Fractional Metric Dimension of Generalized Sunlet Networks

Let N � (V(N), E(N)) be a connected network with vertex V(N) and edge set E(N)⊆(V(N), E(N)). For any two vertices a and b, the distance d(a, b) is the length of the shortest path between them. (e local resolving neighbourhood (LRN) set for any edge e � ab of N is a set of all those vertices whose distance varies from the end vertices a and b of the edge e. A real-valued functionΦ from V(N) to [0, 1] is called a local resolving function (LRF) if the sum of all the labels of the elements of each LRN set remains greater or equal to 1. (us, the local fractional metric dimension (LFMD) of a connected network N is dimlf(N) � min |Φ|: Φ isminimal LRF of N { }. In this study, LFMD of various types of sunlet-related networks such as sunlet network (Sm), middle sunlet network (MSm), and total sunlet network (TSm) are studied in the form of exact values and sharp bounds under certain conditions. Furthermore, the unboundedness and boundedness of all the obtained results of LFMD of the sunlet networks are also checked.


Introduction
e problem to find the location number for the connected networks was firstly introduced by Slater in 1975 [1]. Later on, Melter and Harary also studied the concept of location number in networking theory, but they used different term called by metric dimension (MD) [2]. It has been investigated that computing MD is an NP-hard problem [3]. e concept of MD being a graph theoretic parameter is a useful tool in the discovery and verification of the networks [4], allocation of different destinations to robots [5], investigation of percolation in a hierarchical lattice [6], and configuration of the chemical compounds in chemistry [7].
Chartrand et al. established the sharp bounds of MD for the unicyclic networks; they proved that MD of a connected network N is 1 iff N is path network. Furthermore, under certain conditions, by using the concept of MD on the integer programming problem (IPP), they also found the integral solutions [8]. For the study of the various computational results of MD for the different connected networks such as Toeplitz, Mobius ladder, lexicographic product of networks, gear networks, and barycentric subdivision of Cayley networks, we refer to [9][10][11][12][13]. In addition, for the study of constant MD of some families of regular, cycle, and prism-related networks and unbounded MD of nanotubes and convex polytopes, see [6,[14][15][16][17][18].
Later on, Currie and Ollerman defined the fractional version of MD to study the nonintegral solution of IPP [19]. Saddiqi and Imran obtained optimal solution of cretin IPP by using this new fractional technique in the field of metricbased dimensions [17]. Arumugam and Matthew formally introduced the term fractional metric dimension (FMD) in graph theory, and they found exact values of FMD for certain connected networks. Moreover, they also characterized all the networks with FMD equal to half of their order. Feng et al. developed computational criteria to compute FMD of the vertex transitive networks in its general form [20]. Recently, Khalidi et al. established sharp bounds of FMD for the connected networks [21]. To study the latest developed results on FMD for trees, unicyclic, permutation networks, and product networks obtained under the operation of product (hierarchical, comb, corona, and lexicographic), see [22][23][24][25][26].
e new invariant of MD called by local FMD is defined by Aisyah et al. [27]. Liu et al. computed upper bounds of LFMD for the symmetric and planar networks [28]. For all the connected networks, Javaid et al. established upper bound and improved the lower bound of nonbipartite networks from unity. ey also characterized bipartite networks with LFMD as unity [29,30]. Moreover, Moshin et al. studied LFMD of generalized Petersen networks [31][32][33][34]. For more study, we refer to in [34][35][36].
In this note, our main objective is to compute the sharp bounds and exact values of LFMD for the different generalized sunlet networks such as sunlet network (S m ), middle sunlet network (MS m ), and total sunlet network (TS m ), where m is some integral value. In addition, the boundedness and unboundedness of all the obtained results are also investigated. e remaining study is organised as follows. Section 2 contains basic notions. Section 3 has main findings involving LRN sets of LFMD. Section 4 contains the conclusion of this paper.

Preliminaries
For vertex set V(N) and edge set E(N)⊆(V(N) × V(N)), the network (N � V(N), E(N)) is constructed as a simple and connected network. For u, v ∈ V(N), the distance between u and v denoted by d(u, v) is length (number of edges) of the shortest path between them. If each pair of vertices of N is expressed by some path, then N is called connected network. For the further study of preliminary concepts of the subject graph theory, we refer [35].
. . , v m ⊆V(N) and u ∈ V(N); then, m-tuple representation of u with respect to A is r(u|A) � (r(u, v 1 ), r(u, v 2 ), r(u, v 3 ), . . . , r(u, v m )). If the distinct vertices of N have different representations with respect to A, then A is called a resolving set of N. us, MD of N can be defined by where A is the resolving set of N.
Let uv ∈ V(N); then, local resolving neighbourhood (LRN) is defined as A local resolving function (LRF) is a real-valued function us, local fractional metric dimension (LFMD) is defined as follows: where |Φ| � u∈V(N) Φ(u). By using the technique used in [36], now, we define sunlet network (S m ), middle sunlet network (MS m ), and total sunlet network (TS m ) as follows.
Let S m be a sunlet network with order and size 2 m, respectively, where m ≥ 3. It consists of the inner cycle of order m, having inner vertices v i : Figure 1. Middle sunlet network is obtained from sunlet network S m of order 4m and size 5m as for details, see Figure 2.
e sunlet network TS m is obtained from middle sunlet network MS m by adding new edges order 4 m and size 6 m, respectively; for further details, see Figure 3. Now, we define some important results which will be frequently used in the main results as follows.
Theorem 1 (see [29]). For a connected network N and where Theorem 2 (see [30]). For a connected network N and LRN set R ′ (e) of the edge e of N, we have where β � max |R ′ (e)|: e ∈ E(N) and 2 ≤ β ≤ |V(N)|.

Main Result
In this particular section, we computed LRN sets of generalized sunlet networks and LFMD in the form of exact values and sharp bounds.

LFMD of Sunlet
Network. e resolving neighbourhood sets for each pair of adjacent vertices are classified.

Journal of Mathematics
Proof. To prove the result, we have following cases.
e cardinalities of LRN sets other than R ′ (u i ′ v i ) are classified in Table 1.
Hence, by eorem 2, dim lf (MS m ) ≥ 1. Since MS m is a nonbipartite network, therefore, dim lf (MS m ) must be greater than 1. Consequently,

LFMD of Total Sunlet
Network. e LRN sets of each pair of adjacent vertices are classified.

Lemma 3.
Let TS m with m ≥ 3 be a total sunlet network.
Proof. Assume that u i pendent and u i ′ , v i ′ , and v i are other vertices, respectively, of TS m , where 1 ≤ i ≤ m.
e cardinalities of the LRN sets other than R ′ (u i ′ v i ) are classified in Table 2.
It is clear from Table 2

LRN Set
Cardinality

Conclusion
In this article, we studied the LFMD of some families generalized sun let networks and formed bounds of LFMDs and computed the exact values of LFMD in some cases as well.
Exact values of LFMD is attained by total sunlet network, TS m � m.
Bounded and unboundedness of LFMDs are illustrated through Table 3.

Data Availability
e data used to support the findings of this study are included within this article and can be obtained from the corresponding author upon request for more details on the data.
Journal of Mathematics