Metric-Based Fractional Dimension of Rotationally-Symmetric Line Networks

: 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 3 D shapes.


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 LFMDs for certain rotationally symmetric networks.In this manuscript, our aim is to compute both the upper and lower bounds of LFMDs of rotationally symmetric line networks of wheels and prism networks.The detail of line networks prism and wheel network is given from Figures 1 and 2, 3D representation of all the obtained results is given from Figures 3-9.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.

Preliminaries
Let B be a connected network with vertex set V = V(B) and edge set E = E(B).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 ∈ V(B) resolves a pair (v, w) if the distance from u to v is not equal to the distance from u to w (d(u, v) = d(u, w)).Let L = {u 1 , u 2 , u 3 , ..., u t } ⊆ V(B), 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 ), ..., d(v, u t )).If distinct elements of 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 B, thus MD of B is defined as follows: In a connected network B for uv ∈ E(B) a vertex x ∈ V(B) is said to resolve adjacent pairs of vertices as L r (uv) = {x ∈ V(B) : d(x, u) = d(x, u)} and it is called a local resolving neighborhood (LRN) set of an edge uv ∈ E(B).A real-valued function λ : Thus, the LFMD of B is defined as follows: Ldim F (B) = min{|λ| : λ is a minimal local resolving f unction o f B}.
A line network L(B) of B is a network whose vertices are the edges of B, and two vertices u, v ∈ L(B) are connected iff they have a common end vertex in B. For more results about line networks and their MD, we refer [33,34].

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 LFMDs 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.

LRN Sets and LFMD of Line Network of Wheel Network
In this subsection, our aim to compute the LRN sets and the LFMDs of the line network of wheel networks.The network is defined as follows: Let LW t be a line network of a wheel network with a vertex set 2 .For more information about LW t , see Figure 1.
Lemma 1.Let LW t be the line network of wheel network, where t ∼ = 1 (mod 2).Then Proof.Consider a i inner and b i are outer vertices of LW t , where 1 ≤ i ≤ t and t The comparison among the cardinalities of all the LRN sets of LW t is given in Table 1.
Table 1.Comparison between the cardinalities of LRN sets of LW t .

LRN Set Cardinality
Theorem 1.Let LW 3 be a line network of generalized wheel network, then Proof.The LRN sets of LW 3 are as follows: From above, LRN sets the cardinality of all the LRN sets as 4, therefore, we define a constant mapping λ(V(LW 3 )) → [0, 1] as 1  4 to each v ∈ V(LW 3 ), hence Theorem 2. Let LW t be a line network of a wheel network, where t ∼ = 1 (mod 2).Then Proof.To prove the theorem, we have divided it in two cases: Case 1: For t = 5, we have following LRN sets: From above, LRN sets the minimum cardinality of LRN set L r (a i a i+1 ) as 4, where 1 ≤ i ≤ 5; therefore, we define a minimal LRF λ(V(LW 5 )) → [0, 1] as 1  4 to each v ∈ The maximum cardinality of (LRN) set L r (a i b i ) is 7; therefore, we define a maximal LRF λ (V(LW 5 )) → [0, 1] as 1  7 to each v ∈ V(LW In the same context, by Lemma 1, the maximum cardinality of Lemma 2. Let LW t be the line network of wheel network then, where t ∼ = 0(mod 2).Then Proof.Consider a i inner and b i are outer vertices of LW t , where 1 ≤ i ≤ t and The comparison among the cardinalities of all the LRN sets is given in Table 2.

LRN Set Cardinality
, where L r (x) are the other LRN sets of LW t .Theorem 3. Let LW 4 be the line network of generalized wheel network then Proof.The LRN sets of LW 4 are given as follows: From the above, LRN sets the minimum cardinality of LRN set L r (a i a i+1 ) as 4, where 1 ≤ i ≤ 4; therefore, we define a minimal LRF λ(V(LW 4 )) → [0, 1] as 1  4 to Theorem 4. Let LW t be a line network of a generalized wheel network, where t ∼ = 0 (mod2), Proof.To prove the theorem, we have divided it in two cases: Case 1: For t = 6, we have the following possible LRN sets: From above, LRN sets the cardinality of LRN set L r (a i a i+1 ) as 4, where 1 ≤ i ≤ 6; therefore, we define a minimal LRF λ(V(LW 6 )) → [0, 1] as 1  4 to each v ∈ V(LW 6 ), hence Ldim F (LW 6 ) ≤

Line Network of Prism Network LD 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 LD t be the line network of prism network with vertex set V(LD t ) = {a i , b i , c i : 1 ≤ i ≤ t} and edge set E(LD t ) = {a i a i+1 , a i b i , b i a i+1 , c i b i , c i c i+1 : 1 ≤ i ≤ t} with order 3t and size 6t.For more information see Figure 2. Lemma 3. Let LD t be the line network of prism network, where t ∼ = 1 (mod 2).Then Proof.Consider a i inner, b i middle, and c i are outer vertices of LD t , where 1 ≤ i ≤ t and t + 1 ∼ = 1 (mod t).
The comparison among the cardinalities of all the LRN sets is given in Table 3.

LRN Set Cardinality
Theorem 5. Let LD t be a line network of prism network then Proof.The LRN sets of LD 5 are given by: L r (a Since the cardinality of each LRN set of LD 5 is 10, we define a constant LRF λ(V(LD 5 )) → [0, 1] as 1  10 to each v ∈ V(LD 5 ), hence Case 2: For t ≥ 7, in the view of Lemma 3, the cardinality of t+3 .In the same context by Lemma 3 the maximum cardinality of LRN set L r (a i a i+1 ) is 3t − 5 and |L r (a i a i+1 )| > |L r (x)|, where L r (x) are other LRN sets of LW t , where 1 ≤ i ≤ t.Therefore, we define a maximal LRF λ (V(LW t )) The bounds of LFMD of LD t are given as follows: Lemma 4. Let LD t be the line network of prism network then, where t ∼ = 0 (mod 2) then Proof.Consider a i inner, b i middle, and c i are outer vertices of LD t , where 1 ≤ i ≤ t and t parison among the cardinalities of all the LRN sets is given in Table 4.
Proof.For t = 4, we have following LRN sets: Since the cardinality of each LRN set of LD 4 is 10, therefore, we define a constant LRF λ(V(LD 4 )) → [0, 1] as 1  8 to each v ∈ V(LD 4 ), hence Theorem 8. Let LD t be a line network of prism network, where t ∼ = 0 (mod 2).Then Proof.To prove the theorem, we have divided it in two cases: Case 1: The LRN sets of LD 6 are: The LRN sets with a minimum cardinality are L r (b i c i ), L r (a i b i ), L r (b i a i+1 ), L r (b i c i+1 ) and the cardinality of each of them is 11 , where 1 ≤ i ≤ 6.Therefore, we define a minimal LRF λ(V(LD 6 )) → [0, 1] as 1  11 to each v ∈ V(LD

Conclusions
In this manuscript, we have established sharp bounds of the LFMD of the rotationally symmetric line networks of the wheel (LW t ) and prism (LD t ).It is proved that for t = 3, LW t attains the exact value of LFMD which is 3  2 and for t = 4, 5 the LFMD of LD t is 3 2 as well.It has been observed that the LW t remains unbounded and LD t remains bounded under LFMD, when the order of these networks approaches ∞.The boundedness and unboundedness of these networks is illustrated in Table 5.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.

Figure 1 .
Figure 1.Wheel network W t and its line network LW t .

Figure 2 .
Figure 2. Prism network D t and its line network LD t .

Table 2 .
Comparison between the cardinalities of LRN sets of LW t .
1  8to each v ∈ V(LW 6 ), hence Ldim F (LW 6 ) ≥ For t ≥ 8, in the view of Lemma 2, the cardinality of LRN set L r (a i a i+1 ) is 4 and |L r (a i a i+1 )| < |L r (x)|, where L r (x) are other LRN sets of LW t .Therefore, we define a minimal LRF λ(V(LW t )) → [0, 1] as1  4to each v ∈ V(LW t ), hence Ldim F (LW t ) ≤ In the same context, by Lemma 2, the maximum cardinality of LRN set L r (a i b i ) is 2t − 4 and |L r (a i b i )| > |L r (x)|, where L r (x) are the other LRN sets of LW t .Therefore, we define a maximal LRF λ

Table 3 .
Comparison between the cardinalities of LRN sets of LD t .

Table 4 .
Comparison among the cardinalities of LRN sets of LD t .
6), hence Ldim F (LD 6 ) ≤ The LRN set having maximum cardinality is L r (a i a i+1 ), and its cardinality is 15; therefore, we define a maximal LRN λ(V(LD 6 )) → [0, 1] as1  15to each v ∈ V(LD 6 ), hence Ldim F (LD 6 ) ≥ For t ≥ 6, in the view of Lemma 4, the cardinalities of the LRN sets L r (ai b i ), L r (b i a i+1 ), L r (b i c i ), and L r (b i c i+1 ) is 3t+4 2 and |L r (a i b i )| ≤ |L r (x)|, where L r (x) are other LRN sets of LD t , where 1 ≤ i ≤ t.Therefore, we define a minimal LRF λ(V(LD t )) → [0, 1] as 2 3t+4 to each v ∈ V(LD t ), hence Ldim F (LD t ) ≤ In the same context by Lemma 4 the maximum cardinalities of the LRN sets are L r (a i a i+1 ) and L r (c i c i+1 ) is 3t − 3 and |L r (a i a i+1 )| ≥ |L r (x)|, where L r (x) are other LRN sets of LD t , where 1 ≤ i ≤ t.Therefore, we define a maximal LRF λ (V(LD t )) → [0, 1] as 1 3t−3 to each v ∈ V(LD t ), hence Ldim F (LD t ) ≥ .Hence, the bounds of LFMD of LD t are given as follows: