# Fractional Metric Dimension of Generalized Sunlet Networks

Authors: Muhammad Javaid; Hassan Zafar; Ebenezer Bonyah
Journal: Journal of Mathematics
 (2021)

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## Abstract

LetN=VN,EN be a connected network with vertex VN and edge set EN⊆VN,EN. For any two vertices a and b, the distance da,b is the length of the shortest path between them. The 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 function Φ from VN 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. Thus, the local fractional metric dimension (LFMD) of a connected network N is dimlfN=minΦ:Φis minimal LRF ofN. 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.


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## Body

## 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 networkN 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 [9–13]. 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 (Sm), middle sunlet network (MSm), and total sunlet network (TSm), where m 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 setVN and edge set EN⊆VN×VN, the network N=VN,EN is constructed as a simple and connected network. For u,v∈VN, the distance between uandv denoted by du,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].A vertexv∈VN resolves a pair w,z of vertices in N if dw,v≠dz,v. Let A=v1,v2,v3,…,vm⊆VN and u∈VN; then, m-tuple representation of u with respect to A is ru|A=ru,v1,ru,v2,ru,v3,…,ru,vm. If the distinct vertices of N have different representations with respect to A, then A is called a resolving set of N. Thus, MD of N can be defined by(1)dimN=minA,where A is the resolving set of N.Letuv∈VN; then, local resolving neighbourhood (LRN) is defined as(2)ℝ′uv=z∈VN:du,z≠dv,z.A local resolving function (LRF) is a real-valued functionΦ:VN⟶0,1 such that Φℝ′uv≥1 for each ℝ′uv of N, where Φℝuv=∑x∈ℝ′uvΦx. A LRF Φ of N is called minimal if there exists some other function Φ′:VN⟶0,1 such that Φ′≤Φ and Φu≠Φ′u for at least one u∈VN that is not LRF of N. Thus, local fractional metric dimension (LFMD) is defined as follows:(3)dimlfN=minΦ,where Φ = ∑u∈VNΦu.By using the technique used in [36], now, we define sunlet network Sm, middle sunlet network MSm, and total sunlet network TSm as follows.LetSm be a sunlet network with order and size 2m, respectively, where m≥3. It consists of the inner cycle of order m, having inner vertices vi:1≤i≤m, pendent vertices ui:1≤i≤m, and edge set of ESm=vivi+1,uivi:1≤i≤m, see Figure 1. Middle sunlet network is obtained from sunlet network Sm of order 4m and size 5m as VMSm=vi,ui,ui′,vi′:1≤i≤m and EMSm=uiui′,vivi′,vi′ui+1,vi′vi+1′,ui′vi:1≤i≤m; for details, see Figure 2. The sunlet network TSm is obtained from middle sunlet network MSm by adding new edges uivi,vivi+1:1≤i≤m 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.Figure 1
Sunlet networkSm.Figure 2
Middle sunlet networkMSm.Figure 3
Total sunlet networkTSm.Theorem 1 (see [29]).
For a connected networkN and LRN set ℝ′e of the edge e of N if ℝ′e∩A≥γ,∀e∈EN,(4)1≤dimlfN≤Aγ,where A=∪ℝ′e:ℝ′e=γ, γ=minℝ′e:e∈EN, and 2≤γ≤VN.Theorem 2 (see [30]).
For a connected networkN and LRN set ℝ′e of the edge e of N, we have(5)VNβ≤dimlfN,where β=maxℝ′e:e∈EN and 2≤β≤VN.Corollary 1 (see [30]).
For a connected networkN, ℝ′e as LRN of e∈EN, δ=maxℝ′e:e∈EN, γ=minℝ′e|:e∈EN, and X=∪ℝ′e:ℝ′e=γ. If γ=δ and X=VN, then(6)dimlfN=VNδ.Theorem 3 (see [29]).
IfN is a connected bipartite network, then dimlfN=1.


## 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.
LetSm with m≥3 and m≅1mod2 be a sunlet network. Then, for 1≤i≤m, we have(a)
ℝ′vivi+1=2m−2, ℝ′uivi=2m, ∪i=1mℝ′vivi+1=2m, and ℝ′vivi+1<ℝ′uivi(b)
ℝ′uivi∩∪i=1mℝ′vivi+1=ℝ′vivi+1Proof.
Assume thatui pendent and vi are internal vertices, respectively, of Sm and vm+1≅v1, where 1≤i≤m.(a)
Sinceℝ′vivi+1=VSm−um+2i+1/2,vm+2i+1/2, this implies that ℝ′vivi+1=2m−2 and ℝ′uivi=VSm; therefore, ℝ′viui=VSm=2m, where 1≤i≤m. Furthermore, ∪i=1mℝ′vivi+1=2m.(b)
Since∪i=1mℝ′vivi+1=VSm, therefore, ℝ′uivi∩∪i=1mℝ′vivi+1=ℝ′vivi+1.Theorem 4.
LetSm be a sunlet network, where m≥3 and m≅1mod2. Then,(7)1<dimlfMSm≤m/m−1.Proof.
To prove the result, we have following cases.Case 1.
Form=3, the LRN sets of Sm areℝ1′=ℝ′v1v2=VS3−v3,u3ℝ2′=ℝ′v2v3=VS3−v1,u1ℝ3′=ℝ′v3v1=VS3−v2,u2ℝ4′=ℝ′u1v1=VS3ℝ5′=ℝ′u2v2=VS3ℝ6′=ℝ′u3v3=VS3
From the above LRN sets,ℝ1′=ℝ2′=ℝ3′=3 and their cardinalities less than the cardinalities of other LRN sets of S3. Hence, the function h:VS3⟶0,1 is defined as hv=1/4∀v∈VS3 which is an upper LRF; therefore, by Theorem 1, dimlfS3≤3/2.
From the above LRN sets,ℝ4′=ℝ5′=ℝ6′=6, and their cardinalities are greater than the cardinalities of other LRN sets of S3. Now, the function h′:VS3⟶0,1 is a lower LRF defined as h′v=1/6; hence, by Theorem 2, dimlfS3≥1. Since S3 is a nonbipartite network, therefore, dimlfS3 must be greater then 1. Consequently,(8)1<dimlfS3≤3/2.Case 2.
In this case,ℝ′vivi+1=2m−2 and ∪i=1mℝ′vivi+1=2m. Furthermore, ℝ′y∩∪i=1mℝ′vivi+1≥ℝ′y∀y∈VSm; hence, the constant function h:VSm⟶0,1 is defined as hv=1/2m−2∀v∈VSm which is an upper LRF; therefore, by Theorem 1, dimlfSm≤m/m−1.
Sinceℝ′uivi=2m and ℝ′uivi≥ℝ′y∀y∈ESm, therefore, the function h′:VSm⟶0,1 is a lower LRF which is defined as h′v=1/2m∀h∈VSm. Hence, by Theorem 2, dimlf≥1. Since Sm is a nonbipartite network, therefore, dimlfMSm must be greater than 1. Consequently,(9)1<dimlfSm≤mm−1.Theorem 5.
LetSm be a sunlet network, where m≥4 and m≅0mod2. Then,(10)dimlfSm=1.Proof.
Since no cycle inSm is of odd length, therefore, Sm is a bipartite network, where m≅0mod2; hence, by Theorem 3, dimlfSm=1.


### 3.2. LFMD of Middle Sunlet Network

The LRN sets for each pair of adjacent vertices are classified.Lemma 2.
LetMSm with m≥3 be a middle sunlet network. Then, for 1≤i≤m,(a)
ℝ′viui′=3, ℝ′uiui+1′=4m, and ∪i=1mℝ′viui′=3m(b)
ℝ′viui′≤ℝ′y and ℝ′y∩∪i=1mℝ′viui′≥ℝ′viui′∀y∈EMSmProof.
Assume thatui pendent and ui′, vi′, and vi are internal vertices, respectively, of MSm, where 1≤i≤m.(a)
Sinceℝ′ui′vi=ui,ui′,vi, this implies that ℝ′ui′vi=3 and ℝ′uiui+1′=VMSm; therefore, ℝ′uiui+1′=VMSm=4m, where 1≤i≤m. Furthermore, ∪i=1mℝ′viui′=3m.(b)
Considerℝ′vivi′=vi,vi+1,vi+2,…,vm+2i−1/2,vi′,vi+1′,vi+2′,…,vm+2i−1/2′,ui+1,ui+2,…,um+2i−1/2,ui+1′,ui+2′,…,um+2i−1/2′, ℝ′vivi−1′=vi,vm+2i+1/2,vm+2i+3/2,…,vm,vm+2i+1/2′,vm+2i+3/2′,vm+2i+5/2′,………,vm′,um+2i+1/2,um+2i+3/2,um+2i+5/2…,um,um+2i+1/2′,um+2i+3/2′,um+2i+5/2′,…,um′,ℝ′ui′vi′=vi+1,vi+2,vi+3,….,vm+2i−1/2,vi′,vi+1′,vi+2′,vi+3,…,vm+2i−1/2′,ui,ui+1,ui+2,ui+3,…,um+2i−1/2,ui′,ui+1′,ui+2′,ui+3′,…,um+2i−1/2′,ℝ′ui′vi−1′=vm+2i+1/2,vm+2i+3/2,vm+2i+5/2…,vm,vm+2i+1/2′,vm+2i+3/2′,vm+2i+5/2′,…,vm′,ui,um+2i+1/2,um+2i+3/2,…,um,um+2i+1/2′,um+2i+3/2′,…,um′, and ℝ′vi′vi+1′=VMSm−vi,ui′,ui.
The cardinalities of LRN sets other thanℝ′ui′vi are classified in Table 1.
It is clear from Table1 that ℝ′ui′vi≤ℝ′y and ℝ′y∩∪i=1mℝ′viui′≥ℝ′viui′∀y∈EMSm.Table 1
Cardinality of each LRN set.
LRN SetCardinalityℝ′vivi′2m>ℝ′ui′viℝ′vivi−1′2m>ℝ′ui′viℝ′ui′vi′2m+1>ℝ′ui′viℝ′ui′vi−1′2m>ℝ′ui′viTheorem 6.
LetMSm be a middle sunlet network, where m≥3.
Then,(11)1<dimlfMSm≤m.Proof.
To prove the result, we have the following cases.Case 3.
Form=3, the LRN sets of MSm areℝ1′=ℝ′u1′v1=u1′,v1,u1ℝ2′=ℝ′u2′v2=u2′,v2,u2ℝ3′=ℝ′u3′v3=u3′,v3,u3ℝ4′=ℝ′v1v1′=v1,v2,v1′,v2′,v2,u2′ℝ5′=ℝ′v2v2′=v2,v3,v2′,v3′,v3,u3′ℝ6′=ℝ′v3v3′=v3,v1,v3′,v1′,v1,u1′ℝ7′=ℝ′v1v3′=v1,v3,v2′,v3′,u3,u3′ℝ8′=ℝ′v2v2′=v2,v2′,v3′,v3,u3′,u3ℝ9′=ℝ′v3v3′=v3,v3′,v1′,v1,u1′,u1ℝ10′=ℝ′v1′v3′=v1′,v3′,v2,u2′,u2,v3,u3′,u3ℝ11′=ℝ′v1′v2′=v2′,v1′,v3,u3′,u3,v1,u1′,u1ℝ12′=ℝ′v2′v3′=v2′,v1′,v3,u3′,u3,v1,u1′,u1ℝ13′=ℝ′u1′u1=VMS3ℝ14′=ℝ′u2′u2=VMS3ℝ15′=ℝ′u3′u3=VMS3
From the above LRN sets,ℝ1′=ℝ2′=ℝ3′=3 and ℝ′ui′vi≤ℝ′y∀y∈EMS3 and ℝ′y∩∪i=13ℝ′viui′≥ℝ′viui′∀y∈EMS3. Hence, the function h:VMS3⟶0,1 is defined as(12)hv=13ifv∈∪i=13ℝ′ui′vi,0ifv∈VMS3−∪i=13ℝ′ui′vi.
Therefore, by Theorem1, dimlfMS3≤3.
Sinceℝ13′=ℝ14′=ℝ15′=12 and cardinality of these LRN sets is greater than all other LRN sets, therefore, we define a constant function h′:VMS3⟶0,1 which is a lower LRF defined by h′v=1/12; hence, by Theorem 2, dimlf≥1. Also, MS3 is a nonbipartite network; therefore, dimlfMS3 must be greater then 1. Consequently,(13)1<dimlfMS3≤3.Case 4.
In this case,ℝ′ui′vi=3 and ∪i=1mℝ′uivi=3m. Furthermore, ℝ′y∩∪i=1mℝ′uivi≥ℝ′uivi∀y∈VMSm; hence, the constant function h:VMS3⟶0,1 is defined as(14)hv=13ifv∈∪i=1mℝ′ui′vi,0ifv∈VMS3−∪i=1mℝ′ui′vi.
Therefore, by Theorem1, dimlfMSm≤m.
Sinceℝ′ui′ui=4m and ℝ′ui′ui≥ℝ′y∀y∈EMSm, therefore, a function h′:VMSm⟶0,1 is a lower LRF and is defined as h′v=1/4m∀v∈VMSm. Hence, by Theorem 2, dimlfMSm≥1. Since MSm is a nonbipartite network, therefore, dimlfMSm must be greater than 1. Consequently,(15)1<dimlfMSm≤m.


### 3.3. LFMD of Total Sunlet Network

The LRN sets of each pair of adjacent vertices are classified.Lemma 3.
LetTSm with m≥3 be a total sunlet network. Then, for 1≤i≤m, we have(a)
ℝ′viui′=2, ℝ′uiui+1′=4m−1, and ∪i=1mℝ′viui′=2m(b)
ℝ′viui′≤ℝ′y and ℝ′y∩∪i=1mℝ′ui≥ℝ′ui′vi∀y∈EMSmProof.
Assume thatui pendent and ui′, vi′, and vi are other vertices, respectively, of TSm, where 1≤i≤m.(a)
Sinceℝ′ui′vi=ui′,vi, this implies that ℝ′ui′vi=2 and ℝ′ui′ui′=VTSm−viandℝ′uivi=VTSm−ui′; therefore, ℝ′uiui′=VTSm=4m−1, where 1≤i≤m. Furthermore, ∪i=1mℝ′viui′=2m.(b)
Considerℝ′vivi′=vi,vi+1,vi+2,…,vm+2i−1/2,vi′,vi+1′,vi+2′,…,vm+2i−1/2′,ui+1,ui+2,…,um+2i−1/2,ui+1′,ui+2′,…,um+2i−1/2′, ℝ′vi′vi+1=vi−1,vi,vm+2i+3/2,vm+2i+5/2,…,vm,vi′,vm+2i+1/2′,vm+2i+3/2′,vm+2i+5/2′,…,vm′,ui+1,um+2i+3/2,um+2i+5/2,…..,um,ui′,ui+1′,um+2i+3/2′,um+2i+5/2′,…,um′, ℝ′ui′vi′=vi+1,vi+2,vi+3,…,vm+2i−1/2,vi′,vi+1′,vi+2′,vi+3,…,vm+2i−1/2′,ui,ui+1,ui+2,ui+3,…,um+2i−1/2,ui′,ui+1′,ui+2′,ui+3′,….,um+2i−1/2′,ℝ′ui′vi−1′=vm+2i+1/2,vm+2i+3/2,vm+2i+5/2…,vm,vm+2i+1/2′,vm+2i+3/2′,vm+2i+5/2′,…,vm′,ui,um+2i+1/2,um+2i+3/2,…,um,um+2i+1/2′,um+2i+3/2′,…,um′, and ℝ′vi′vi+1′=VTSm−vi,ui′,ui.
The cardinalities of the LRN sets other thanℝ′ui′vi are classified in Table 2.
It is clear from Table2 that ℝ′ui′vi≤ℝ′y∀y∈ETSm. Furthermore, ℝ′y∩∪i=1mℝ′ui′vi≥ℝ′ui′vi.Table 2
Cardinality of each LRN set.
LRN SetCardinalityℝ′vivi′2m>2ℝ′vivi−1′2m>2ℝ′ui′vi′2m+1>2ℝ′ui′vi−1′2m>2Theorem 7.
LetTSm be a total sunlet network, where m≥3. Then,(16)dimlfTSm=m.Proof.
To prove the result, we have the following cases.Case 5.
Form=3, the LRN sets of TSm areℝ1′=ℝ′u1′v1=u1′,v1ℝ2′=ℝ′u2′v2=u2′,v2ℝ3′=ℝ′u3′v3=u3′,v3ℝ4′=ℝ′v1v1′=v1,v2,v1′,v2′,u1,u2,u2′ℝ5′=ℝ′v2v2′=v2,v3,v2′,v3′,u2,u3,u3′ℝ6′=ℝ′v3v3′=v3,v1,v3′,v1′,u3,u1,u1′ℝ7′=ℝ′v1v3′=v1,v3,v2′,v3′,u1,u3,u3′ℝ8′=ℝ′v2v1′=v2,v1,v1′,v3′,u2,u1,u1′ℝ9′=ℝ′v3v2′=v3,v2,v2′,v1′,u3,u2,u2′ℝ10′=ℝ′v1′v3′=v1′,v3′,v2,u2′,u2,v3,u3′,u3ℝ11′=ℝ′v1′v2′=v2′,v1′,v3,u3′,u3,v1,u1′,u1ℝ12′=ℝ′v2′v3′=v2′,v1′,v3,u3′,u3,v1,u1′,u2ℝ13′=ℝ′u1′u1=VTS3−v1ℝ14′=ℝ′u2′u2=VTS3−v2ℝ15′=ℝ′u3′u3=VTS3−v3ℝ16′=ℝ′u1v1=VTS3−u1′ℝ17′=ℝ′u2v2=VTS3−u2′ℝ18′=ℝ′u3v3=VTS3−u3′
From the above LRN setsℝ1′=ℝ2′=ℝ3′=2 and the cardinalities of these LRN sets which are less than the cardinalities of all other LRN sets, furthermore, ℝy∩∪i=13ℝui′vi≥ℝui′vi∀y∈ETS3. Hence, the function h:VTS3⟶0,1 is defined as(17)hv=12ifv∈∪i=13ℝ′ui′vi,0ifv∈VTS3−∪i=13ℝ′ui′vi.
Therefore,dimlfTS3=3.Case 6.
In this case,ℝ′ui′vi=2 and ∪i=1mℝ′uivi=2m. Furthermore, ℝ′y∩∪i=1mℝ′ui′vi≥ℝ′ui′vi∀y∈ETSm. Hence, a constant function h:VTSm⟶0,1 is defined as(18)hv=12ifv∈∪i=1mℝ′ui′vi,0ifv∈VTSm−∪i=1mℝ′ui′vi.
Therefore,(19)dimlfTSm=m.


## 3.1. LFMD of Sunlet Network

The resolving neighbourhood sets for each pair of adjacent vertices are classified.Lemma 1.
LetSm with m≥3 and m≅1mod2 be a sunlet network. Then, for 1≤i≤m, we have(a)
ℝ′vivi+1=2m−2, ℝ′uivi=2m, ∪i=1mℝ′vivi+1=2m, and ℝ′vivi+1<ℝ′uivi(b)
ℝ′uivi∩∪i=1mℝ′vivi+1=ℝ′vivi+1Proof.
Assume thatui pendent and vi are internal vertices, respectively, of Sm and vm+1≅v1, where 1≤i≤m.(a)
Sinceℝ′vivi+1=VSm−um+2i+1/2,vm+2i+1/2, this implies that ℝ′vivi+1=2m−2 and ℝ′uivi=VSm; therefore, ℝ′viui=VSm=2m, where 1≤i≤m. Furthermore, ∪i=1mℝ′vivi+1=2m.(b)
Since∪i=1mℝ′vivi+1=VSm, therefore, ℝ′uivi∩∪i=1mℝ′vivi+1=ℝ′vivi+1.Theorem 4.
LetSm be a sunlet network, where m≥3 and m≅1mod2. Then,(7)1<dimlfMSm≤m/m−1.Proof.
To prove the result, we have following cases.Case 1.
Form=3, the LRN sets of Sm areℝ1′=ℝ′v1v2=VS3−v3,u3ℝ2′=ℝ′v2v3=VS3−v1,u1ℝ3′=ℝ′v3v1=VS3−v2,u2ℝ4′=ℝ′u1v1=VS3ℝ5′=ℝ′u2v2=VS3ℝ6′=ℝ′u3v3=VS3
From the above LRN sets,ℝ1′=ℝ2′=ℝ3′=3 and their cardinalities less than the cardinalities of other LRN sets of S3. Hence, the function h:VS3⟶0,1 is defined as hv=1/4∀v∈VS3 which is an upper LRF; therefore, by Theorem 1, dimlfS3≤3/2.
From the above LRN sets,ℝ4′=ℝ5′=ℝ6′=6, and their cardinalities are greater than the cardinalities of other LRN sets of S3. Now, the function h′:VS3⟶0,1 is a lower LRF defined as h′v=1/6; hence, by Theorem 2, dimlfS3≥1. Since S3 is a nonbipartite network, therefore, dimlfS3 must be greater then 1. Consequently,(8)1<dimlfS3≤3/2.Case 2.
In this case,ℝ′vivi+1=2m−2 and ∪i=1mℝ′vivi+1=2m. Furthermore, ℝ′y∩∪i=1mℝ′vivi+1≥ℝ′y∀y∈VSm; hence, the constant function h:VSm⟶0,1 is defined as hv=1/2m−2∀v∈VSm which is an upper LRF; therefore, by Theorem 1, dimlfSm≤m/m−1.
Sinceℝ′uivi=2m and ℝ′uivi≥ℝ′y∀y∈ESm, therefore, the function h′:VSm⟶0,1 is a lower LRF which is defined as h′v=1/2m∀h∈VSm. Hence, by Theorem 2, dimlf≥1. Since Sm is a nonbipartite network, therefore, dimlfMSm must be greater than 1. Consequently,(9)1<dimlfSm≤mm−1.Theorem 5.
LetSm be a sunlet network, where m≥4 and m≅0mod2. Then,(10)dimlfSm=1.Proof.
Since no cycle inSm is of odd length, therefore, Sm is a bipartite network, where m≅0mod2; hence, by Theorem 3, dimlfSm=1.


## 3.2. LFMD of Middle Sunlet Network

The LRN sets for each pair of adjacent vertices are classified.Lemma 2.
LetMSm with m≥3 be a middle sunlet network. Then, for 1≤i≤m,(a)
ℝ′viui′=3, ℝ′uiui+1′=4m, and ∪i=1mℝ′viui′=3m(b)
ℝ′viui′≤ℝ′y and ℝ′y∩∪i=1mℝ′viui′≥ℝ′viui′∀y∈EMSmProof.
Assume thatui pendent and ui′, vi′, and vi are internal vertices, respectively, of MSm, where 1≤i≤m.(a)
Sinceℝ′ui′vi=ui,ui′,vi, this implies that ℝ′ui′vi=3 and ℝ′uiui+1′=VMSm; therefore, ℝ′uiui+1′=VMSm=4m, where 1≤i≤m. Furthermore, ∪i=1mℝ′viui′=3m.(b)
Considerℝ′vivi′=vi,vi+1,vi+2,…,vm+2i−1/2,vi′,vi+1′,vi+2′,…,vm+2i−1/2′,ui+1,ui+2,…,um+2i−1/2,ui+1′,ui+2′,…,um+2i−1/2′, ℝ′vivi−1′=vi,vm+2i+1/2,vm+2i+3/2,…,vm,vm+2i+1/2′,vm+2i+3/2′,vm+2i+5/2′,………,vm′,um+2i+1/2,um+2i+3/2,um+2i+5/2…,um,um+2i+1/2′,um+2i+3/2′,um+2i+5/2′,…,um′,ℝ′ui′vi′=vi+1,vi+2,vi+3,….,vm+2i−1/2,vi′,vi+1′,vi+2′,vi+3,…,vm+2i−1/2′,ui,ui+1,ui+2,ui+3,…,um+2i−1/2,ui′,ui+1′,ui+2′,ui+3′,…,um+2i−1/2′,ℝ′ui′vi−1′=vm+2i+1/2,vm+2i+3/2,vm+2i+5/2…,vm,vm+2i+1/2′,vm+2i+3/2′,vm+2i+5/2′,…,vm′,ui,um+2i+1/2,um+2i+3/2,…,um,um+2i+1/2′,um+2i+3/2′,…,um′, and ℝ′vi′vi+1′=VMSm−vi,ui′,ui.
The cardinalities of LRN sets other thanℝ′ui′vi are classified in Table 1.
It is clear from Table1 that ℝ′ui′vi≤ℝ′y and ℝ′y∩∪i=1mℝ′viui′≥ℝ′viui′∀y∈EMSm.Table 1
Cardinality of each LRN set.
LRN SetCardinalityℝ′vivi′2m>ℝ′ui′viℝ′vivi−1′2m>ℝ′ui′viℝ′ui′vi′2m+1>ℝ′ui′viℝ′ui′vi−1′2m>ℝ′ui′viTheorem 6.
LetMSm be a middle sunlet network, where m≥3.
Then,(11)1<dimlfMSm≤m.Proof.
To prove the result, we have the following cases.Case 3.
Form=3, the LRN sets of MSm areℝ1′=ℝ′u1′v1=u1′,v1,u1ℝ2′=ℝ′u2′v2=u2′,v2,u2ℝ3′=ℝ′u3′v3=u3′,v3,u3ℝ4′=ℝ′v1v1′=v1,v2,v1′,v2′,v2,u2′ℝ5′=ℝ′v2v2′=v2,v3,v2′,v3′,v3,u3′ℝ6′=ℝ′v3v3′=v3,v1,v3′,v1′,v1,u1′ℝ7′=ℝ′v1v3′=v1,v3,v2′,v3′,u3,u3′ℝ8′=ℝ′v2v2′=v2,v2′,v3′,v3,u3′,u3ℝ9′=ℝ′v3v3′=v3,v3′,v1′,v1,u1′,u1ℝ10′=ℝ′v1′v3′=v1′,v3′,v2,u2′,u2,v3,u3′,u3ℝ11′=ℝ′v1′v2′=v2′,v1′,v3,u3′,u3,v1,u1′,u1ℝ12′=ℝ′v2′v3′=v2′,v1′,v3,u3′,u3,v1,u1′,u1ℝ13′=ℝ′u1′u1=VMS3ℝ14′=ℝ′u2′u2=VMS3ℝ15′=ℝ′u3′u3=VMS3
From the above LRN sets,ℝ1′=ℝ2′=ℝ3′=3 and ℝ′ui′vi≤ℝ′y∀y∈EMS3 and ℝ′y∩∪i=13ℝ′viui′≥ℝ′viui′∀y∈EMS3. Hence, the function h:VMS3⟶0,1 is defined as(12)hv=13ifv∈∪i=13ℝ′ui′vi,0ifv∈VMS3−∪i=13ℝ′ui′vi.
Therefore, by Theorem1, dimlfMS3≤3.
Sinceℝ13′=ℝ14′=ℝ15′=12 and cardinality of these LRN sets is greater than all other LRN sets, therefore, we define a constant function h′:VMS3⟶0,1 which is a lower LRF defined by h′v=1/12; hence, by Theorem 2, dimlf≥1. Also, MS3 is a nonbipartite network; therefore, dimlfMS3 must be greater then 1. Consequently,(13)1<dimlfMS3≤3.Case 4.
In this case,ℝ′ui′vi=3 and ∪i=1mℝ′uivi=3m. Furthermore, ℝ′y∩∪i=1mℝ′uivi≥ℝ′uivi∀y∈VMSm; hence, the constant function h:VMS3⟶0,1 is defined as(14)hv=13ifv∈∪i=1mℝ′ui′vi,0ifv∈VMS3−∪i=1mℝ′ui′vi.
Therefore, by Theorem1, dimlfMSm≤m.
Sinceℝ′ui′ui=4m and ℝ′ui′ui≥ℝ′y∀y∈EMSm, therefore, a function h′:VMSm⟶0,1 is a lower LRF and is defined as h′v=1/4m∀v∈VMSm. Hence, by Theorem 2, dimlfMSm≥1. Since MSm is a nonbipartite network, therefore, dimlfMSm must be greater than 1. Consequently,(15)1<dimlfMSm≤m.


## 3.3. LFMD of Total Sunlet Network

The LRN sets of each pair of adjacent vertices are classified.Lemma 3.
LetTSm with m≥3 be a total sunlet network. Then, for 1≤i≤m, we have(a)
ℝ′viui′=2, ℝ′uiui+1′=4m−1, and ∪i=1mℝ′viui′=2m(b)
ℝ′viui′≤ℝ′y and ℝ′y∩∪i=1mℝ′ui≥ℝ′ui′vi∀y∈EMSmProof.
Assume thatui pendent and ui′, vi′, and vi are other vertices, respectively, of TSm, where 1≤i≤m.(a)
Sinceℝ′ui′vi=ui′,vi, this implies that ℝ′ui′vi=2 and ℝ′ui′ui′=VTSm−viandℝ′uivi=VTSm−ui′; therefore, ℝ′uiui′=VTSm=4m−1, where 1≤i≤m. Furthermore, ∪i=1mℝ′viui′=2m.(b)
Considerℝ′vivi′=vi,vi+1,vi+2,…,vm+2i−1/2,vi′,vi+1′,vi+2′,…,vm+2i−1/2′,ui+1,ui+2,…,um+2i−1/2,ui+1′,ui+2′,…,um+2i−1/2′, ℝ′vi′vi+1=vi−1,vi,vm+2i+3/2,vm+2i+5/2,…,vm,vi′,vm+2i+1/2′,vm+2i+3/2′,vm+2i+5/2′,…,vm′,ui+1,um+2i+3/2,um+2i+5/2,…..,um,ui′,ui+1′,um+2i+3/2′,um+2i+5/2′,…,um′, ℝ′ui′vi′=vi+1,vi+2,vi+3,…,vm+2i−1/2,vi′,vi+1′,vi+2′,vi+3,…,vm+2i−1/2′,ui,ui+1,ui+2,ui+3,…,um+2i−1/2,ui′,ui+1′,ui+2′,ui+3′,….,um+2i−1/2′,ℝ′ui′vi−1′=vm+2i+1/2,vm+2i+3/2,vm+2i+5/2…,vm,vm+2i+1/2′,vm+2i+3/2′,vm+2i+5/2′,…,vm′,ui,um+2i+1/2,um+2i+3/2,…,um,um+2i+1/2′,um+2i+3/2′,…,um′, and ℝ′vi′vi+1′=VTSm−vi,ui′,ui.
The cardinalities of the LRN sets other thanℝ′ui′vi are classified in Table 2.
It is clear from Table2 that ℝ′ui′vi≤ℝ′y∀y∈ETSm. Furthermore, ℝ′y∩∪i=1mℝ′ui′vi≥ℝ′ui′vi.Table 2
Cardinality of each LRN set.
LRN SetCardinalityℝ′vivi′2m>2ℝ′vivi−1′2m>2ℝ′ui′vi′2m+1>2ℝ′ui′vi−1′2m>2Theorem 7.
LetTSm be a total sunlet network, where m≥3. Then,(16)dimlfTSm=m.Proof.
To prove the result, we have the following cases.Case 5.
Form=3, the LRN sets of TSm areℝ1′=ℝ′u1′v1=u1′,v1ℝ2′=ℝ′u2′v2=u2′,v2ℝ3′=ℝ′u3′v3=u3′,v3ℝ4′=ℝ′v1v1′=v1,v2,v1′,v2′,u1,u2,u2′ℝ5′=ℝ′v2v2′=v2,v3,v2′,v3′,u2,u3,u3′ℝ6′=ℝ′v3v3′=v3,v1,v3′,v1′,u3,u1,u1′ℝ7′=ℝ′v1v3′=v1,v3,v2′,v3′,u1,u3,u3′ℝ8′=ℝ′v2v1′=v2,v1,v1′,v3′,u2,u1,u1′ℝ9′=ℝ′v3v2′=v3,v2,v2′,v1′,u3,u2,u2′ℝ10′=ℝ′v1′v3′=v1′,v3′,v2,u2′,u2,v3,u3′,u3ℝ11′=ℝ′v1′v2′=v2′,v1′,v3,u3′,u3,v1,u1′,u1ℝ12′=ℝ′v2′v3′=v2′,v1′,v3,u3′,u3,v1,u1′,u2ℝ13′=ℝ′u1′u1=VTS3−v1ℝ14′=ℝ′u2′u2=VTS3−v2ℝ15′=ℝ′u3′u3=VTS3−v3ℝ16′=ℝ′u1v1=VTS3−u1′ℝ17′=ℝ′u2v2=VTS3−u2′ℝ18′=ℝ′u3v3=VTS3−u3′
From the above LRN setsℝ1′=ℝ2′=ℝ3′=2 and the cardinalities of these LRN sets which are less than the cardinalities of all other LRN sets, furthermore, ℝy∩∪i=13ℝui′vi≥ℝui′vi∀y∈ETS3. Hence, the function h:VTS3⟶0,1 is defined as(17)hv=12ifv∈∪i=13ℝ′ui′vi,0ifv∈VTS3−∪i=13ℝ′ui′vi.
Therefore,dimlfTS3=3.Case 6.
In this case,ℝ′ui′vi=2 and ∪i=1mℝ′uivi=2m. Furthermore, ℝ′y∩∪i=1mℝ′ui′vi≥ℝ′ui′vi∀y∈ETSm. Hence, a constant function h:VTSm⟶0,1 is defined as(18)hv=12ifv∈∪i=1mℝ′ui′vi,0ifv∈VTSm−∪i=1mℝ′ui′vi.
Therefore,(19)dimlfTSm=m.


## 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,TSm=m.Bounded and unboundedness of LFMDs are illustrated through Table3.Table 3
Bounded and unboundedness of LFMDs.
NetworkLFMDsLower bound of LFMD whenm⟶∞Upper bound of LFMDCommentSm1<dimlfSm≤mm−11limm⟶3mm−1=1.5BoundedMSm1<dimlfMSm≤m1limm⟶∞m=∞Unbounded



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