Abstract
Let be a connected network with vertex and edge set . For any two vertices and , the distance is the length of the shortest path between them. The local resolving neighbourhood (LRN) set for any edge of is a set of all those vertices whose distance varies from the end vertices and of the edge . A real-valued function from to 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. Thus, the local fractional metric dimension (LFMD) of a connected network is . In this study, LFMD of various types of sunlet-related networks such as sunlet network (), middle sunlet network (), and total sunlet network () 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.
1. Introduction
The 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]. The 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 is 1 iff 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 [9–13]. 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, 14–18].
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 metric-based 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 [22–26].
The 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. They also characterized bipartite networks with LFMD as unity [29, 30]. Moreover, Moshin et al. studied LFMD of generalized Petersen networks [31–34]. For more study, we refer to in [34–36].
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 (), middle sunlet network (), and total sunlet network (), where is some integral value. In addition, the boundedness and unboundedness of all the obtained results are also investigated. The 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.
2. Preliminaries
For vertex set and edge set , the network is constructed as a simple and connected network. For , the distance between denoted by is length (number of edges) of the shortest path between them. If each pair of vertices of is expressed by some path, then is called connected network. For the further study of preliminary concepts of the subject graph theory, we refer [35].
A vertex resolves a pair of vertices in if . Let and ; then, -tuple representation of with respect to is . If the distinct vertices of have different representations with respect to , then is called a resolving set of . Thus, MD of can be defined bywhere A is the resolving set of .
Let ; then, local resolving neighbourhood (LRN) is defined as
A local resolving function (LRF) is a real-valued function such that for each of , where . A LRF of is called minimal if there exists some other function such that and for at least one that is not LRF of . Thus, local fractional metric dimension (LFMD) is defined as follows:where = .
By using the technique used in [36], now, we define sunlet network , middle sunlet network , and total sunlet network as follows.
Let be a sunlet network with order and size , respectively, where . It consists of the inner cycle of order , having inner vertices , pendent vertices , and edge set of , see Figure 1. Middle sunlet network is obtained from sunlet network of order and size as and ; for details, see Figure 2. The sunlet network is obtained from middle sunlet network by adding new edges with 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 and LRN set of the edge of if ,where , , and .
Theorem 2 (see [30]). For a connected network and LRN set of the edge of , we havewhere and .
Corollary 1 (see [30]). For a connected network , as LRN of , , , and . If and , then
Theorem 3 (see [29]). If is a connected bipartite network, then .
3. 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.
3.1. LFMD of Sunlet Network
The resolving neighbourhood sets for each pair of adjacent vertices are classified.
Lemma 1. Let with and be a sunlet network. Then, for , we have(a), , , and (b)
Proof. Assume that pendent and are internal vertices, respectively, of and , where .(a)Since , this implies that and ; therefore, , where . Furthermore, .(b)Since , therefore, .
Theorem 4. Let be a sunlet network, where and . Then,
Proof. To prove the result, we have following cases.
Case 1. For , the LRN sets of are From the above LRN sets, and their cardinalities less than the cardinalities of other LRN sets of . Hence, the function is defined as which is an upper LRF; therefore, by Theorem 1, .
From the above LRN sets, , and their cardinalities are greater than the cardinalities of other LRN sets of . Now, the function is a lower LRF defined as ; hence, by Theorem 2, . Since is a nonbipartite network, therefore, must be greater then 1. Consequently,
Case 2. In this case, and . Furthermore, ; hence, the constant function is defined as which is an upper LRF; therefore, by Theorem 1, .
Since and , therefore, the function is a lower LRF which is defined as . Hence, by Theorem 2, . Since is a nonbipartite network, therefore, must be greater than 1. Consequently,
Theorem 5. Let be a sunlet network, where and . Then,
Proof. Since no cycle in is of odd length, therefore, is a bipartite network, where ; hence, by Theorem 3, .
3.2. LFMD of Middle Sunlet Network
The LRN sets for each pair of adjacent vertices are classified.
Lemma 2. Let with be a middle sunlet network. Then, for ,(a), , and (b) and
Proof. Assume that pendent and , , and are internal vertices, respectively, of , where .(a)Since , this implies that and ; therefore, , where . Furthermore, .(b)Consider , , and .The cardinalities of LRN sets other than are classified in Table 1.
It is clear from Table 1 that and .
Theorem 6. Let be a middle sunlet network, where .
Then,
Proof. To prove the result, we have the following cases.
Case 3. For , the LRN sets of are From the above LRN sets, and and . Hence, the function is defined asTherefore, by Theorem 1, .
Since and cardinality of these LRN sets is greater than all other LRN sets, therefore, we define a constant function which is a lower LRF defined by ; hence, by Theorem 2, . Also, is a nonbipartite network; therefore, must be greater then 1. Consequently,
Case 4. In this case, and . Furthermore, ; hence, the constant function is defined asTherefore, by Theorem 1, .
Since and , therefore, a function is a lower LRF and is defined as . Hence, by Theorem 2, . Since is a nonbipartite network, therefore, must be greater than 1. Consequently,
3.3. LFMD of Total Sunlet Network
The LRN sets of each pair of adjacent vertices are classified.
Lemma 3. Let with be a total sunlet network. Then, for , we have(a), , and (b) and
Proof. Assume that pendent and , , and are other vertices, respectively, of , where .(a)Since , this implies that and ; therefore, , where . Furthermore, .(b)Consider , , , and .The cardinalities of the LRN sets other than are classified in Table 2.
It is clear from Table 2 that . Furthermore, .
Theorem 7. Let be a total sunlet network, where . Then,
Proof. To prove the result, we have the following cases.
Case 5. For , the LRN sets of are From the above LRN sets and the cardinalities of these LRN sets which are less than the cardinalities of all other LRN sets, furthermore, . Hence, the function is defined asTherefore, .
Case 6. In this case, and . Furthermore, . Hence, a constant function is defined asTherefore,
4. 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, .
Bounded and unboundedness of LFMDs are illustrated through Table 3.
Data Availability
The 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.
Conflicts of Interest
The authors declare no conflicts of interest regarding this article.