Distance-Based Fractional Dimension of Certain Wheel Networks

Metric dimension is one of the distance-based parameters which are used to fnd the position of the robot in a network space by utilizing lesser number of notes and minimum consumption of time. It is also used to characterize the chemical compounds. Te metric dimension has a wide range of applications in the feld of computer science such as integer programming, radar tracking, pattern recognition, robot navigation, and image processing. A vertex x in a network W resolves the adjacent pair of vertices uv if x attains an unequal distance from end points of uv . A local resolving neighbourhood set R L ( uv ) is a set of vertices of W which resolve uv . A mapping α : V ( W ) ⟶ [ 0 , 1 ] is called local resolving function of W if α ( R L ( uv )) ≥ 1 for any adjacent pair of vertices of uv of W and the minimal value of α ( R L ( uv )) for all local resolving functions α of W is called local fractional metric dimension of W . In this paper, we have studied the local fractional metric dimension of wheel-related networks such as web-wheel network, subdivision of wheel network, line network of subdivision of wheel network, and double-wheel network and also examined their boundedness.


Introduction and Preliminaries
Te notion of metric dimension (MD) was introduced in the 1970s independently by Slater and Harary [1,2].NPhardness and complexity of the MD problem were briefy studied in [3,4].MD is substantially applied in diferent felds such as robot navigation [5], in pharmaceutical chemistry [6], image processing [1], and in computer science [7].In 2000, Chartrand et al. characterized all the connected networks that have a specifc value of MD [6].Liu et al. computed MD of tenser product of path, cycles, and the constant MD of Toeplitz networks [8,9].Barragán-Ramírezet al. defned the concept of local MD, and they also computed the local MD of the strong product of some connected networks [10].
Te term fractional metric dimension (FMD) is defned by Currie and Oellermann to fnd the solution of certain IPP [11] and Feher et al. computed the optimal solution of IPP by using FMD [12].In 2011, Arumugam and Mathew introduced the term FMD in networking theory [13], and the notion of local fractional metric dimension (LFMD) is defned by Aisyah et al. [14], for more about FMD see [15,16].Javaid et al. played an important role in the feld of LFMD as they have established bounds of LFMD and characterized some connected networks those obtain the exact value of LFMD.Furthermore, they developed a computational technique to evaluate the lower bound of LFMD [17,18].
A network W is an ordered pair (V, E), where the set V composing of the nodes called the vertex set V(W) and E is the set of the links among these nodes is called the edge set E(W).A path is a sequence of vertices in which each one adjacent to the next.Te number of edges in the minimal path between two vertices u and v is called distance between them donated by d (u, v).
Te line network L(W) of a network W is defned to have as its vertices the edges of W, with two nodes are adjacent if the corresponding edges share a node in [19].A subdivision of a network S(W) is obtained by adding an additional vertex into each edge of W. Since Javaid et al. [17,18] have established the bounds of LFMD of general networks and they have also computed the exact value of LFMD of specifc networks.In this context, we have developed bounds of LFMD of some special class of generalized wheel networks.Furthermore, the bounds and exact values of LFMD are depends upon the cardinalities of the LRN of each network.
In this article, our objective is to compute the LFMD of wheel-related networks such as web-wheel, subdivision of wheel, line network of subdivision of wheel, and doublewheel networks.Tese networks attain diferent values of LFMD at diferent levels; therefore, it is very interesting to investigate their LFMD.In the end, a comprehensive conclusion is given as well.Te article is organised as follows: Section 2 contains the preliminary concepts involving of the concepts involved in the article; in Section 3, all the main results are given in detail; and Section 4 deals with the conclusion.

Main Results
In this current section, we are interested in determining the LFMD of wheel-related networks, such as web-wheel network, subdivision of wheel network, and line network of subdivision of the wheel network.

LRN Set and LFMD of Subdivision of Wheel Network.
Te subdivision of wheel network (SW k ) is obtained by adding a vertex w i and v i to each edge of wheel network W k , where 1 ≤ i ≤ k.For more details, see Figure 1.
Theorem 1.Let SW k be a subdivision of wheel network.Ten, ( Proof.Since SW k is a bipartite network and the cardinality of each LRN set of SW k is equal to its vertex.Hence, ( For more information, see Figure 2.

Lemma . Let LSW k be a line network of subdivision of wheel network. Ten, (a) |R
Proof.Let LSW k be a web wheel network, where , and Proof.In order to prove the theorem, we have divided into particular case (Case A) and general case (Case B).
Case A.
Te possible LRN sets of LSW 3 are as follows: ( It is clear from above LRN sets that |R L (x i w i+1 )| � 10; now consider α: Case 2.
For k ≥ 3 with the reference of Lemma 2 Moreover, the cardinality of each LRN set is not same.Terefore, we consider a maximal LLRF α: For more details about double-wheel network, see Figure 3.  Table 1: LRN sets and their comparison.
Table 2 clears the order of each R L (y).

□
Theorem 5. Let DW 3 be a double-wheel network.Ten, Proof.Te possible LRN sets of DW 3 are It is clear from above LRN sets that |R L (u i x)| � 7 now consider a maximal LRF α: □ Theorem 6.Let DW 5 be a double-wheel network.Ten, Proof.Te possible LRN sets of DW 5 are

□
Theorem 7. Let DW k be a double-wheel network.Ten, Journal of Mathematics Proof.In order to prove the theorem, we have divided into a particular case (Case A) and general case (Case B).
From above LRN sets that □

Conclusion
In this article, we have obtained the sharp bounds of the local fractional metric dimension of wheel-related networks such as the web-wheel network, subdivision of wheel network, line network of subdivision of wheel network, and doublewheel network.It has been proved that link networks of subdivision of wheel network (LSW k ) and double-wheel network (DW k ) remain unbounded when the order of these networks approaches to ∞.Moreover, the LFMD of subdivision of wheel network is exactly 1, and in future, it would be very interesting to investigate the LFMD of all the wheelrelated networking attaining an exact value.Te boundedness and unboundedness other than the subdivision of wheel networks is also obtained in Table 3.

Lemma 4 .
Let DW k be a double-wheel network.Ten, (a) |R L

Table 2 :
LRN sets and their comparison.