# Improved Lower Bound of LFMD with Applications of Prism-Related Networks

Authors: Muhammad Javaid; Hassan Zafar; Q. Zhu; Abdulaziz Mohammed Alanazi
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

The different distance-based parameters are used to study the problems in various fields of computer science and chemistry such as pattern recognition, image processing, integer programming, navigation, drug discovery, and formation of different chemical compounds. In particular, distance among the nodes (vertices) of the networks plays a supreme role to study structural properties of networks such as connectivity, robustness, completeness, complexity, and clustering. Metric dimension is used to find the locations of machines with respect to minimum utilization of time, lesser number of the utilized nodes as places of the objects, and shortest distance among destinations. In this paper, lower bound of local fractional metric dimension for the connected networks is improved from unity and expressed in terms of ratio obtained by the cardinalities of the under-study network and the local resolving neighbourhood with maximum order for some edges of network. In the same context, the LFMDs of prism-related networks such as circular diagonal ladder, antiprism, triangular winged prism, and sun flower networks are computed with the help of obtained criteria. At the end, the bounded- and unboundedness of the obtained results is also shown numerically.


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## Body

## 1. Introduction

For a connected networkG, Salter introduced the concept of resolving (locating) set with the cardinality of minimum resolving set which is called the location number of G [1]. Harary and Melter introduced the concept of metric dimension for the connected networks [2]. The concept of metric independence number miG of a graph G is introduced by Currie and Oellermann [3]. The metric dimension has been applied to solve the problems involving percolation in hierarchical lattice [4], coin weighting, and robot navigation [5]. It is also applied in subject of chemistry to find the structures of chemical compounds having similar characteristics in functional groups. These functional groups play a vital role in chemical and pharmaceutical industries to predict the various chemical properties of the molecular compounds that are used in the drug discovery [6]. Metric dimension of graph was formulated as integer programming problem by Charterand et al. [6]. Fehar et al. studied the metric dimension of Cayley digraphs [7]. For further studies of metric dimension of convex polytopes, Caylay and Toeplitz networks, see [8–11].Currie and Oellermann defined the concept of fractional metric dimension (FMD) as an optimal solution of the linear relaxation of the integer programming problem (IPP) [3]. Later, Faher et al. presented the identical calculation of IPP with the help of FMD [7]. Arguman and Matthew introduced many different properties of FMD for connected networks with respect to their order [12]. FMDs of hierarchical product of graphs were computed by Feng and Wang [13]. Liu et al. [14] computed the FMD of generalized Jahangir graph. The concept of local fractional metric dimension (LFMD) is introduced by Aisyah et al. [15]. They also computed it for the connected networks which are obtained by the operation of the corona product. The results for the LFMDs of some cycle-related networks and rotationally symmetric and planar networks can be found in [16, 17]. Javaid et al. (2020) computed the sharp bounds of LFMD of connected networks and illustrated the obtained results with the help of wheel-related networks. They also compared the bounded- and unboundedness of the obtained results [18].In this paper, lower bound of LFMD for connected networks is improved from unity and expressed in terms of ratio obtained by the cardinalities of the under-study network and the local resolving neighbourhood with maximum order for some edges of network. In the outcome of the obtained result, the LFMDs of prism-related networks as exact values and sharp bounds are computed. The rest of the article is organised as follows: Section2 consists the preliminaries, Section 3 consists of main results of LFMD of connected networks, Section 4 deals with the local resolving neighbourhoods of prism-related networks, Section 5 presents LFMD of prism-related networks and Section 6 consists of conclusion and comparison among the main results.


## 2. Preliminaries

LetG=VG,EG be a network with VG and EG as set of vertices and edges, respectively. A walk is defined as a sequence of alternating vertices and edges. A walk in which the vertices are all distinct is a path between vertices a and b and a closed path is called a cycle. For any two vertices a and b of G, the distance da,b is the length of shortest path a∼b in G. A pair of vertices a and b in a network is a connected pair if there is a path between them and the network is a connected network. For a connected network G and e=ab∈EG, a vertex x∈VG distinguishes two vertices a and b if dx,a=dx,b is known as symmetric vertex. Moreover, x resolves the edge e in G if dx,a≠dx,b. For D=a1,a2,a3,…,an⊆VG and x∈VG, the k-tuple metric form of D in terms of x is given by rx|D=dx,a1,dx,a2,dx,d3,…,dx,an. The set D becomes resolving set having n elements of graph G if each pair of vertices in G bears a distinct metric form with respect to D. The resolving set with least number of vertices is referred as metric basis for G and cardinality of such resolving set is called metric dimension of G defined by(1)dimG=minD:Dis resolving set ofG.For an edgeab∈EG, the local resolving neighbourhood LRN is defined as LRab=x∈VG:dx,a≠dx,b, where x∈VG. A function is called an upper local resolving function (ULRF) if f:VG⟶0,1 and fLRab≥1 for each LRab of G, where fLRab=∑x∈LRabfx. On the other hand, a function is called lower local resolving function (LLRF) if Ψ:VG⟶0,1 and ΨLRab≤1 for each LRab of G, where ΨLRab=∑x∈LRabΨx. Then, LFMD is defined as dimlfG=τ, where τ is min{f:f which is the upper local minimal resolving function of G} or max{g:g is the lower local maximal resolving function of G}.For1≤i≤n, now we present some prism-related networks. The circular diagonal ladder CDLn is obtained from prism network Dn of order 2n and size 5n by adding some double crossing edges aibi+1 and ai+1bi, as shown in Figure 1. The antiprism network An of order 2n and size 4n is obtained by prism network Dn by adding some crossing edges aibi+1, see Figure 2. The sun flower network SFn of order 2n and size 3n, we mean a network, is isomorphic to the network obtained from An by deleting edges bibi+1, see Figure 3 [19].Figure 1
Circular diagonal ladderCDLn.Figure 2
AntiprismAn.Figure 3
Sun flower networkSFn.Theorem 1.
(see [18]). Let G=VG,EG be a connected network. Let LRe be a local resolving neighbourhood for the edge e of G. If LRe∩Z≥α,∀e∈EG, then(2)1≤dimlfG≤Zα,where Z=∪LRe:LRe=α, α=minLRe:e∈EG, and 2≤α≤VG.Proposition 1.
(see [18]). Let G=VG,EG be a connected network. For each e∈EG, if LRe∩Z≥2, then dimlf=Z/2, where Z=∪LRe:LRe=2, and LRe is a LRN set of e∈EG.


## 3. Main Results

Main results of LFMD are as follows.Proposition 2.
LetG=VG,EG be a connected network and LRe be the local resolving neighbourhood set of the edge e of G. For β=maxLRe:e=EG, if Y=∪LRe:LRe=β⊆VG, then LRe∩Y≤β for each local resolving neighbourhood LRe of G.Theorem 2.
LetG=VG,EG be a connected network and LRe be the local resolving neighbourhood set. Then,(3)VGβ≤dimlfG,where β=maxLRe:e∈EG and 2≤β≤VG.Proof.
DefineΦ:VG⟶0,1 as ϕv=1/β for v∈VG. By Proposition 2, for e∈EG, we have(4)ϕLRe=∑x∈LReϕx=∑x∈LRe∩VG1β=LRe∩VG1β≤1.
This shows thatϕ is a lower local resolving function (LLRF). To show that ϕ is maximal, suppose on contrary, there exists another LLRF Ψ such that Ψx≥Φx, where Ψx≠Φx, for at least one x∈VG. ∀x∈LRe such that LRe=β, we have(5)ΨLRe=∑x∈LReΨx>∑x∈LReϕx=1.
Thus,ΨLRe>1. This shows that Ψ is not LLRF and consequently ϕ is maximal LLRF. Let ϕ′ be another maximal LLRF of G. Then,(6)ϕ′=∑x∈VGϕ′x.
Now, we consider three cases (i)ϕ′x>1/β for each x∈VG, (ii) ϕ′x≤1/β for each x∈VG, and (iii) ϕ′x>1/β for some x∈VG.Case 1: ifϕ′x>1/β, for each x∈VG. For LRe⊆Y such that LRe=β, we have ϕ′LRe>1. This shows ϕ′ is not LLRF. Thus, this case does not hold.Case 2: letϕ′x≤1/β∀x∈VG. Then,(7)ϕ′=∑x∈VGϕ′x≤VGβ=ϕ.Consequently,(8)dimlfG=VGβ.Case 3: assume thatϕ′x>1/β for some x∈VG. Suppose that S=t∈VG:ϕ′t>1/β and Y=∪LRe:LRe=β. We note that S∩Y=∅; otherwise, for LRe=β, ϕ′LRx>1 which implies that ϕ′ is not a LLRF. Consider(9)ϕ′=∑x∈VGϕ′x=∑x∈Yϕ′x+∑x∈VG−Yϕ′x.
Since∑x∈VG−Yϕ′x≥∑x∈VG−Yϕx, therefore,(10)ϕ′=∑x∈VGϕ′x=∑x∈Yϕ′x+∑x∈VG−Yϕ′x≥∑x∈Yϕx+∑x∈VG−Yϕx=VGβ=ϕ.
Consequently,(11)dimlfG=ϕ′≥ϕ=VGβ.
Thus, from all the cases,(12)VGβ≤dimlfG,which completes the proof.
Now, we present the following two corollaries as the direct consequences of the above result.Corollary 1.
LetG=VG,EG be a connected network, LRe be LRN of e∈EG, β=maxLRe:e∈EG, α=minLRe:e∈EG, and X=∪LRe:LRe=α. If α=β and X=VG, then(13)dimlfG=VGβ.Proof.
Sinceα=β and X=VG, therefore, by Theorem A, dimlfG=VG/β. Also, by Theorem 1, dimlfG=VG/β. Consequently,(14)dimlfG=VGβ.Corollary 2.
LetG=VG,EG be a connected network, LRe be the LRN of e∈EG, β=maxLRe:e∈EG, α=minLRe:e∈EG, and X=∪LRe:LRe=α. If α=X and β=VG, then dimlfG=1.Proof.
Sinceβ=VG, then by Theorem 2, 1≤dimlfG. Also, as α=X, therefore, by Theorem 1, dimlfG≤1. Consequently, dimlfG=1.Remark 1.
Corollary2 strengthens the result proved in [18].


## 4. LRNs of the Prism-Related Networks

In this section, the local resolving neighbourhoods of prism-related networks are classified.Lemma 1.
Let CDLn with n≥4 be a circular diagonal ladder network, where n≅0mod2 and VCDLn=2n, where 1≤i≤n. We have(a)
LRei=LRaibi=2 and ∪i=1nLRei=2n,(b)
LRei=LRbibi+1 and LRbibi+1∩∪i=1nLRei≥LRei,(c)
LRei=LRaiai+1 and LRaiai+1∩∪i=1nLRei≥LRei,(d)
LRei=LRaibi+1 and LRaibi+1∩∪i=1nLRei≥LRei,(e)
LRei=LRbiai+1 and LRbiai+1∩∪i=1nLRei≥LRei,(f)
LRei=LRaibi−1 and LRaibi−1∩∪i=1nLRei≥LRei.Proof.
Assume thatai and bi are inner and outer vertices, respectively, for 1≤i≤n and n+1≅1modn. We have the following:(a)
ConsiderLRaibi=ai,bi with LRei=LRaibi=2. Moreover, ∪i=1nLRei=VCDLn=2n.(b)
LRbibi+1=VCDLn−ai,ai+1 with LRbi+1bi=2n−2>2=LRei and LRbibi+1∩∪i=1nLRei=2n−2>2=LRei.(c)
LRaiai+1=VCDLn−bi,bi+1 with LRaiai+1=2n−2>2=LRei and LRaiai+1∩∪i=1nLRei=2n−2>2=LRei.(d)
LRaibi+1=VCDLn−ai+1,bi with LRaibi+1=2n−2>2=LRei and LRaibi+1∩∪i=1nLRei=2n−2>2=LRei.(e)
LRbiai+1=VCDLn−bi+1,ai with LRbiai+1=2n−2>2=LRei and LRbiai+1∩∪i=1nLRei=2n−2>2=LRei.(f)
LRbiai−1=VCDLn−ai−1,bi+1 with LRbiai+1=2n−2>2=LRei and LRbiai−1∩∪i=1nLRei=2n−2>2=LRei.Lemma 2.
LetCDLn with n≥5 be a circular diagonal ladder network, where n≅1mod2 and VCDLn=2n, where 1≤i≤n. We have(a)
LRei=LRaibi=2 and ∪i=1nLRei=2n,(b)
LRei<LRbibi+1 and LRbibi+1∩∪i=1nLRei≥LRei,(c)
LRei<LRaiai+1 and LRaiai+1∩∪i=1nLRei≥LRei,(d)
LRei<LRaibi+1 and LRaibi+1∩∪i=1nLRei≥LRei,(e)
LRei<LRbiai+1 and LRbiai+1∩∪i=1nLRei≥LRei,(f)
LRei<LRaibi−1 and LRaibi−1∩∪i=1nLRei≥LRei.Proof.
Assume thatai and bi are inner and outer vertices, respectively, for 1≤i≤n and n+1≅1modn, 3n+1/2≅1modn. We have the following:(a)
ConsiderLRaibi=ai,bi with LRei=LRaibi=2. Moreover, ∪i=1nLRei=VCDLn=2n.(b)
AsLRbibi+1=VCDLn−bn+2i+1/2,an+2i+1/2,ai,ai+1 with LRbibi+1=2n−4>2=LRei and LRbibi+1∩∪i=1nLRei=2n−4>2=LRei.(c)
AsLRaiai+1=VCDLn−an+2i+1/2,bn+2i+1/2,bi,bi+1 with LRaiai+1=2n−4>2=LRei and LRaiai+1∩∪i=1nLRei=2n−4>2=LRei.(d)
AsLRaibi+1=VCDLn−an+2i+1/2,bn+2i+1/2,ai+1,bi with LRaibi+1=2n−4>2=LRei and LRaibi+1∩∪i=1nLRei=2n−4>2=LRei.(e)
AsLRbiai+1=VCDLn−ai,bi+1,an+2i+1/2,bn+2i+1/2 with LRbiai+1=2n−4>2=LRei and LRbiai+1∩∪i=1nLRei=2n−4>2=LRei.(f)
AsLRaibi−1=VCDLn−ai−1,bi−1,an+2i+1/2,bn+2i+1/2 with LRbiai+1=2n−4>2=LRei and LRbiai+1∩∪i=1nLRei=2n−4>2=LRei.Lemma 3.
LetAn with n≥4 be an antiprism network, where n≅0mod2 and VAn=2n and 1≤i≤n. We have(a)
LRei=LRaibi and LRaibi+1=n and ∪i=1nLRei=2n,(b)
LRei=LRbibi+1 and LRbibi+1∩∪i=1nLRei≥LRei,(c)
LRei=LRaiai+1 and LRaiai+1∩∪i=1nLRei≥LRei.Proof.
Assume thatai and bi are inner and outer vertices, respectively, for 1≤i≤n, where n+1≅1mod2, 3n+2/2≅1modn, and 3n/2≅1modn. We have(a)
LRei=LRaibi=bi,bn+2i+2/2,bn+2i+2/2+1,bn+2i+2/2+2,…,bn+i−1,ai,ai+1,ai+2,…,an+2i−2/2 and LRaibi+1=bi+1,bi+2,bi+3,…,b2i+n/2,ai,an+2i+2/2,an+2i+2/2+1,an+2i+2/2+2,…,an+i−1 with LRei=n. Moreover, ∪i=1nLRei=VAn=2n,(b)
AsLRbibi+1=VAn−ai,an+2i/2 with LRbibi+1=2n−2>n=LRei and LRbibi+1∩∪i=1nLRei=2n−2>n=LRei,(c)
AsLRaiai+1=VAn−bi+2,bn+2i+2/2 with LRaiai+1=2n−2>n=LRei and LRaiai+1∩∪i=1nLRei=2n−2>n=LRei.Lemma 4.
LetAn with n≥3 be an antiprism network, where VAn=2n with n≅1mod2. For 1≤i≤n, we have(a)
LRei=LRaibi and LRaibi+1=n+1 and ∪i=1nLRei=2n,(b)
LRei=LRbibi+1 and LRbibi+1∩∪i=1nLRei≥LRei,(c)
LRei=LRaiai+1 and LRaiai+1∩∪i=1nLRei≥LRei.Proof.
Assume thatai and bi are inner and outer vertices, respectively, for 1≤i≤n, where n+1≅1mod2, 3n+1/2≅1modn, 3n−1/2≅1modn, and 2n−1≅1mod2. We have(a)
LRaibi=ai,ai+1,ai+2,…,an+2i−1/2,bi,bn+2i+1/2,bn+2i+3/2,…,bn+i−1 and LRaibi+1=bi+1,bi+2,…,bn+2i+1/2,ai,an+2i+1/2,an+2i+1/2,…,an+i−1 with LRei=n+1. Moreover, ∪i=1nLRei=VAn=2n.(b)
LRaiai+1 = VAn−an+2i+1/2,bi+1 and LRaiai+1=2n−2>n+1=LRei.(c)
LRbibi+1 = VAn−ai,bn+2i+1/2 and LRbibi+1=2n−2>n+1=LRei.Lemma 5.
LetSFn with n≥3 be a sun flower network, where VSFn=2n and n≅1mod2. For 1≤i≤n, we have(a)
LRei=LRaibi and LRbiai+1=n+1 and ∪i=1nLRei=2n,(b)
LRei=LRaiai+1 and LRaiai+1∩∪i=1nLRei≥LRei.Proof.
Assume thatai and bi are inner and outer vertices, respectively, for 1≤i≤n, where n+1≅1modn, n+2≅1modn, 3n+1/2≅1modn, 3n−1/2≅1modn, and 2n−1≅1modn. Now, we have(a)
LRaibi=ai,an+2i+1/2,an+2i+1/2+1,…,an+i−1,bi,bn+2i+1/2,bn+2i+1/2+1,…,bn+i−1 and LRbiai+1=bi,bi+1,bi+2,….,bn+2i+1/2,ai+1,ai+2,ai+3,…,an+i−1 with LRei=n+1. Moreover, ∪i=1nLRei=VSFn=2n.(b)
LRaiai+1=VSFn−an+2i+1/2,bi and LRaiai+1=2n−2 with LRaiai+1=2n−2>n+1=LRei and LRaiai+1∩∪i=1nLRei=2n−2>n+1=LRei.Lemma 6.
LetSFn with n≥4 be a sun flower network, where VSFn=2n and n≅0mod2. For 1≤i≤n, we have(a)
LRei=LRaibi and LRbiai+1=n+1 and ∪i=1nLRei=2n,(b)
LRei=LRaiai+1 and LRaiai+1∩∪i=1nLRei≥LRei.Proof.
Assume thatai and bi are inner and outer vertices, respectively, for 1≤i≤n, where n+1≅1modn, n+2≅1modn, 3n+1/2≅1modn, 3n/2≅1modn, and 2n−1≅1modn. We have(a)
2n−1≅1modn and LRbiai+1=bi,bi+1,bi+2,…,bn+2i/2,ai+1,ai+2,ai+3,…,an+2i/2 with LRei=n+1. Moreover, ∪i=1nLRei=VSFn=2n.(b)
LRaiai+1=VSFn−bn+2i/2,bi and LRaiai+1=2n−2 with LRaiai+1=2n−2>n+1=LRei and LRaiai+1∩∪i=1nLRei=2n−2>n+1=LRei.


## 5. LFMD of Prism-Related Networks

The LFMD of prism-related networks is computed as follows.Theorem 3.
LetCDLn with n≥4 be a circular diagonal ladder network and VCDLn=2n. Then, dimlfCDLn=n.Proof.
Case 1: Forn=4, LRNs are as follows:LR1=LRb1b2=VCDL4−a1,a2,LR2=LRb2b3=VCDL4−a2,a3,LR3=LRb3b4=VCDL4−a3,a4,LR1=LRb1b2=VCDL4−a1,a2,LR5=LRa1a2=VCDL4−b1,b2,LR6=LRa2a3=VCDL4−b2,b3,LR7=LRa3a4=VCDL4−b3,b4,LR8=LRa4a1=VCDL4−b4,b1,LR9=LRa1b2=VCDL4−a2,b1,LR10=LRa2b3=VCDL4−a3,b2,LR11=LRa3b4=VCDL4−a4,b3,LR12=LRa4b1=VCDL4−a1,b4,LR13=LRa2b1=VCDL4−a1,b2,LR14=LRa3b2=VCDL4−a2,b3,LR15=LRa4b3=VCDL4−a3,b4,LR16=LRa1b4=VCDL4−a4,b1,LR17=LRb1a2=VCDL4−b2,a1,LR18=LRb2a3=VCDL4−b3,a2,LR19=LRb3a4=VCDL4−b4,a3,LR20=LRb4a1=VCDL4−b1,a4,LR21=LRa1b1=a1,b1,LR22=LRa2b2=a2,b2,LR23=LRa3b3=a3,b3,LR24=LRa4b4=a4,b4.
As, for1≤i≤4, LRei is 2 such that LRei<LRes, where 1≤s≤20. Moreover, ∪i=14LRei=8 and LRs∩∪i=14LRei=LRei=2. Define ϕ:VCDL4⟶0,1 as ϕx=1/2 is a LRF with minimum cardinality for each x∈VCDL4. Consequently, by Preposition 1, dimlfCDL4=VCDL4/2=4.Case 2: Forn=5, LRNs are as follows:LR1=LRb1b2=VCDL5−b4,a4,a1,a2,LR2=LRb2b3=VCDL5−b5,a5,a2,a3,LR3=LRb3b4=VCDL5−b6,a6,a3,a4,LR4=LRb4b5=VCDL5−b2,a2,a4,a5,LR5=LRb5b1=VCDL5−b3,a1,a3,a5,LR6=LRa1a2=VCDL5−b4,a4,b1,b2,LR7=LRa2a3=VCDL5−b5,a5,b2,b3,LR8=LRa3a4=VCDL5−b1,a1,b3,b4,LR9=LRa4a5=VCDL5−b2,a2,b4,b5,LR10=LRa5a1=VCDL5−b1,a3,b3,b5,LR11=LRa1b2=VCDL5−b4,a4,b1,a2,LR12=LRa2b3=VCDL5−b5,a5,b2,a3,LR12=LRa3b4=VCDL5−b1,a4,b3,a4,LR11=LRa1b2=VCDL5−b4,a4,b1,a2,LR15=LRa5b1=VCDL5−b3,a3,b5,a6,LR16=LRa1b5=a1,a2,a4,b2,b4,b4,LR17=LRa2b1=a2,a3,a5,b1,b3,b5,LR18=LRa3b2=a2,a3,a4,b1,b2,b4,LR19=LRa4b3=a2,a4,a5,b2,b3,b5,LR20=LRa5b4=a1,a3,a5,b1,b3,b4,LR21=LRb1a2=a2,a3,a5,b1,b3,b5,LR22=LRb2a3=a1,a3,a4,b1,b2,b4,LR23=LRb3a4=a2,a4,a5,b2,b3,b5,LR24=LRb4a5=a1,a3,a5,b1,b3,b4,LR25=LRb5a1=a1,a2,a4,b2,b4,b5,LR26=LRa1b1=a1,b1,LR27=LRa2b2=a2,b2,LR28=LRa3b3=a3,b3,LR29=LRa4b4=a4,b4,LR30=LRa5b5=a5,b5.
As, for1≤i≤5, LRNei=2 such that LRei<LRes, where 1≤s≤25. Moreover, ∪i=15LRei=VCDL5; this implies ∪i=15LRei=10 and |LRs∩∪i=15LRei|≥|LRei|=2. Define ϕ:VCDL5⟶0,1 such that ϕx=1/2 is the LRF with minimum cardinality for each x∈VCDL5. Consequently, by Proposition 1, dimlfCDL5=VCDL5/2=5.Case 3: Forn≥6 and 1≤i≤n by Lemma 1, LRei=2=α and R∩∪i=1nLRei≥2, where R are all other LRNs. Define ϕ:VCDLn⟶0,1 such that ϕ is the LRF with minimum cardinality and ϕ<ϕ′, as ϕv=1/2∀v∈VCDLn. VCDLn=∪LRe:|LRe=2. Consequently, by Proposition 1, dimlfCDLn=VCDLn/2=2n/2=n.Theorem 4.
LetAn with n≥3 be an antiprism network, where n≅1mod2 and VAn=2n. Then, dimlfAn,(15)nn−1≤dimlfAn≤2nn+1.Proof.
Case 1: Forn=3, LRNs are as follows:LR1=LRa1a2=a1,a2,b1,b3,LR2=LRa2a3=a2,a3,b1,b2,LR3=LRa3a1=a3,a1,b2,b3,LR4=LRa1b1=a1,a2,b1,b3,LR5=LRa2b2=a2,a3,b1,b2,LR6=LRa3b3=a1,a3,b2,b3,LR7=LRa1b2=a1,a3,b1,b2,LR8=LRa2b3=a1,a2,b1,b3,LR9=LRa3b1=a2,a3,b1,b2.
As, for1≤i≤9, LRei=4. Moreover, ∪i=19LRei=VA3=8 such that ∪i=19LRei=6 and LRNs∩∪i=19LRei>LRei=4. Define ϕ:VA3⟶0,1 as ϕx=1/4 is the LRF with minimum cardinality. Consequently, by Corollary 1, dimlfA3=VA3/4=3/2.Case 2: Forn=5, LRNs are as follows:LR1=LRb1b2=VA5−a1,b4,LR2=LRb2b3=VA5−a3,b5,LR3=LRb3b4=VA5−a3,b1,LR4=LRb4b5=VA5−a4,b2,LR5=LRb5b1=VA5−a5,b3,LR6=LRa1a2=VA5−a4,b2,LR7=LRa2a3=VA5−a5,b3,LR8=LRa3a4=VA5−a1,b4,LR9=LRa4a5=VA5−a2,b5,LR10=LRa5a1=VA5−a3,b1,LR11=LRb1a1=b1,b4,b5,a1,a2,a3,LR12=LRb2a2=b2,b5,b1,a2,a3,a4,LR13=LRb3a3=b3,b1,b2,a3,a4,a5,LR14=LRb4a4=b4,b2,b1,a4,a5,a1,LR15=LRb5a5=b5,b2,b3,a5,a1,a2,LR16=LRa1b2=a1,a4,a5,b2,b3,b4,LR17=LRa2b3=a2,a5,a1,b3,b4,b5,LR18=LRa3b4=a3,a1,a2,b4,b5,b1,LR19=LRa4b5=a4,a2,a1,b5,b1,b2,LR20=LRa5b1=a5,a2,a3,b1,b2,b3.
As, for1≤i≤10, LRei=6 such that LRei<LRes and 1≤s≤10. ∪i=110LRei=VA5; this implies ∪i=110LRei=10 and LRN∩∪i=110LRei>LRei=6. Define ϕ:VA5⟶0,1 as ϕx=1/6 is the LRF with minimum cardinality for each x∈VA5. Consequently, by Theorem 1, dimlfA5≤VA5/6≤5/3.
As, for1≤i≤10, LRei=8, LRei>LRes, and 1≤s≤10. Moreover, ∪i=110LRei=VA5. Define ϕ′:VA5⟶0,1 as ϕ′x=1/8 is the LRF with minimum cardinality for each x∈VA5. Therefore, by Theorem 2, dimlfA5≥VA5/8≥5/4. Consequently,(16)54≤dimlfA5≤53.Case 3: Forn≥6 and 1≤i≤n by Lemma 4,LRei=n+1=α and R∩∪i=1nLRei≥n+1, where R are all other LRNs. There exists an ULRF ϕ:VAn⟶0,1, ϕ<ϕ′ which is defined as ϕv=1/α∀v∈VAn, as VAn=∪LRe:|LRe=α. Consequently, by Theorem 1, dimlfAn≤VAn/n+1≤2n/n+1.Case 4: ForLRbibi+1=2n−2≥LRei, moreover, ∪i=12nLRei=VAn. There exists a LLRF such that ϕ′:VAn⟶0,1 is defined as ϕ′x=1/2n−2 for each x∈VAn. Therefore, by Theorem 2, we have dimlfAn≥n/n−1. Consequently,(17)nn−1≤dimlfAn≤2nn+1.Theorem 5.
LetAn with n≥4 be an antiprism network, where n≅0mod2 and VAn=2n. Then,(18)nn−1≤dimlfA4≤2.Proof.
Case 1: Forn=4, LRNs are as follows:LR1=LRb1b2=VA4−a1,a3,LR2=LRb2b3=VA4−a2,a4,LR3=LRb3b4=VA4−a1,a3,LR4=LRb4b1=VA4−a2,a4,LR5=LRa1a2=VA4−b2,b4,LR6=LRa2a3=VA4−b3,b1,LR7=LRa3a4=VA4−b2,b4,LR8=LRa4a1=VA4−b1,b3,LR9=LRb1a1=b1,b4,a1,a2,LR10=LRb2a2=b1,b2,a2,a3,LR11=LRb3a3=b2,b3,a3,a4,LR12=LRb4a4=b3,b4,a1,a4,LR13=LRb2a1=b2,b3,a1,a4,LR14=LRb3a2=b3,b4,a1,a2,LR15=LRb4a3=b1,b4,a2,a3,LR16=LRb1a4=b1,b2,a3,a4.
As, for1≤i≤8, |LRei|=4 such that LRei<LRes, where 1≤s≤8. Moreover, ∪i=18LRei=VA4 and LRs∩∪i=18LRei>LRei=4. There exits an ULRF ϕ:VA4⟶0,1 defined as ϕx=1/4 which is the LRF with minimum cardinality ∀x∈VA4. Consequently, by Theorem 1, dimlfA4≤VA4/4≤2.
As, for1≤i≤8, LRei=6 such that LRei>LRes, where1≤s≤8. Moreover, ∪i=18LRei=VA4. There exists a LLRF sϕ′:VA4⟶0,1 which is defined as ϕ′x=1/6 for each x∈VA4. Therefore, by Theorem 1, we have dimlfA4≥VA4/64/3. Consequently,(19)43≤dimlfA4≤2.Case 2: Forn≥5, 1≤i≤n, by Lemma 3, LRei=n=α and R∩∪i=1nLRei≥n, where R are all other LRNs.There exists an ULRFϕ:VAn⟶0,1 which is defined as ϕv=1/α∀v∈VAn, as V(G) =∪LRe:|LRe=α. Consequently, by Theorem 1, dimlfAn≤VAn/n≤2n/n=2.Case 3: As,LRaiai+1=2n−2, which is greater or equal to the cardinalities of all other LRNs, moreover, ∪i=12nLRei=VAn. There exists a LLRF ϕ′:VAn⟶0,1 which is defined as ϕ′x=1/2n−2∀v∈VAn. Therefore, by Theorem 2, dimlfAn≥n/n−1. Consequently,(20)nn−1≤dimlfAn≤2.Theorem 6.
LetSFn with n≥3 be a sun flower network and VSFn=2n. Then,(21)nn−1≤dimlfSFn≤2nn+1.Proof.
Case 1: Forn=3, LRNs are as follows:LR1=LRa1a2=a1,a2,b2,b3,LR2=LRa2a3=a2,a3,b3,b1,LR3=LRa3a1=a3,a1,b1,b2,LR4=LRa1b1=a1,a3,b1,b3,LR5=LRa2b2=a1,a2,b1,b2,LR6=LRa3b3=a2,a3,b2,b3,LR7=LRa2b1=b1,b2,a2,a3,LR8=LRa3b2=a1,a3,b2,b3,LR9=LRa1b3=a1,a2,b1,b3.
As, for1≤i≤9, LRei=4 and LRNs∩∪i=19LRei=LRei=4. There exists an ULRF ϕ:VSF3⟶0,1 which is defined by ϕx=1/4, ∀v∈VSF3. Consequently, by Corollary 1, dimlfSF3=3/2.Case 2: Forn=4, LRNs are as follows:LR1=LRa1a2=a1,a2,a3,a4,b2,b4,LR2=LRa2a3=a1,a2,a3,a4,b1,b3,LR3=LRa3a4=a1,a2,a3,a4,b2,b4,LR4=LRa4a1=a1,a2,a3,a4,b1,b3,LR5=LRa1b1=a1,a4,b1,b2,b4,LR6=LRa2b2=a1,a2,b1,b2,b4,LR7=LRa3b3=a2,a3,b1,b2,b3,LR8=LRa4b4=a3,a4,b2,b3,b4,LR9=LRa2b1=a2,a3,b1,b2,b3,LR10=LRa3b2=a3,a4,b2,b3,b4,LR11=LRa4b3=a1,a4,b2,b3,b4,LR12=LRa1b4=a1,a2,b1,b2,b4.
As, for1≤i≤8, LRei=5, such that LRei<LRes, where 1≤s≤4. Moreover, ∪i=18LRei=VSF4. There exists ϕ:VSF4⟶0,1 defined as ϕx=1/5 which is the LRF ∀v∈VSF4, as VSF4=∪LRei:|LRe=5. Consequently, by Theorem 1, dimlfSF4≤8/5.
As, for1≤i≤4, LRei=6 such that LRei>LRes, where 1≤s≤8. There exists ϕ′:VSF4⟶0,1 defined as ϕv=1/6∀v∈VSF4, which is the LLRF with maximum cardinality, where ϕ′<ϕ″, as VSF4=∪LRe:|LRe=6. By Theorem 1, dimlfSF4≥4/3. Consequently,(22)43≤dimlfSF4≤85.Case 3: Asn≥5, 1≤i≤n, LRei=n+1=α, and R∩∪i=1nLRei≥n+1, where R are all other LRNs. There exists an ULRF ϕ:VSFn⟶0,1 defined as ϕv=1/α∀v∈VSFn, as V(G) =∪LRe:|LRe=α. Consequently, by Theorem 1, dimlfSFn≤VSFn/n+1≤2n/n+1.Case 4: AsLRaiai+1=2n−2 which is greater than the cardinalities of all other LRNs of SFn, there exists a LLRF ϕ′:VSFn⟶0,1 defined by ϕ′v=1/2n−2, ∀v∈VSFn, as V(G) =∪LRe:|LRe=2n. By Theorem 2, dimlfSFn≥n/n−1. Consequently,(23)nn−1≤dimlfSFn≤2nn+1.


## 6. Conclusion

In this article, we studied the various aspects of the LFMDs for the different connected networks including the existence of the exact values, lower and upper bonds and bounded- and unbounded behaviours. Mainly, the lower bond of LFMD for the arbitrary connected networks is improved from unity. As the applications of the main result, LFMDs of the prism-related networks are as follows:(i)
The exact value LFMD ofCDLn is n and of A3 is 3/2, where n≥4.(ii)
The LFMDs of antiprism and sun flower networks are computed in bounds, see Table1.Table 1
LFMD of networks.
NetworksLFMDsCommentAntiprism network [Anforn≅0mod2]n/n−1≤dimlfAn≤2.BoundedAntiprism networkAn, where n≅1mod2n/n−1≤dimlfAn≤2n/n+1.BoundedSun flower networkSFn.n/n−1≤dimlfSFn≤2n/n+1.BoundedMoreover, we note that, forn⟶∞, the computed bounds are convergent which show their boundedness. Now, we close our discussion by proposing the following open problem: open problem. Characterize the connected networks whose LFMDs is |VG|/a, where a<VG is some integral value.



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