t-1}\le Ldi{m}_F\left(L{D}_t\right)\le \frac{6t}{3t+4}.$$

Proof. 

To prove the theorem, we have divided it in two cases:

Case 1:

The $LRN$ sets of $L{D}_6$ are:

$L_r\left(a_1{a}_2\right)=V\left(L{D}_6\right)-\{b_1,c_1,c_2\}$,

$L_r\left(a_2{a}_3\right)=V\left(L{D}_6\right)-\{b_2,c_2,c_3\}$,

$L_r\left(a_3{a}_4\right)=V\left(L{D}_6\right)-\{b_3,c_3,c_4\}$,

$L_r\left(a_4{a}_5\right)=V\left(L{D}_6\right)-\{b_4,c_4,c_5\}$,

$L_r\left(a_5{a}_6\right)=V\left(L{D}_6\right)-\{b_5,c_5,c_6\}$,

$L_r\left(a_6{a}_1\right)=V\left(L{D}_6\right)-\{b_6,c_6,c_1\}$,

$L_r\left(c_1{c}_2\right)=V\left(L{D}_6\right)-\{a_1,a_2,b_1,b_4\}$,

$L_r\left(c_2{c}_3\right)=V\left(L{D}_6\right)-\{a_2,a_3,b_2,b_5\}$,

$L_r\left(c_3{c}_4\right)=V\left(L{D}_6\right)-\{a_3,a_4,b_3,b_6\}$,

$L_r\left(c_4{c}_5\right)=V\left(L{D}_6\right)-\{a_4,a_5,b_4,b_1\}$,

$L_r\left(c_5{c}_6\right)=V\left(L{D}_6\right)-\{a_5,a_6,b_5,b_2\}$,

$L_r\left(c_6{c}_1\right)=V\left(L{D}_6\right)-\{a_6,a_1,b_6,b_3\}$,

$L_r\left(a_1{b}_1\right)=V\left(L{D}_6\right)-\{a_2,a_3,a_4,b_2,b_3,c_5,c_6\}$,

$L_r\left(a_2{b}_2\right)=V\left(L{D}_6\right)-\{a_3,a_4,a_5,b_3,b_4,c_6,c_1\}$,

$L_r\left(a_3{b}_3\right)=V\left(L{D}_6\right)-\{a_4,a_5,a_6,b_4,b_5,c_1,c_2\}$,

$L_r\left(a_4{b}_4\right)=V\left(L{D}_6\right)-\{a_5,a_6,a_1,b_5,b_6,c_2,c_3\}$,

$L_r\left(a_5{b}_5\right)=V\left(L{D}_6\right)-\{a_6,a_1,a_2,b_6,b_1,c_3,c_4\}$,

$L_r\left(a_6{b}_6\right)=V\left(L{D}_6\right)-\{a_1,a_2,a_3,b_1,b_2,c_4,c_5\}$,

$L_r\left(b_1{a}_2\right)=V\left(L{D}_6\right)-\{a_1,a_5,a_6,b_5,b_6,c_3,c_4\}$,

$L_r\left(b_2{a}_3\right)=V\left(L{D}_6\right)-\{a_2,a_6,a_1,b_6,b_1,c_4,c_5\}$,

$L_r\left(b_3{a}_4\right)=V\left(L{D}_6\right)-\{a_2,a_3,a_4,b_2,b_3,c_5,c_6\}$,

$L_r\left(b_4{a}_5\right)=V\left(L{D}_6\right)-\{a_3,a_4,a_5,b_3,b_4,c_6,c_1\}$,

$L_r\left(b_5{a}_6\right)=V\left(L{D}_6\right)-\{a_4,a_5,a_6,b_4,b_5,c_1,c_2\}$,

$L_r\left(b_6{a}_1\right)=V\left(L{D}_6\right)-\{a_5,a_6,a_1,b_5,b_6,c_2,c_3\}$,

$L_r\left(b_1{c}_1\right)=V\left(L{D}_6\right)-\{a_5,a_6,b_2,b_3,c_2,c_3,c_4\}$,

$L_r\left(b_2{c}_2\right)=V\left(L{D}_6\right)-\{a_6,a_1,b_3,b_4,c_3,c_4,c_5\}$,

$L_r\left(b_3{c}_3\right)=V\left(L{D}_6\right)-\{a_1,a_2,b_4,b_5,c_4,c_5,c_6\}$,

$L_r\left(b_4{c}_4\right)=V\left(L{D}_6\right)-\{a_2,a_3,b_5,b_6,c_5,c_6,c_1\}$,

$L_r\left(b_5{c}_5\right)=V\left(L{D}_6\right)-\{a_3,a_4,b_6,b_1,c_6,c_1,c_2\}$,

$L_r\left(b_6{c}_6\right)=V\left(L{D}_6\right)-\{a_4,a_5,b_1,b_2,c_1,c_2,c_3\}$,

$L_r\left(b_1{c}_2\right)=V\left(L{D}_6\right)-\{a_3,a_4,b_5,b_6,c_1,c_5,c_6\}$,

$L_r\left(b_1{c}_2\right)=V\left(L{D}_6\right)-\{a_3,a_4,b_5,b_6,c_1,c_5,c_6\}$,

$L_r\left(b_2{c}_3\right)=V\left(L{D}_6\right)-\{a_4,a_5,b_6,b_1,c_2,c_6,c_1\}$,

$L_r\left(b_3{c}_4\right)=V\left(L{D}_6\right)-\{a_5,a_6,b_1,b_2,c_3,c_1,c_2\}$,

$L_r\left(b_4{c}_5\right)=V\left(L{D}_6\right)-\{a_6,a_1,b_2,b_3,c_4,c_2,c_3\}$,

$L_r\left(b_5{c}_6\right)=V\left(L{D}_6\right)-\{a_1,a_2,b_3,b_4,c_5,c_3,c_4\}$,

$L_r\left(b_6{c}_1\right)=V\left(L{D}_6\right)-\{a_2,a_3,b_4,b_5,c_6,c_4,c_5\}$.

The $LRN$ sets with a minimum cardinality are $L_r\left(b_i{c}_i\right)$, $L_r\left(a_i{b}_i\right)$, $L_r\left(b_i{a}_{i+1}\right)$, $L_r\left(b_i{c}_{i+1}\right)$ and the cardinality of each of them is 11, where $1\le i\le 6$. Therefore, we define a minimal $LRF$ $\lambda \left(V\left(L{D}_6\right)\right)\to [0,1]$ as $\frac{1}{11}$ to each $v\in V\left(L{D}_6\right)$, hence $Ldi{m}_F\left(L{D}_6\right)\le {\displaystyle \sum _{i=1}^{18}}\frac{1}{11}=\frac{18}{11}$. The $LRN$ set having maximum cardinality is $L_r\left(a_i{a}_{i+1}\right)$, and its cardinality is 15; therefore, we define a maximal $LRN$ $\lambda \left(V\left(L{D}_6\right)\right)\to [0,1]$ as $\frac{1}{15}$ to each $v\in V\left(L{D}_6\right)$, hence $Ldi{m}_F\left(L{D}_6\right)\ge {\displaystyle \sum _{i=1}^{18}}\frac{1}{15}=\frac{6}{5}.$ The bounds of $LFMD$ of $L{D}_6$ is given as follows:

$$\frac{6}{5}\le Ldi{m}_F\left(L{D}_6\right)\le \frac{18}{11}.$$

Case 2:

For $t\ge 6$, in the view of Lemma 4, the cardinalities of the $LRN$ sets $L_r\left(a_i{b}_i\right)$, $L_r\left(b_i{a}_{i+1}\right),$ $L_r\left(b_i{c}_i\right)$, and $L_r\left(b_i{c}_{i+1}\right)$ is $\frac{3t+4}{2}$ and $|L_r\left(a_i{b}_i\right)|\le |L_r\left(x\right)|,$ where $L_r\left(x\right)$ are other $LRN$ sets of $L{D}_t$, where $1\le i\le t.$ Therefore, we define a minimal $LRF$ $\lambda \left(V\left(L{D}_t\right)\right)\to [0,1]$ as $\frac{2}{3t+4}$ to each $v\in V\left(L{D}_t\right)$, hence $Ldi{m}_F\left(L{D}_t\right)\le {\displaystyle \sum _{i=1}^{3t}}\frac{2}{3t+4}=\frac{6t}{3t+4}$. In the same context by Lemma 4 the maximum cardinalities of the $LRN$ sets are $L_r\left(a_i{a}_{i+1}\right)$ and $L_r\left(c_i{c}_{i+1}\right)$ is $3t-3$ and $|L_r\left(a_i{a}_{i+1}\right)|\ge |L_r\left(x\right)|,$ where $L_r\left(x\right)$ are other $LRN$ sets of $L{D}_t$, where $1\le i\le t.$ Therefore, we define a maximal $LRF$ ${\lambda }^{\prime }\left(V\left(L{D}_t\right)\right)\to [0,1]$ as $\frac{1}{3t-3}$ to each $v\in V\left(L{D}_t\right)$, hence $Ldi{m}_F\left(L{D}_t\right)\ge {\displaystyle \sum _{i=1}^{3t}}\frac{1}{3t-3}=\frac{t}{t-1}$. Hence, the bounds of $LFMD$ of $L{D}_t$ are given as follows:

$$\frac{t}{t-1}\le Ldi{m}_F\left(L{D}_t\right)\le \frac{6t}{3t+4}.$$

□

4. Conclusions

In this manuscript, we have established sharp bounds of the $LFMD$ of the rotationally symmetric line networks of the wheel $\left(L{W}_t\right)$ and prism $\left(L{D}_t\right)$. It is proved that for $t=3$, $L{W}_t$ attains the exact value of $LFMD$ which is $\frac{3}{2}$ and for $t=4,5$ the $LFMD$ of $L{D}_t$ is $\frac{3}{2}$ as well. It has been observed that the $L{W}_t$ remains unbounded and $L{D}_t$ remains bounded under $LFMD$, when the order of these networks approaches ∞. The boundedness and unboundedness of these networks is illustrated in

Table 5. Boundedness and unboundedness of $L{W}_t$ and $L{D}_t$ via $LFMD$.

Network	LFMD	Lower Bound	Upper Bound	Comment
$L{W}_t$, $t\cong$ 1(mod 2)	$\frac{2t}{2t-3}\le Ldi{m}_F\left(L{W}_t\right)\le \frac{t}{2}.$	1	∞	Unbounded
$L{W}_t$, $t\cong$ 0(mod 2)	$\frac{t}{t-2}\le Ldi{m}_F\left(L{W}_t\right)\le \frac{t}{2}.$	1	∞	Unbounded
$L{D}_t$, $t\cong$ 1(mod 2)	$\frac{3t}{3t-5}\le Ldi{m}_F\left(L{D}_t\right)\le \frac{3t}{t+3}.$	1	3	Bounded
$L{D}_t$, $t\cong$ 0(mod 2)	$\frac{t}{t-1}\le Ldi{m}_F\left(L{D}_t\right)\le \frac{6t}{3t+4}.$	1	2	Bounded

. Furthermore, the results are more precise as the both lower and upper bounds $LFMD$ of these line networks have been established. Now in the end of our discussion, we suggest an open problem that characterizes all the rotationally symmetric networks having the exact value of $LFMD$.

$3D$ representation of lower and upper bounds of LFMD of rotationally symmetric line networks