# Computing LF-Metric Dimension of Generalized Gear Networks

Authors: Hassan Zafar; Muhammad Javaid; Ebenezer Bonyah
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

The parameter of distance in the theory of networks plays a key role to study the different structural properties of the understudy networks or graphs such as symmetry, assortative, connectivity, and clustering. For the purpose, with the help of the parameter of distance, various types of metric dimensions have been defined to find the locations of machines (or robots) with respect to the minimum consumption of time, the shortest distance among the destinations, and the lesser number of utilized nodes as places of the objects. In this article, the latest derived form of metric dimension called as LF-metric dimension is studied, and various results for the generalized gear networks are obtained in the form of exact values and sharp bounds under certain conditions. The LF-metric dimension of some particular cases of generalized gear networks (called as generalized wheel networks) is also illustrated. Moreover, the bounded and unboundedness of the LF-metric dimension for all obtained results is also presented.


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## Body

## 1. Introduction

Slater defined the idea of the resolving sets for a connected network to find the reference or location number for a connected network [1]. Harary and Melter studied the same concept of location number with the different term called metric dimension (MD) for connected networks [2]. Melter and Tomescu investigated metric basis in digital geometry those often used in city block distance [3], and Slater studied the dominating and reference sets in a network [4]. Garey and Johnson proved that finding MD of all the connected networks in general form is an NP complete problem [5]. Moreover, Kelenca et al. defined the idea of edge MD, they showed that computing edge MD is NP-complete problem [6], and the directed distance dimension of oriented networks is computed by Chattrand et al. [7]. For further studies of MD of Caylay, mobius ladders, lexicographic product, and Toeplitz networks, we refer to [8–11].The concepts of various metric dimensions used in engineering are robot navigation, image processing, and pattern recognition [12]. It is used to solve problems involving percolation in hierarchical lattice [13] and to study the structural properties of chemical compounds. More importantly, Chartrand et al. used the concept of MD to solve an integer programming problem (IPP) [14]. The generalized Jahangir network Jm,k is defined by Tomescu and Javaid [15], and they also computed MD for J2n. Furthermore, MD of generalized certain gear networks J2n,m and J3n was computed by Imran et al. [16].Recently, Currie and Oellermann defined the concept of fractional metric dimension (FMD) to improve the solution of the linear relaxation of the IPP [17]. Fehr et al. used FMD to obtain an optimal solution of IPP [18]. Arumugam and Matthew calculated the exact values of FMD of some connected networks [19]. The bounds and exact values of FMD of vertex-transitive and distance-regular networks are computed by Feng et al. [20]. Saputro et al. computed the FMD of comb product of connected networks [21]. For further studies of FMD for hierarchical product, trees, and unicyclic and permutation networks, see [22–24]. Raza et al. computed the bounds of FMD of metal organic frameworks [25]. The FMD of generalized Jahangir network Jm,k for m=5 is computed by Liu et al. [26].The idea of LF- metric dimension is defined by Aisyah et al., and they also computed exact values for LF- metric dimension for the corona product of connected networks [27]. Liu et al. [28] studied the LF-metric dimension of triangular circular, quadrangular circular, and pentagonal circular ladders. Javaid et al. established the criteria to compute the upper bounds of LF-metric dimension of connected networks and demonstrated the main results by using LF-metric dimension such as wheel-related networks, flowers, and antiweb gear networks [29]. Recently, Javaid et al. improved the lower bound of LF-metric dimension from unity and established the sharp bounds and computed the exact values of LFMD of some prism-related networks [30].In this article, we have computed LF-metric dimension of generalized gear networks in the form of exact values and sharp bounds. Furthermore, Sections2 and 3 consist of preliminaries and LRN sets of Jm,k, respectively. In Section 4, we have computed LF-metric dimension of generalized gear networks, and Section 5 represents the conclusion and limiting values of LF-metric dimension.


## 2. Preliminaries

LetG=VG,EG be a network with VG and EG⊆VG×VG as vertex and edge set, respectively. A walk is a sequence uo,e1,u1,…,um−1,em,um of vertices and edges such that the edge ei has end points ui−1 and ui for 1≤i≤m. A path between two vertices u and v is called a walk if the repetition of vertices does not exist [31]. For any two vertices u,v∈VG, the distance du,v is the length of the shortest path between them in G. A network is said to be connected if there exists a path between any pair of vertices. A vertex x∈VG is said to resolve a pair a,b⊆VG if dx,a≠dx,b. Let S=v1,v2,v3,…,vm⊆VG be an ordered set and x∈VG, then the m tuple representation of x with respect to S is dx|S=x,v1,x,v2,x,v3,…,x,vm. If the distinct vertices of G have distinct representations with respect to S, then S is called resolving (locating) set. The resolving set with minimum cardinality is called the metric basis of G, and the cardinality of metric basis is called metric dimension of G defined as(1)dimG=minS:Sis the resolving set ofG.For an edgeab∈EG, the local resolving neighbourhood set of G is LRab=z∈VG:dz,a≠dz,b. A real-valued function f:VG⟶0,1 such that fLRab≥1 for each LR(ab) is called a local resolving function (LRF) of G, where fLRab=∑x∈LRabfx. A LRF f is called minimal if there exists a function g:VG⟶0,1 such that g≤f and gx≠fx for at least one x∈G that is not LRF of G. LF-metric dimension of G is donated by dimlfG and defined as(2)dimlfG=minf:fis minimal local resolving function ofG.Now, we present some important results frequently used in this paper.Theorem 1 (see [29]).
LetG be a connected network. If G is bipartite network, then dimlfG=1.Theorem 2 (see [29]).
LetG be a connected network and LRe be a LRN set for some e∈EG. If LRe∩A≥α,∀e∈EG, then(3)1≤dimlfG≤Aα,where A=∪LRe:LRe=α, α=minLRe:e∈EG, and 2≤α≤VG.Theorem 3 (see [30]).
LetG be a connected network and LRe be the LRN set. Then,(4)VGλ≤dimlfG,λ=maxLRe:e∈EG,2≤λ≤VG.Corollary 1 (see [30]).
LetG=VG,EG be a connected network, LRe be LRN of e∈EG, λ=maxLRe:e∈EG, α=minLRe:e∈EG, and X=∪LRe:LRe=α. If α=λ and X=VG, then(5)dimlfG=VGλ.Proposition 1 (see [28]).
LetG=VG,EG be a connected network X=∪LRe:LRe=2. If LRe∩X≥2 for all e∈EG, then dimIf=X/2.Now, we define generalized gear network as follows.Definition 1.
Generalized gear network sometimes known as bipartite wheel network is obtained from wheel network. LetJm,k be the generalized gear network of order mk+1+1, where m≥3 and k≥1. There are three type of vertices in Jm,k such as major vertices ai, minor vertices bki, where 1≤i≤m and a central vertex b. Moreover, EJm,k=aibki,1≤i≤m∪bkibki+1,1≤i≤m∪aib,1≤i≤m. Central vertex is adjacent to m major vertices and contains an outer cycle cmk+1 as shown in Figure 1. For more details of Jm,k, see [16].Figure 1
Generalized gear networkJm,k.


## 3. LRN Sets of the Generalized Gear NetworkJm,k

The resolving neighbourhood sets for each pair of adjacent vertices are classified.Lemma 1.
LetJm,0 with m≥6 be a generalized gear network, where VJm,0=m+1. Then, for 1≤i≤m, we have(a)
LRei=LRaiai+1=4 and ∪i=1mLRei=m(b)
LRei≤LRx, and LRx∩∪i=1mLRei≥LRei∀x∈EJm,0Proof.
Assume thatai (major) vertices for 1≤i≤m and b (centre) vertex of Jm,0 such that am=a−1 and am+1=a1.(a)
ConsiderLRei=LRaiai+1=ai,ai+1,ai+2,ai−1, this implies that LRei=4, where 1≤i≤m. Furthermore, ∪i=1mLRei=m.(b)
ConsiderLRaib=VJm,0−ai−1,ai+1, therefore LRaib=m−1. Moreover, LRaib>LRei and LRx∩∪i=1mLRei≥LRei,∀x∈EJm,0.Lemma 2.
LetJm,k with m≥4 be a generalized gear network, where m≅0mod2, and k≅0mod2. Then, for 1≤i≤m, we have(a)
LRei=LRbk/2ibk/2+1i=2k+4 and ∪i=1mLRei=VJm,k−1(b)
LRei≤LRx and LRx∩∪i=1mLRei≥LRei, ∀x∈EJm,kProof.
Assume thatai (major), b (center), and bki be a minor vertex, respectively, where 1≤i≤k and i+P≅Pmodk. Now, we have,(a)
ConsiderLRei=LRbk/2ibk/2+1i=bk/2i,bk/2+1i,…,bk/2+1i+1∪bk/2m,bk/2+ii,…,bk/2−1i, LRei=2k+4 and ∪i=1mLRei=VJm,k−1(b)
LRNs other thanLRei areLRaib=VJm,k−bk+2/2i,bk/2i+k+1,LRaib1i=VJm,k−bk+4/2i,LRaibk=VJm,k−bk/2−1i,LRb1ib2i=VJm,k−bk+2/2m,bk+2/2+1m,…,ai,fork=2LRb1ib2i=VJm,2−ai+1,b1i+1,bki+1
The cardinality of each LRN set is shown in Table1.
From Table1, it is clear that LRei≤LRx,∀x∈EJm,k. Furthermore, ∪i=1mLRei=VJm,k−1; therefore, LRx∩∪i=1mLRei≥LRei∀x∈EJm,k.Table 1
Cardinality of each LRN set.
LRN setCardinalityLRaibVJm,k−2>LReiLRaib1iVJm,k−4>LReiLRaib1ik=2VJm,k−3>LReiLRaibkVJm,k−1>LReiLemma 3.
LetJm,k be a generalized gear network, where k≅0mod2 and m≅1mod2. Then, for 1≤i≤m, we have(a)
LRei=LRbk/2ibk/2+1i=2k+4 and ∪i=1mLRei=VJm,k−1(b)
LRei≤LRx and LRx∩∪i=1mLRei≥LRei,∀x∈EJm,kProof.
Assume thatai (major), b (center) and bki be the minor vertex, for 1≤i≤m,m+1modm=1. Now, we have(a)
LRei=LRbk/2ibk/2+1i=bk/2i,bk/2+1i,…,bk/2+1i+1∪bk/2m,bk/2+im,…,bk/2−1i, LRei=2k+4 and ∪i=1mLRei=VJm,k−1.(b)
LRNs other thanLRei are LRaibki=VJm,4−bki, LRaibki for k≥6=VJm,k−bk/2+1i, LRaib=VJm,k−bk/2+1i,bk/2m, LRb1ib2i=VJm,4−ai+1,b1i+1,b2i+1, LRb1ib2ik≥6=VJm,k−bk+6/2i, LRb3ib4i=VJm,4−ai−1,b4i−1,b4i−1, LRbk−1ibki=VJm,k−bk−4/2iLRbk−2ibk−1i=VJm,k−bk−6/2i, and LRb2ib3i=VJm,k−bk+8/2i.
The cardinality of each LRN set is shown in Table2.
From Table2, it is clear that LRei≤LRx,∀x∈EJm,k. Furthermore, ∪i=1mLRei=VJm,k−1. Therefore, LRx∩∪i=1mLRei≥LRei∀x∈EJm,k.Table 2
Cardinality of each LRN set.
LRN setCardinalityLRaibkiVJm,k−1>LReiLRaibVJm,k−1>LReiLRb1ib2ik=4VJm,4−3>LReiLRb1ib2ik≥6VJm,k−3>LReiLRaib1ik=2VJm,k−3>LReiLRaibkVJm,k−1>LReiLRb3ib4ik=4VJm,4−3>LReiLRbk−1ibkiVJm,k−1>LReiLRbk−2ibk−1iVJm,k−1>LRei


## 4. LFMD of Generalized Gear NetworksJm,k

In this section, we compute LF-metric dimension of generalized gear network in the form of exact values and bounds under certain conditions.


### 4.1. Fork=0 and m≥3

In this particular section, we determine the LF-metric dimension of the generalized wheel network, which is the special case of the generalized gear network. Fork=0, let A=a1,a2,a3,a4,…,am be the set of outer vertices and b is the central vertex of Jm,0.Theorem 4.
LetJ3,0 be a generalized gear network, then dimlfJ3,0=2.Proof.
Form=3 and k=0. The LRN sets are given as follows.
The LRN sets of major vertices to central vertex are as follows:LR1=LRa1b=a1,bLR2=LRa2b=a2,bLR3=LRa3b=a3,b
The LRN sets of adjacent pair of major vertices are as follows:LR4=LRa1a2=a1,a2LR5=LRa2a3=a2,a3LR6=LRa3a1=a1,a3
From above LRN sets,LRei=2, where 1≤i≤6 and ∪i=16LRei=VJ3,0=4. Hence, a function Γ:VJ3,0⟶0,1 is an LRF defined by Γv=1/2 for each v∈VJ3,0. Therefore, by Proposition 1,(6)dimlfJ3,0=VJ3,02=2.Theorem 5.
LetJ4,0 be a generalized gear network, then(7)54≤dimlfJ4,0≤53.Proof.
Form=4 and k=0, the LRN sets are given as follows.
The LRN sets of adjacent pair of major vertices are as follows:LR1=LRa1a2=a1,a2,a3,a4LR2=LRa2a3=a1,a2,a3,a4LR3=LRa3a4=a1,a2,a3,a4LR4=LRa4a5=a1,a2,a3,a4
The LRN sets of major vertices with central vertex are as follows:LR5=LRa1b=a1,a3,bLR6=LRa2b=a2,a4,bLR7=LRa3b=a1,a3,bLR8=LRa4b=a2,a4,b
From above LRN sets,LRaib=3, where 1≤i≤4 and LRaib<LRx∀x∈EJ4,0. Moreover, ∪i=14LRaib=VJ4,0; this implies ∪i=14LRaib=4 and LRx∩∪i=14LRaib=4. Therefore, a function Γ:VJ4,0⟶0,1 defined by Γv=1/3 for each v∈VJ4,0 is an upper LRF. Consequently, by Theorem 2, dimlfJ4,0≤5/3.
From above LRN sets,LRaiai+1=4, where 1≤i≤4 and LRaiai+1≥LRx, ∀x∈EJ4,0. Therefore, a function Γ′:VJ4,0⟶0,1 is defined by Γ′v=1/4 which is a lower LRF for each v∈VJ4,0. Hence, by Theorem 3, dimlfJ4,0≥5/4.
Consequently,(8)54≤dimlfJ4,0≤53.Theorem 6.
LetJ5,0 be the generalized gear network, then(9)dimlfJ5,0=32.Proof.
Form=5 and k=0, the LRN sets are given as follows:
The LRN sets of major vertices with central vertex are as follows:LR1=LRa1b=a1,a3,a4,bLR2=LRa2b=a2,a4,a5,bLR3=LRa3b=a3,a1,a5,bLR4=LRa4b=a1,a2,a4,bLR5=LRa5b=a2,a3,a5,b
The LRN sets of major vertices with central vertex are as follows:LR6=LRa1a2=a1,a2,a3,a5LR7=LRa2a3=a1,a2,a3,a4LR8=LRa3a4=a2,a3,a4,a5LR9=LRa4a5=a1,a3,a4,a5LR10=LRa1a5=a1,a2,a4,a5
From above LRN sets,LRei=4, where 1≤i≤10. Hence, by Corollary 1,(10)dimlfJ5,0=32.Theorem 7.
LetJm,0 with m≥6 be a generalized gear network, then(11)m+1m−1≤dimlfJm,0≤m4.Proof.
To prove the result, we have the following cases:Case 1: fork = 0 and m=6, the LRN sets are given as follows:The LRN sets of major vertices with central vertex are as follows:LR1=LRa1b=a1,a3,a4,a5,bLR2=LRa2b=a2,a4,a5,a6,bLR3=LRa3b=a3,a5,a6,a1,bLR4=LRa4b=a4,a6,a1,a2,bLR5=LRa5b=a5,a1,a2,a3,bLR6=LRa6b=a6,a2,a3,a4,bThe LRN sets of adjacent pair of major vertices:LR7=LRa1a2=a1,a2,a3,a6LR8=LRa2a3=a2,a3,a1,a4LR9=LRa3a4=a3,a4,a5,a2LR10=LRa4a5=a4,a5,a6,a3LR11=LRa5a6=a4,a5,a6,a1LR12=LRa6a1=a1,a5,a6,a2From above LRN sets,LRaiai+1=4, where 1≤i≤6 and LRaiai+1≤LRx, ∀x∈EJ4,0. Moreover, ∪i=16LRaiai+1=6 and LRx∩∪i=16LRaiai+1≥LRx. Therefore, a function Γ:VJ6,0⟶0,1 defined as an upper LRF function Γx=1/4 for each x∈∪i=16LRaiai+1 and 0 otherwise. Therefore, by Theorem 2, dimlfJ6,0≤3/2.SinceLRaib=5, where 1≤i≤6 and LRaib≥LRx, ∀x∈EJ4,0. Therefore, a function Γ′:VJ6,0⟶0,1 defined by Γ′v=1/5 for each v∈VJ6,0 is a lower LRF, Consequently, by Theorem 3, dimlfJ6,0≥7/5.(12)75≤dimlfJ6,0≤32.Case 2: fork = 0 and m≥7.Form≥7: by Lemma 1, LRei=LRaiai+1=4 and LRx∩∪i=1mLRaiai+1≥LRaiai+1, ∀x∈EJm,0. Furthermore, ∪i=1mLRei=m. Hence, a function Γ:VJm,0⟶0,1 is an upper LRF with minimum cardinality defined as(13)Γv=14,forv∈∪i=1mLRei,0,forv∈VJm,0−∪i=1mLRei.Consequently, by Theorem2dimlfJm,0≤∑i=1m1/4=m/4.By Lemma1, LRaib=m−1, where 1≤i≤m and LRaib≥|LRx, ∀x∈EJm,0. Furthermore, ∪i=1mLRaib=VJm,0; this implies ∪i=1mLRaib=m+1. Hence, function γ:VJm,0⟶0,1 is lower LRF defined by γv=1/m−1 for each v∈VJm,0; therefore, by Theorem 3, dimlfJm,0≥m+1/m−1.Consequently,(14)m+1m−1≤dimlfJm,0≤m4.


### 4.2. Fork=2 and m≅1mod2

Theorem 8.
LetJ3,2 be a generalized gear network, then dimlfJ3,2=5/4.Proof.
LetA=a1,a2,a3 be the set of major, B=b11,b21,b12,b22,b13,b13,b23 is the set of minor vertices, respectively, and b is the central vertex. For k=2 and m=3, the LRN sets are given as follows.
The LRN sets of major vertices to central vertex are as follows:LR1=LRa1b=a1,a2,a3,b11,b12,b22,b23,bLR2=LRa2b=a1,a2,a3,b21,b12,b23,b13,bLR3=LRa3b=a1,a2,a3,b11,b21,b22,b23,b
The LRN sets of major to minor vertices are as follows:LR4=LRa1b11=a1,a3,b11,b21,b22,b13,b23,bLR5=LRa2b12=a1,a2,b11,b21,b12,b22,b23,bLR6=LRa3b13=a2,a3,b21,b12,b22,b13,b23,bLR7=LRa1b23=a1,a2,b11,b21,b12,b13,b23,bLR8=LRa2b21=a2,a3,b11,b21,b12,b22,b13,bLR9=LRa3b22=a1,a3,b11,b12,b22,b23,b13,b
The LRN sets of adjacent pair of minor vertices are as follows:LR10=LRb11b21=a1,a2,b11,b21,b12,b22,b13,b23LR11=LRb12b22=a2,a3,b11,b21,b12,b22,b13,b23LR12=LRb13b23=a1,a3,b11,b21,b12,b22,b13,b23
Since eachLRei=8, where 1≤i≤12, by Corollary 1, dimlfJ3,2=VJ3,2/α=5/4.Theorem 9.
LetJm,2, where m≅1mod2, be a generalized gear network. Then, for 1≤i≤m,(15)3m+13m−1≤dimlfJm,2≤3m8.Proof.
To prove the result, we have the following cases:Case 1: for,k=2 and m=5, the LRN sets are given as follows.The LRN sets of major vertices with central vertex are as follows:LR1=LRa1b=VJ2,5−b22,b15LR2=LRa2b=VJ2,5−b21,b22LR3=LRa3b=VJ2,5−b22,b23LR4=LRa4b=VJ2,5−b13,b24LR5=LRa4b=VJ2,5−b24,b25The LRN sets of adjacent pair of minor vertices are as follows:LR6=LRa1b11==VJ2,5−a2,b12LR7=LRa2b12==VJ2,5−a3,b13LR8=LRa3b13==VJ2,5−a4,b14LR9=LRa4b14==VJ2,5−a5,b15LR10=LRa5b15==VJ2,5−a1,b11LR11=LRa2b21==VJ2,5−a1,b25LR12=LRa3b22==VJ2,5−a2,b22LR13=LRa4b23==VJ2,5−a3,b22LR14=LRa5b24==VJ2,5−a4,b23LR15=LRa1b25==VJ2,5−a5,b24The LRN sets of adjacent pair of minor vertices are as follows:LR16=LRb11b21=a1,a2,b11,b21,b12,b22,b13,b23LR17=LRb12b22=a2,a3,b11,b21,b12,b22,b13,b23LR18=LRb13b23=a1,a3,b12,b22,b15,b25,b14,b24LR19=LRb14b24=a1,a3,b11,b21,b13,b23,b15,b25LR20=LRb15b25=a1,a3,b11,b21,b13,b23,b14,b24From above LRN sets,LRei=LRbiibi+1i=8, where 1≤i≤5 and LRei≤LRx∀x∈EJ5,2. Furthermore, ∪i=15LRei=15 and LRx∩∪i=15LRei≥LRei=8. Hence, a function γ:VJ5,2⟶0,1 defined by γv=1/8, if v∈∪i=15LRei and 0 otherwise is an upper LRF. Consequently, by Theorem 2, dimlfJ5,2≤15/8.From above LRN sets,LRei′=14, where 1≤i≤15 and LRei′≥LRx∀x∈EJ5,2. Therefore, there exists a lower LRF A function γ′:VJ5,2⟶0,1 defined by γ′v=1/12 is an LRF of maximum cardinality for each v∈VJ5,2. Hence, by Theorem 3, dimlfJ5,2≥8/7.Consequently,(16)87≤dimlfJ5,2≤158.Case 2: fork=2, m≥5, and m≅1mod2,LRei=LRbk/2ibk/2+1i, where 1≤k≤n. By Lemma 2, we have the following:(i)
LRei=LRbk/2ibk/2+1i=8(ii)
LRx∩∪i=1nLRei≥8∀x∈EJm,2Therefore, a functionΓ:VJm,2⟶0,1 is an upper LRF with minimum cardinality defined by(17)Γv=18,ifv∈∪i=1mLRei,0,ifv∈VJm,2−∪i=1mLRei.Consequently, by Theorem2, dimlfJm,2≤3m/8.SinceLRaibi=VJm,2−2 and LRaibi≥LRx, ∀x∈EJm,2. Hence, the function Γ′:VJm,2⟶0,1 is a lower LRF with maximum cardinality, defined by Γ′v=1/3m−1∀v∈VJm,2. Therefore, by Theorem 3, dimlfJm,2≥3m+1/3m−1.Consequently,(18)3m+13m−1≤dimlfJm,2≤3m8.Theorem 10.
LetJm,k be a generalized gear network with m≥4, k≥2, where k≅0mod2. Then,(19)mk+1+1mk+1≤dimlfJm,k≤mk+12k+4.Proof.
To prove the result, we have the following cases:Case 1: for,m = 4 and k = 2, the LRN sets are given as follows.Adjacent pair of minor vertices is as follows:LR1=LRb11b21=VJ4,2−a3,b13,b23,a4,bLR2=LRb12b22=VJ4,2−b14,a1,b,b24,b14LR3=LRb13b23=VJ4,2−a1,b11,b21,a2,bLR4=LRb14b24=VJ4,2−b12,a2,b,a3,b22Major with central vertex.LR5=LRa1b=VJ4,2−b21,b14LR6=LRa2b=VJ4,2−b11,b22LR7=LRa3b=VJ4,2−b23,b12LR8=LRa4b=VJ4,2−b24,b21Major with minor vertex.LR9=LRa1b11=VJ4,2−a2LR10=LRa1b24=VJ4,2−a4LR11=LRa2b21=VJ4,2−a1LR12=LRa2b12=VJ4,2−a3LR13=LRa4b23=VJ4,2−a3LR14=LRa4b23=VJ4,2−a3LR15=LRa4b14=VJ4,2−a1From above LRN sets,LRei=LRb1ib2i=8, where 1≤i≤4 and LRei≤LRx∀x∈EJ4,2. Furthermore, ∪i=14LRei=12 and LRx∩∪i=14LRei≥LRei=8. Hence, a function γ:VJ4,2⟶0,1 defined by γv=1/8 if v∈∪i=14LRei and 0 otherwise is a lower LRF with minimum cardinality. Consequently, by Theorem 2, dimlfJ4,2≤3/2.From above LRN sets,LRei′=12, where 1≤i≤6 and LRei′≥LRx∀x∈EJ4,2. Therefore, a function γ′:VJ4,2⟶0,1 defined by γ′v=1/12 is a lower LRF with maximum cardinality for each v∈VJ4,2.(20)1312≤dimlfJ4,2≤32.Case 2: fork = 4 and m = 3, the possible LRN sets are as follows.Central to major vertex.LR1a1b=VJ3,4−b31,b23LR2a2b=VJ3,4−b32,b21LR3a3b=VJ3,4−b33,b23Major with minor vertex.LR4a1b11=VJ3,4−b41LR5a2b12=VJ3,4−b42LR6a3b13=VJ3,4−b43LR7a1b43=VJ3,4−b13LR8a2b41=VJ3,4−b11LR9a3b42=VJ3,4−b12Adjacent pair of internal vertices.LR10b11b21=VJ3,4−a2,b12,b22LR11b31b41=VJ3,4−a1,b33,b43LR12b12b22=VJ3,4−a3,b13,b23LR13b32b42=VJ3,4−a2,b41,b31LR14b13b23=VJ3,4−a1,b11,b21LR15b23b33=VJ3,4−a2,b12,b41LR16b33b43=VJ3,4−a3,b42,b32LRN sets with minimum cardinality are as follows:LR17b21b31=VJ3,4−b,b42,b13,a3LR18b22b32=VJ3,4−b,b43,b11,a1LR19b23b33=VJ3,4−b,b12,b41,a2From above LRN sets,LRei=LRb2ib3i=12, where 1≤i≤3 and LRei≤LRx∀x∈EJ3,4. Moreover, ∪i=13LRei=VJ3,4−1, and this implies ∪i=13LRei=15 and LRx∩∪i=112LRei≥LRei=12. Hence, a function Γ:VJ3,4⟶0,1 is defined by Γv=1/12 which is an upper LRF with minimum cardinality for all v∈VJ3,4 and 0 otherwise. Therefore, dimlfJ3,4≤5/4.From above LRN setsLRei=6, where 1≤i≤6 and LRei≥LRx, ∀x∈EJ3,4. Therefore, a function Γ′:VJ3,4⟶0,1 is defined by Γ′v=1/6∀v∈VJ3,4; hence, by Theorem 3, dimlfJ3,4≥4/3. Consequently,(21)43≤dimlfJ3,4≤85.Case 3: fork≅0mod2 and m≥4, the possible LRN sets are LRbk/2ibk/2+1i, where 1≤k≤n. By Lemma 2 and 3, we have(i)
LRei=LRbk/2ibk/2+1i=2k+4 and LRei≤LRx∀x∈EJm,k(ii)
LRx∩∪i=1mLRei≥LRei∀x∈EJm,kTherefore,Γ:VJm,k⟶0,1 is an upper LRF defined by(22)Γv=12k+4,ifv∈⋃i=1mLRei,0,ifv∈VJm,k−∪i=1mLRei.Consequently, by Theorem2, dimlfJm,k≤mk+1/2k+4.Case 4: sinceLRaibi=VJm,k−1 and LRaibi≥LRx, ∀x∈EJm,k. Therefore, Γ′:VJm,k⟶0,1 defined by Γ′v=1/mk+1,∀v∈VJm,k is lower LRF. Hence, by Theorem 3, dimlfJm,k≥mk+1+1/mk+1.Consequently,(23)mk+1+1mk+1≤dimlfJm,k≤mk+12k+4.


### 4.3. Fork≅1mod2 and m≥3

Theorem 11.
Fork≥3, kk≅1mod2 and G≅Jm,k is a generalized gear network, and then dimlfG=1.Proof.
As there is no cycle of odd length inG≅Jm,k, where k≅1mod2. Therefore, G is bipartite network; hence, by Theorem 1, dimlfG=1.


## 4.1. Fork=0 and m≥3

In this particular section, we determine the LF-metric dimension of the generalized wheel network, which is the special case of the generalized gear network. Fork=0, let A=a1,a2,a3,a4,…,am be the set of outer vertices and b is the central vertex of Jm,0.Theorem 4.
LetJ3,0 be a generalized gear network, then dimlfJ3,0=2.Proof.
Form=3 and k=0. The LRN sets are given as follows.
The LRN sets of major vertices to central vertex are as follows:LR1=LRa1b=a1,bLR2=LRa2b=a2,bLR3=LRa3b=a3,b
The LRN sets of adjacent pair of major vertices are as follows:LR4=LRa1a2=a1,a2LR5=LRa2a3=a2,a3LR6=LRa3a1=a1,a3
From above LRN sets,LRei=2, where 1≤i≤6 and ∪i=16LRei=VJ3,0=4. Hence, a function Γ:VJ3,0⟶0,1 is an LRF defined by Γv=1/2 for each v∈VJ3,0. Therefore, by Proposition 1,(6)dimlfJ3,0=VJ3,02=2.Theorem 5.
LetJ4,0 be a generalized gear network, then(7)54≤dimlfJ4,0≤53.Proof.
Form=4 and k=0, the LRN sets are given as follows.
The LRN sets of adjacent pair of major vertices are as follows:LR1=LRa1a2=a1,a2,a3,a4LR2=LRa2a3=a1,a2,a3,a4LR3=LRa3a4=a1,a2,a3,a4LR4=LRa4a5=a1,a2,a3,a4
The LRN sets of major vertices with central vertex are as follows:LR5=LRa1b=a1,a3,bLR6=LRa2b=a2,a4,bLR7=LRa3b=a1,a3,bLR8=LRa4b=a2,a4,b
From above LRN sets,LRaib=3, where 1≤i≤4 and LRaib<LRx∀x∈EJ4,0. Moreover, ∪i=14LRaib=VJ4,0; this implies ∪i=14LRaib=4 and LRx∩∪i=14LRaib=4. Therefore, a function Γ:VJ4,0⟶0,1 defined by Γv=1/3 for each v∈VJ4,0 is an upper LRF. Consequently, by Theorem 2, dimlfJ4,0≤5/3.
From above LRN sets,LRaiai+1=4, where 1≤i≤4 and LRaiai+1≥LRx, ∀x∈EJ4,0. Therefore, a function Γ′:VJ4,0⟶0,1 is defined by Γ′v=1/4 which is a lower LRF for each v∈VJ4,0. Hence, by Theorem 3, dimlfJ4,0≥5/4.
Consequently,(8)54≤dimlfJ4,0≤53.Theorem 6.
LetJ5,0 be the generalized gear network, then(9)dimlfJ5,0=32.Proof.
Form=5 and k=0, the LRN sets are given as follows:
The LRN sets of major vertices with central vertex are as follows:LR1=LRa1b=a1,a3,a4,bLR2=LRa2b=a2,a4,a5,bLR3=LRa3b=a3,a1,a5,bLR4=LRa4b=a1,a2,a4,bLR5=LRa5b=a2,a3,a5,b
The LRN sets of major vertices with central vertex are as follows:LR6=LRa1a2=a1,a2,a3,a5LR7=LRa2a3=a1,a2,a3,a4LR8=LRa3a4=a2,a3,a4,a5LR9=LRa4a5=a1,a3,a4,a5LR10=LRa1a5=a1,a2,a4,a5
From above LRN sets,LRei=4, where 1≤i≤10. Hence, by Corollary 1,(10)dimlfJ5,0=32.Theorem 7.
LetJm,0 with m≥6 be a generalized gear network, then(11)m+1m−1≤dimlfJm,0≤m4.Proof.
To prove the result, we have the following cases:Case 1: fork = 0 and m=6, the LRN sets are given as follows:The LRN sets of major vertices with central vertex are as follows:LR1=LRa1b=a1,a3,a4,a5,bLR2=LRa2b=a2,a4,a5,a6,bLR3=LRa3b=a3,a5,a6,a1,bLR4=LRa4b=a4,a6,a1,a2,bLR5=LRa5b=a5,a1,a2,a3,bLR6=LRa6b=a6,a2,a3,a4,bThe LRN sets of adjacent pair of major vertices:LR7=LRa1a2=a1,a2,a3,a6LR8=LRa2a3=a2,a3,a1,a4LR9=LRa3a4=a3,a4,a5,a2LR10=LRa4a5=a4,a5,a6,a3LR11=LRa5a6=a4,a5,a6,a1LR12=LRa6a1=a1,a5,a6,a2From above LRN sets,LRaiai+1=4, where 1≤i≤6 and LRaiai+1≤LRx, ∀x∈EJ4,0. Moreover, ∪i=16LRaiai+1=6 and LRx∩∪i=16LRaiai+1≥LRx. Therefore, a function Γ:VJ6,0⟶0,1 defined as an upper LRF function Γx=1/4 for each x∈∪i=16LRaiai+1 and 0 otherwise. Therefore, by Theorem 2, dimlfJ6,0≤3/2.SinceLRaib=5, where 1≤i≤6 and LRaib≥LRx, ∀x∈EJ4,0. Therefore, a function Γ′:VJ6,0⟶0,1 defined by Γ′v=1/5 for each v∈VJ6,0 is a lower LRF, Consequently, by Theorem 3, dimlfJ6,0≥7/5.(12)75≤dimlfJ6,0≤32.Case 2: fork = 0 and m≥7.Form≥7: by Lemma 1, LRei=LRaiai+1=4 and LRx∩∪i=1mLRaiai+1≥LRaiai+1, ∀x∈EJm,0. Furthermore, ∪i=1mLRei=m. Hence, a function Γ:VJm,0⟶0,1 is an upper LRF with minimum cardinality defined as(13)Γv=14,forv∈∪i=1mLRei,0,forv∈VJm,0−∪i=1mLRei.Consequently, by Theorem2dimlfJm,0≤∑i=1m1/4=m/4.By Lemma1, LRaib=m−1, where 1≤i≤m and LRaib≥|LRx, ∀x∈EJm,0. Furthermore, ∪i=1mLRaib=VJm,0; this implies ∪i=1mLRaib=m+1. Hence, function γ:VJm,0⟶0,1 is lower LRF defined by γv=1/m−1 for each v∈VJm,0; therefore, by Theorem 3, dimlfJm,0≥m+1/m−1.Consequently,(14)m+1m−1≤dimlfJm,0≤m4.


## 4.2. Fork=2 and m≅1mod2

Theorem 8.
LetJ3,2 be a generalized gear network, then dimlfJ3,2=5/4.Proof.
LetA=a1,a2,a3 be the set of major, B=b11,b21,b12,b22,b13,b13,b23 is the set of minor vertices, respectively, and b is the central vertex. For k=2 and m=3, the LRN sets are given as follows.
The LRN sets of major vertices to central vertex are as follows:LR1=LRa1b=a1,a2,a3,b11,b12,b22,b23,bLR2=LRa2b=a1,a2,a3,b21,b12,b23,b13,bLR3=LRa3b=a1,a2,a3,b11,b21,b22,b23,b
The LRN sets of major to minor vertices are as follows:LR4=LRa1b11=a1,a3,b11,b21,b22,b13,b23,bLR5=LRa2b12=a1,a2,b11,b21,b12,b22,b23,bLR6=LRa3b13=a2,a3,b21,b12,b22,b13,b23,bLR7=LRa1b23=a1,a2,b11,b21,b12,b13,b23,bLR8=LRa2b21=a2,a3,b11,b21,b12,b22,b13,bLR9=LRa3b22=a1,a3,b11,b12,b22,b23,b13,b
The LRN sets of adjacent pair of minor vertices are as follows:LR10=LRb11b21=a1,a2,b11,b21,b12,b22,b13,b23LR11=LRb12b22=a2,a3,b11,b21,b12,b22,b13,b23LR12=LRb13b23=a1,a3,b11,b21,b12,b22,b13,b23
Since eachLRei=8, where 1≤i≤12, by Corollary 1, dimlfJ3,2=VJ3,2/α=5/4.Theorem 9.
LetJm,2, where m≅1mod2, be a generalized gear network. Then, for 1≤i≤m,(15)3m+13m−1≤dimlfJm,2≤3m8.Proof.
To prove the result, we have the following cases:Case 1: for,k=2 and m=5, the LRN sets are given as follows.The LRN sets of major vertices with central vertex are as follows:LR1=LRa1b=VJ2,5−b22,b15LR2=LRa2b=VJ2,5−b21,b22LR3=LRa3b=VJ2,5−b22,b23LR4=LRa4b=VJ2,5−b13,b24LR5=LRa4b=VJ2,5−b24,b25The LRN sets of adjacent pair of minor vertices are as follows:LR6=LRa1b11==VJ2,5−a2,b12LR7=LRa2b12==VJ2,5−a3,b13LR8=LRa3b13==VJ2,5−a4,b14LR9=LRa4b14==VJ2,5−a5,b15LR10=LRa5b15==VJ2,5−a1,b11LR11=LRa2b21==VJ2,5−a1,b25LR12=LRa3b22==VJ2,5−a2,b22LR13=LRa4b23==VJ2,5−a3,b22LR14=LRa5b24==VJ2,5−a4,b23LR15=LRa1b25==VJ2,5−a5,b24The LRN sets of adjacent pair of minor vertices are as follows:LR16=LRb11b21=a1,a2,b11,b21,b12,b22,b13,b23LR17=LRb12b22=a2,a3,b11,b21,b12,b22,b13,b23LR18=LRb13b23=a1,a3,b12,b22,b15,b25,b14,b24LR19=LRb14b24=a1,a3,b11,b21,b13,b23,b15,b25LR20=LRb15b25=a1,a3,b11,b21,b13,b23,b14,b24From above LRN sets,LRei=LRbiibi+1i=8, where 1≤i≤5 and LRei≤LRx∀x∈EJ5,2. Furthermore, ∪i=15LRei=15 and LRx∩∪i=15LRei≥LRei=8. Hence, a function γ:VJ5,2⟶0,1 defined by γv=1/8, if v∈∪i=15LRei and 0 otherwise is an upper LRF. Consequently, by Theorem 2, dimlfJ5,2≤15/8.From above LRN sets,LRei′=14, where 1≤i≤15 and LRei′≥LRx∀x∈EJ5,2. Therefore, there exists a lower LRF A function γ′:VJ5,2⟶0,1 defined by γ′v=1/12 is an LRF of maximum cardinality for each v∈VJ5,2. Hence, by Theorem 3, dimlfJ5,2≥8/7.Consequently,(16)87≤dimlfJ5,2≤158.Case 2: fork=2, m≥5, and m≅1mod2,LRei=LRbk/2ibk/2+1i, where 1≤k≤n. By Lemma 2, we have the following:(i)
LRei=LRbk/2ibk/2+1i=8(ii)
LRx∩∪i=1nLRei≥8∀x∈EJm,2Therefore, a functionΓ:VJm,2⟶0,1 is an upper LRF with minimum cardinality defined by(17)Γv=18,ifv∈∪i=1mLRei,0,ifv∈VJm,2−∪i=1mLRei.Consequently, by Theorem2, dimlfJm,2≤3m/8.SinceLRaibi=VJm,2−2 and LRaibi≥LRx, ∀x∈EJm,2. Hence, the function Γ′:VJm,2⟶0,1 is a lower LRF with maximum cardinality, defined by Γ′v=1/3m−1∀v∈VJm,2. Therefore, by Theorem 3, dimlfJm,2≥3m+1/3m−1.Consequently,(18)3m+13m−1≤dimlfJm,2≤3m8.Theorem 10.
LetJm,k be a generalized gear network with m≥4, k≥2, where k≅0mod2. Then,(19)mk+1+1mk+1≤dimlfJm,k≤mk+12k+4.Proof.
To prove the result, we have the following cases:Case 1: for,m = 4 and k = 2, the LRN sets are given as follows.Adjacent pair of minor vertices is as follows:LR1=LRb11b21=VJ4,2−a3,b13,b23,a4,bLR2=LRb12b22=VJ4,2−b14,a1,b,b24,b14LR3=LRb13b23=VJ4,2−a1,b11,b21,a2,bLR4=LRb14b24=VJ4,2−b12,a2,b,a3,b22Major with central vertex.LR5=LRa1b=VJ4,2−b21,b14LR6=LRa2b=VJ4,2−b11,b22LR7=LRa3b=VJ4,2−b23,b12LR8=LRa4b=VJ4,2−b24,b21Major with minor vertex.LR9=LRa1b11=VJ4,2−a2LR10=LRa1b24=VJ4,2−a4LR11=LRa2b21=VJ4,2−a1LR12=LRa2b12=VJ4,2−a3LR13=LRa4b23=VJ4,2−a3LR14=LRa4b23=VJ4,2−a3LR15=LRa4b14=VJ4,2−a1From above LRN sets,LRei=LRb1ib2i=8, where 1≤i≤4 and LRei≤LRx∀x∈EJ4,2. Furthermore, ∪i=14LRei=12 and LRx∩∪i=14LRei≥LRei=8. Hence, a function γ:VJ4,2⟶0,1 defined by γv=1/8 if v∈∪i=14LRei and 0 otherwise is a lower LRF with minimum cardinality. Consequently, by Theorem 2, dimlfJ4,2≤3/2.From above LRN sets,LRei′=12, where 1≤i≤6 and LRei′≥LRx∀x∈EJ4,2. Therefore, a function γ′:VJ4,2⟶0,1 defined by γ′v=1/12 is a lower LRF with maximum cardinality for each v∈VJ4,2.(20)1312≤dimlfJ4,2≤32.Case 2: fork = 4 and m = 3, the possible LRN sets are as follows.Central to major vertex.LR1a1b=VJ3,4−b31,b23LR2a2b=VJ3,4−b32,b21LR3a3b=VJ3,4−b33,b23Major with minor vertex.LR4a1b11=VJ3,4−b41LR5a2b12=VJ3,4−b42LR6a3b13=VJ3,4−b43LR7a1b43=VJ3,4−b13LR8a2b41=VJ3,4−b11LR9a3b42=VJ3,4−b12Adjacent pair of internal vertices.LR10b11b21=VJ3,4−a2,b12,b22LR11b31b41=VJ3,4−a1,b33,b43LR12b12b22=VJ3,4−a3,b13,b23LR13b32b42=VJ3,4−a2,b41,b31LR14b13b23=VJ3,4−a1,b11,b21LR15b23b33=VJ3,4−a2,b12,b41LR16b33b43=VJ3,4−a3,b42,b32LRN sets with minimum cardinality are as follows:LR17b21b31=VJ3,4−b,b42,b13,a3LR18b22b32=VJ3,4−b,b43,b11,a1LR19b23b33=VJ3,4−b,b12,b41,a2From above LRN sets,LRei=LRb2ib3i=12, where 1≤i≤3 and LRei≤LRx∀x∈EJ3,4. Moreover, ∪i=13LRei=VJ3,4−1, and this implies ∪i=13LRei=15 and LRx∩∪i=112LRei≥LRei=12. Hence, a function Γ:VJ3,4⟶0,1 is defined by Γv=1/12 which is an upper LRF with minimum cardinality for all v∈VJ3,4 and 0 otherwise. Therefore, dimlfJ3,4≤5/4.From above LRN setsLRei=6, where 1≤i≤6 and LRei≥LRx, ∀x∈EJ3,4. Therefore, a function Γ′:VJ3,4⟶0,1 is defined by Γ′v=1/6∀v∈VJ3,4; hence, by Theorem 3, dimlfJ3,4≥4/3. Consequently,(21)43≤dimlfJ3,4≤85.Case 3: fork≅0mod2 and m≥4, the possible LRN sets are LRbk/2ibk/2+1i, where 1≤k≤n. By Lemma 2 and 3, we have(i)
LRei=LRbk/2ibk/2+1i=2k+4 and LRei≤LRx∀x∈EJm,k(ii)
LRx∩∪i=1mLRei≥LRei∀x∈EJm,kTherefore,Γ:VJm,k⟶0,1 is an upper LRF defined by(22)Γv=12k+4,ifv∈⋃i=1mLRei,0,ifv∈VJm,k−∪i=1mLRei.Consequently, by Theorem2, dimlfJm,k≤mk+1/2k+4.Case 4: sinceLRaibi=VJm,k−1 and LRaibi≥LRx, ∀x∈EJm,k. Therefore, Γ′:VJm,k⟶0,1 defined by Γ′v=1/mk+1,∀v∈VJm,k is lower LRF. Hence, by Theorem 3, dimlfJm,k≥mk+1+1/mk+1.Consequently,(23)mk+1+1mk+1≤dimlfJm,k≤mk+12k+4.


## 4.3. Fork≅1mod2 and m≥3

Theorem 11.
Fork≥3, kk≅1mod2 and G≅Jm,k is a generalized gear network, and then dimlfG=1.Proof.
As there is no cycle of odd length inG≅Jm,k, where k≅1mod2. Therefore, G is bipartite network; hence, by Theorem 1, dimlfG=1.


## 5. Conclusion

In this article, we studied the LF-metric dimensions of generalized gear networks and established the sharp lower and upper bounds of LF-metric dimensions and computed the exact values in some cases as well.Exact values of LF-metric dimensions of some cases of the generalized gear networks are as follows:J3,0=2, J5,0=3/2 and J3,2=5/4Unboundedness is illustrated in Table3Now, this section is closed by raising the following open problem:Investigate the LF-metric dimension of the nonregular networks such as convex polytopes, Toeplitz, and Prism-related networks.Table 3
Unboundedness of LFMDs.
NetworkLFMDsLower bound of LF-metric dimension whenm⟶∞Upper bound of LF-metric dimension whenm⟶∞CommentJm,0m+1/m−1≤dimlfJm,0≤m/41∞UnboundedJm,23m+1/3m−1≤dimlfJm,2≤3m/81∞UnboundedJm,kmk+1+1/mk+1≤dimlfJm,k≤mk+1/2k+41∞Unbounded



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