# Classification of Upper Bound Sequences of Local Fractional Metric Dimension of Rotationally Symmetric Hexagonal Planar Networks

Authors: Shahbaz Ali; Muhammad Khalid Mahmood; Fairouz Tchier; F. M. O. Tawfiq
Journal: Journal of Mathematics
 (2021)

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## Abstract

The term metric or distance of a graph plays a vital role in the study to check the structural properties of the networks such as complexity, modularity, centrality, accessibility, connectivity, robustness, clustering, and vulnerability. In particular, various metrics or distance-based dimensions of different kinds of networks are used to resolve the problems in different strata such as in security to find a suitable place for fixing sensors for security purposes. In the field of computer science, metric dimensions are most useful in aspects such as image processing, navigation, pattern recognition, and integer programming problem. Also, metric dimensions play a vital role in the field of chemical engineering, for example, the problem of drug discovery and the formation of different chemical compounds are resolved by means of some suitable metric dimension algorithm. In this paper, we take rotationally symmetric and hexagonal planar networks with all possible faces. We find the sequences of the local fractional metric dimensions of proposed rotationally symmetric and planar networks. Also, we discuss the boundedness of sequences of local fractional metric dimensions. Moreover, we summarize the sequences of local fractional metric dimension by means of their graphs.


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## Body

## 1. Introduction

The subject graph theory has a vital role in computer science, medical, chemical, networking, image processing, navigation, pattern recognition, drug discovery, chemical formation, sensors fixing, circuit designing, and many other fields. In graph theory, the term distance or metric plays a crucial role in solving many problems, in which we want to set the minimum distance between different objects to fixed objects, for example, if we want to fix a sensor for recognition of object, then we will fix the sensor to minimum distance from each object for better result. Moreover, when we check the structural properties of the networks such as complexity, modularity, centrality, accessibility, connectivity, robustness, clustering, and vulnerability, then metric or distance is very useful.A graph or network is linked between two sets: one of them is a set of nodes and the other one is a set of edges. The set of vertices is denoted byVG, and EG represents a set of edges which is a subset of VG×VG. The distance or metric between two vertices x and y is defined as the number of minimum edges between x and y; it is denoted as dx,y. For more details and insights into the nexus of graph theory, we refer the readers to [1–3].A subsetS=s1,s2,…,st of VG is known as resolving set if a∈VG, with k− tuple metric form represented as rx|S=dx,s1,dx,s1,…,dx,sk. Then, S will be a resolving set with k elements if rx|S≠ry|S for each pair of distinct vertices x and y. The minimum number set S is known as the basis of G. The cardinality of the set S is called the metric dimension of G; it is denoted by βG in [4, 5].Slatter investigated the term resolving set of connected graphs by using the notion of locating set [6–8]. Harary and Melter [9] investigated individually the aforementioned terminologies and proclaimed them as the metric dimension of a graph. A family of graphs is bounded (constant) with respect to metric dimension if each graph in this family has bounded (constant) metric dimension, respectively. The metric dimension of path and cycle has been investigated in [10]. In [4], it is shown that Peterson, anti-prism, and circulates networks have constant metric dimension. The metric dimension of wheel and Jahangir graphs was found in [11]. In [12], the authors have shown that the generalized Peterson network Pn,3 has a bounded metric dimension. Liu et al. discussed many aspects of graph theory such as indices, metric dimension domination number, and so on in [13–21].In 2000, Chartrand et al. found the solution of integer programming problem (IPP) by means of metric dimension [4]. For a more accurate solution of integer programming problem (IPP), Currie and Oellermann introduced the concept of fractional metric dimension in [22]. Fehr et al. found an optimal solution of a certain linear programming relaxation problem by means of fractional metric dimension [23]. Liu et al. discussed many interesting results on the fractional metric dimension (FMD) in [20]. Many results of fractional metric dimension on several networks such as networks obtained from Cartesian, hierarchical, corona, lexicographic, and comb products of connected networks have been discussed in [4, 20, 24, 25]. The fractional metric dimension of the generalized Jahangir network was computed by Liu et al. in [26]. The local fractional metric dimensions of planar networks were computed in [27]. In 2019, Aisyah et al. introduced the concept of local fractional metric dimension (LFMD) and computed it for the corona product of two graphs [28]. Many researchers discussed this area in [5, 6, 8, 29–31]. A new family of graphs based on totient, super totient, and hyper totient numbers was proposed in [32–36].In this paper, we choose the family of rotationally symmetric hexagonal planar networks with all possible faces and then find their sequences of local fractional metric dimension (LFMD). Moreover, we summarize sequences of local fractional metric dimensions by means of discrete graphs. This paper proceeds in the following sequence. Section2 gives the preliminaries. In Section 3, we define the vertex and edge sets of rotationally symmetric hexagonal planar networks and their subsections; we find the sequences of these networks by introducing lemmas and theorems. In Section 4, we give the boundedness table of sequences of local fractional metric dimension (LFMD) and also plot their discrete graphs.


## 2. Preliminaries

LetG be a connected graph. A pair of vertices s,t⊆VG is a resolved set by a vertex u if ds,u≠dt,u. The set of resolving neighborhood of pair s,t is defined as ℛ=u∈VG|ds,u≠dt,u. Let G be a connected graph of order n. A function ϑ:VG⟶0,1 is termed as resolving function of G if ϑℛs,t≥1,∀s,t∈VG, where(1)ϑℛs,t=∑u∈ℛs,tϑu.A resolving functionϑ′ of G is called a minimal resolving function if there exists u∈VG which is not a resolving function of G and if any other function ϑ:VG⟶0,1 with ϑ≤ϑ′,ϑu≠ϑ′u. The fractional metric dimension (FMD) of the graph G is defined as dimfG=minϑ′:ϑ′is the minimal resolvingfunction ofG, where(2)ϑ′=∑u∈VGϑ′u.The resolving function will be a local resoling function ifϑ′ℛs,t≥1 and the fractional metric dimension becomes a local fractional metric dimension (LFMD) if we assume the pair of adjacent vertices only; it is denoted by DimlfG [20, 28]. The LFMD of given graph in Figure 1 is DimlfG≤12/7.Figure 1
Hexagonal planar graphG with DimlfG≤12/7.The local resolving neighborhoods are(3)ℛx7x13=VG−x2,x5,x8,x12,ℛx1x7=VG−x10,x14,x15,x16,x20,x21,ℛx9x15=VG−x1,x4,x8,x10,ℛx7x6=VG−x10,x16,x17,x18,x22,x23,ℛx8x14=VG−x3,x6,x7,x9,ℛx3x9=VG−x12,x16,x17,x18,x22,x23,ℛx12x18=VG−x1,x4,x7,x11,ℛx2x9=VG−x12,x13,x14,x18,x19,x24,ℛx11x17=VG−x2,x6,x10,x12,ℛx2x8=VG−x11,x15,x16,x17,x21,x22,ℛx10x16=VG−x2,x5,x9,x11,ℛx1x8=VG−x11,x13,x17,x18,x23,x24,ℛx13x18=VG−x3,x6,x21,x24,ℛx6x12=VG−x9,x13,x14,x15,x19,x20,ℛx18x17=VG−x2,x5,x20,x23,ℛx5x12=VG−x9,x15,x16,x17,x21,x22,ℛx16x17=VG−x1,x4,x9,x22,ℛx5x11=VG−x8,x13,x14,x18,x19,x24,ℛx15x16=VG−x3,x6,x21,x24,ℛx4x11=VG−x8,x14,x15,x16,x20,x21,ℛx14x15=VG−x2,x5,x20,x23,ℛx4x10=VG−x7,x13,x17,x18,x23,x24,ℛx13x14=VG−x1,x4,x19,x22,ℛx3x10=VG−x7,x13,x14,x15,x19,x20,ℛx13x24=ℛx15x21=VG−x4,x5,x10,x11,x12,x16,x17,x18,x22,x23,ℛx18x24=ℛx16x21=VG−x1,x2,x7,x8,x9,x13,x14,x15,x19,x20,ℛx18x23=ℛx14x20=VG−x3,x4,x9,x10,x11,x15,x16,x17,x21,x22,ℛx17x23=ℛx15x20=VG−x1,x6,x7,x8,x12,x13,x14,x18,x19,x24,ℛx17x22=ℛx13x19=VG−x2,x3,x8,x9,x10,x14,x15,x16,x20,x21,ℛx16x22=ℛx14x19=VG−x5,x6,x7,x11,x12,x13,x17,x18,x23,x24.The cardinality of each of the local resolving neighborhoods is either 20, 16, or 14. Since 14 is less than the cardinality of all other resolving neighborhoods, we construct a mappingϑ:VG⟶0,1 as(4)ϑxi=ϑxj=114,i∈13,16,17,18,j∈22,23,24.Since 14 is a minimal cardinal number, there must exist a minimal resolving functionϑ′ such that ϑ′≤ϑ. Hence,(5)DimlfG≤∑i=124114=127.


## 3. Rotationally Symmetric Hexagonal Planar Networks

In this section, we define hexagonal circular ladder networks. In Figure2, the different hexagonal networks are given with faces.Figure 2
Hexagonal planar graphs with 2, 3, 4, and 5 faces.The vertex set and edge set of hexagonal circular ladder networks are defined as(6)VHnj=si,ti,ui,vi|1≤i≤n,j=1,2,3,4,5,EHn1=siti,tiui,uivi,uivi−1,siti+1|1≤i≤n,tn+1=t1,v0=vn,EHn2=siti,tiui,uivi,uivi−1,siti+1,sivi|1≤i≤n,tn+1=t1,v0=vn,EHn3=siti,tiui,uivi,uivi−1,siti+1,uiui+1|1≤i≤n,tn+1=t1,un+1=u1,v0=vn,EHn4=siti,tiui,uivi,uivi−1,siti+1,siui|1≤i≤n,tn+1=t1,v0=vn,EHn5=siti,tiui,uivi,uivi−1,siti+1,tiui+1|1≤i≤n,tn+1=t1,un+1=u1,v0=vn,and hexagonal circular ladder networks Hni,1≤i≤5, are shown in Figure 3.Figure 3
Rotationally symmetric hexagonal planar graphs with 2 and 3 faces.The vertex set and edge set of hexagonal circular ladder networksHni,6≤i≤11, are defined as(7)VHnj=si,ti,ui,vi|1≤i≤n,j=6,7,8,9,10,11,EHn6=siti,tiui,uivi,uivi−1,siti+1,sivi,siui|1≤i≤n,tn+1=t1,v0=vn,EHn7=siti,tiui,uivi,uivi−1,siti+1,titi+1,tiui+1|1≤i≤n,tn+1=t1,un+1=u1,v0=vn,EHn8=siti,tiui,uivi,uivi−1,siti+1,tivi,titi+1|1≤i≤n,tn+1=t1,v0=vn,EHn9=siti,tiui,uivi,uivi−1,siti+1,tivi,tiui+1|1≤i≤n,tn+1=t1,un+1=u1,v0=vn,EHn10=siti,tiui,uivi,uivi−1,siti+1,uiui+1,titi+1|1≤i≤n,tn+1=t1,un+1=u1,v0=vn,EHn11=siti,tiui,uivi,uivi−1,siti+1,tivi,siui+1|1≤i≤n,tn+1=t1,un+1=u1,v0=vn,and hexagonal circular ladder networks Hni,6≤i≤11, are shown in Figure 4.Figure 4
Rotationally symmetric hexagonal planar graphs with 4 faces.The vertex set and edge set of hexagonal circular ladder networksHni,12≤i≤17, are defined as(8)VHnj=si,ti,ui,vi|1≤i≤n,j=12,13,14,15,16,17,EHn12=siti,tiui,uivi,uivi−1,siti+1,siui,uiui+1,siui+1|1≤i≤n,tn+1=t1,un+1=u1,v0=vn,EHn13=siti,tiui,uivi,uivi−1,siti+1,titi+1,uiui+1,tiui+1|1≤i≤n,tn+1=t1,un+1=u1,v0=vn,EHn14=siti,tiui,uivi,uivi−1,siti+1,tivi,sivi,siui+1|1≤i≤n,tn+1=t1,un+1=u1,v0=vn,EHn15=siti,tiui,uivi,uivi−1,siti+1,tivi,tiui+1,siui+1|1≤i≤n,tn+1=t1,un+1=u1,v0=vn,EHn16=siti,tiui,uivi,uivi−1,siti+1,siui,sivi,siui+1|1≤i≤n,tn+1=t1,un+1=u1,v0=vn,EHn17=siti,tiui,uivi,uivi−1,siti+1,titi+1,uiti+1,viti+1|1≤i≤n,tn+1=t1,un+1=u1,v0=vn,and hexagonal circular ladder networks Hni,12≤i≤17, are shown in Figure 5.Figure 5
Rotationally symmetric hexagonal planar graphs with 5 faces.


### 3.1. LFMD of Rotationally Symmetric Hexagonal Planar Networks with 2 and 3 Faces

In this section, we find the sequences of upper bounds of local fractional metric dimension of rotationally symmetric hexagonal planar networks with 2 and 3 faces.Javaid et. al. [37] proved that if a network is a bipartite, then its local fractional metric should be one. Since our proposed networks Hnm,m=1,2,5, for each n are bipartite, we have the following result.Theorem 1.
The local fractional metric dimension of rotationally symmetric hexagonal planar networkHnm,m=1,2,5, is 1. That is, DimlfHnm=1.Lemma 1.
LetHn3 be a hexagonal planar circular network for n≥5 with n≡1mod2.(1)
Then, for1≤k≤n, ℛk=ℛukvk=ℛukvk−1=2n+1, with v0=vn and ∪k=1nℛk=4n.(2)
ℛk<ℛsktk and ℛsktk∩∪k=1nℛk≥ℛk.(3)
ℛk<ℛsktk+1 and ℛsktk+1∩∪k=1nℛk≥ℛk with tn+1=t1.(4)
ℛk<ℛtkuk and ℛtkuk∩∪k=1nℛk≥ℛk.(5)
ℛk<ℛukuk+1 and ℛukuk+1∩∪k=1nℛk≥ℛk with un+1=u1.Proof.
(1)
The set of local resolving neighborhoods ofukvk and ukvk−1 is ℛukvk=VHn3−sr|r≡k+1,k+2,…,k−1+n+1/2modn∪tr|r≡k+1,k+2,…,k−1+n+1/2modn∪ur|r≡k+1,k+2,…,k−1+n+1/2modn∪vr|r≡k+1,k+2,…,k−1+n+1/2modn and ℛukvk−1=VHn3−sr|r≡k+n−1/2,k+1+n−1/2,…,k+n−2modn∪tr|r≡k+1+n−1/2,k+2+n−1/2,…,k+n−1modn∪ur|r≡k+1+n−1/2,k+2+n−1/2,…,k+n−1modn∪vr|r≡k+n−1/2,k+1+n−1/2,…,k+n−2modn. Clearly, ∪k=1nℛk=VHn3. Therefore, ∪k=1nℛk=4n.(2)
The set of local resolving neighborhoods ofsktk is ℛsktk=VHn3−sr|r≡k+3,k+4,…,k+n+1/2−1modn,n≥7∪tr|r≡k+3,k+4,…,k+n+1/2−1modn,n≥7∪ur|r≡k+1,k+2,…,k+n+1/2−1modn,n≥5∪vr|r≡k+1,k+2,…,k+n+1/2−1modn,n≥5. Hence, ℛsktk=2n+6. Since ℛk=2n+1<ℛsktk=2n+6, ℛsktk∩∪k=1nℛk=2n+6≥ℛk.(3)
The set of local resolving neighborhoods ofsktk+1, where tn+1=t1, is ℛsktk+1=VHn3−sr|r≡k+1+n−1/2,k+2+n−1/2,…,k+n−3modn,n≥7∪tr|r≡k+2+n−1/2,k+3+n−1/2,…,k+n−2modn,n≥7∪ur|r≡kmodn,r≡k+2+n−1/2,k+3+n−1/2,…,k+n−1modn∪vr|r≡k+1+n−1/2,k+2+n−1/2,…,k+n−1modn Hence, ℛsktk+1=2n+6. Since ℛk=2n+1<ℛsktk+1=2n+6, ℛsktk+1∩∪k=1nℛk=2n+6≥ℛk.(4)
The set of local resolving neighborhoods oftkuk is ℛtkuk=VHn3−sr|r≡k+1modn,r≡k+n−2modn∪tr|r≡k+1modn,r≡k+n−1modn. Hence, ℛtkuk=4n−1. Since ℛk=2n+1<ℛtkuk=4n−1, ℛtkuk∩∪k=1nℛk=4n−1≥ℛk.(5)
The set of local resolving neighborhoods ofukuk+1, where un+1=u1, is ℛukuk+1=VHn3−sr|r≡kmodn∪tr|r≡k+n+3/2−1modn∪ur|r≡k+n+3/2−1modn∪vr|r≡kmodn. Hence, ℛukuk+1=4n−1. Since ℛk=2n+1<ℛukuk+1=4n−1, ℛukuk+1∩∪k=1nℛk=4n−1≥ℛk.Theorem 2.
IfHn3 is a hexagonal planar circular network for n≥3 with n≡1mod2, then DimlfHnm≤2n/n+1.Proof.
The local resolving neighborhoods areCase (a): forn=3,1≤k≤3, by Lemma 1, we have(9)R1¯=ℛukvk=VH33−sr|r≡k+1mod3∪tr|r≡k+1mod3∪ur|r≡k+1mod3∪vr|r≡k+1mod3,R2¯=ℛukvk−1=VH33−sr|r≡k+1mod3∪tr|r≡k+2mod3∪ur|r≡k+2mod3∪vr|r≡k+1mod3,R3¯=ℛukuk+1=VH33−sr|r≡kmod3∪tr|r≡k+2mod3∪ur|r≡k+2mod3∪vr|r≡kmod3,R1=ℛsktk=VH33−ur|r≡k+1mod3∪vr|r≡k+1mod3,R2=ℛsktk+1=VH33−ur|r≡kmod3∪vr|r≡k+1mod3,R3=ℛtkuk=VH33−sr|r≡k+1mod3∪tr|r≡k+1mod3,r≡k+2mod3.The cardinality of each of the local resolving neighborhoodsR1¯,R2¯,R3¯,R1,R2, and R3 for each k=1,2,3 is 8, 8, 8, 10, 10, and 9, respectively. Since cardinality of R1¯, R2¯, and R3¯ is 8 which is less than the cardinality of all other resolving neighborhoods and ∪i=13Ri=VH33, this implies that Rj¯∩∪i=13Ri≥Ri. Now, we construct a mapping ϑ:VH33⟶0,1 as(10)ϑuk=ϑvk−1=ϑvk=ϑuk+1=18.Since 8 is a very small cardinal number, there must exist a minimal resolving functionϑ′ such that ϑ′≤ϑ. Hence,(11)DimlfH33≤∑i=11218=32.Case (b): forn=5,1≤k≤5, by Lemma 2, we have(12)R1¯=ℛukvk=VH53−sr|r≡k+1,k+2mod5∪tr|r≡k+1,k+2mod5∪ur|r≡k+1,k+2mod5∪vr|r≡k+1,k+2mod5,R2¯=ℛukvk−1=VH53−sr|r≡k+2,k+3mod5∪tr|r≡k+3,k+4mod5∪ur|r≡k+3,k+4mod5∪vr|r≡k+2,k+3mod5,R1=ℛsktk=VH53−ur|r≡k+1,k+2mod5∪vr|r≡k+1,k+2mod5,R2=ℛsktk+1=VH53−ur|r≡kmod5,r≡k+4mod5∪vr|r≡k+3,k+4mod5,R3=ℛtkuk=VH53−sr|r≡k+1mod5,r≡k+3mod5∪tr|r≡k+1mod5,r≡k+4mod5,R4=ℛukuk+1=VH53−sr|r≡kmod5,∪tr|r≡k+3mod5∪ur|r≡k+3mod5∪vr|r≡kmod5.The cardinality of each of the local resolving neighborhoodsR1¯,R2¯,R1,R2,R3, and R4 for each 1≤k≤5 is 12, 12, 16, 16, 16, and 16, respectively. Since cardinality of R1¯ and R2¯ is 12 which is less than the cardinality of all other resolving neighborhoods and ∪i=14Ri=VH53, this implies that Rj¯∩∪i=14Ri≥Ri. Now, we construct a mapping ϑ:VH53⟶0,1 as(13)ϑuk=ϑvk−1=ϑvk=112.Since 12 is a very small cardinal number, there must exist a minimal resolving functionϑ′ such that ϑ′≤ϑ. Hence,(14)DimlfH53≤∑i=120112=53.Case (c): forn=7,1≤k≤7, by Lemma 2, we have(15)R1¯=ℛukvk=VH73−sr|r≡k+1,k+2,k+3mod7∪tr|r≡k+1,k+2,k+3mod7∪ur|r≡k+1,k+2,k+3mod7∪vr|r≡k+1,k+2,k+3mod7,R2¯=ℛukvk−1=VH73−sr|r≡k+3,k+4,k+5mod7∪tr|r≡k+4,k+5,k+6mod7∪ur|r≡k+4,k+5,k+6mod7∪vr|r≡k+3,k+4,k+5mod7,R1=ℛsktk=VH73−sr|r≡k+3mod7∪tr|r≡k+3mod7∪ur|r≡k+1,k+2,k+3mod7∪vr|r≡k+1,k+2,k+3mod7,R2=ℛsktk+1=VH73−sr|r≡k+3mod7∪tr|r≡k+5mod7∪ur|r≡kmod7,r≡k+5,k+6mod7∪vr|r≡k+4,k+5,k+6mod7.R3=ℛtkuk=VH73−sr|r≡k+1mod7,r≡k+5mod7∪tr|r≡k+1mod7,r≡k+6mod7,R4=ℛukuk+1=VH73−sr|r≡kmod7,∪tr|r≡k+4mod7∪ur|r≡k+4mod7∪vr|r≡kmod7.The cardinality of each of the local resolving neighborhoodsR1¯,R2¯,R1,R2,R3, and R4 for each 1≤k≤7 is 16, 16, 20, 20, 24, and 24, respectively. Since cardinality of R1¯ and R2¯ is 16 which is less than the cardinality of all other resolving neighborhoods and ∪i=14Ri=VH73, this implies that Rj¯∩∪i=14Ri≥Ri. Now, we construct a mapping ϑ:VH73⟶0,1 as(16)ϑuk=ϑvk−1=ϑvk=116.Since 16 is a very small cardinal number, there must exist a minimal resolving functionϑ′ such that ϑ′≤ϑ. Hence,(17)DimlfH73≤∑i=128116=74.Case (d): forn≥9,1≤k≤n, by Lemma 2, we have(18)R1¯=ℛukvk=VHn3−sr|r≡k+1,k+2,…,k−1+n+12modn∪tr|r≡k+1,k+2,…,k−1+n+12modn∪ur|r≡k+1,k+2,…,k−1+n+12modn∪vr|r≡k+1,k+2,…,k−1+n+12modn,R2¯=ℛukvk−1=VHn3−sr|r≡k+n−12,k+1+n−12,…,k+n−2modn∪tr|r≡k+1+n−12,k+2+n−12,…,k+n−1modn∪ur|r≡k+1+n−12,k+2+n−12,…,k+n−1modn∪vr|r≡k+n−12,k+1+n−12,…,k+n−2modn,R1=ℛsktk=VHn3−sr|r≡k+3,k+4,…,k+n+12−1modn,n≥7∪tr|r≡k+3,k+4,…,k+n+12−1modn,n≥7∪ur|r≡k+1,k+2,…,k+n+12−1modn,n≥5∪vr|r≡k+1,k+2,…,k+n+12−1modn,n≥5,R2=ℛsktk+1=VHn3−sr|r≡k+1+n−12,k+2+n−12,…,k+n−3modn,n≥7∪tr|r≡k+2+n−12,k+3+n−12,…,k+n−2modn,n≥7∪ur|r≡kmodn,r≡k+2+n−12,k+3+n−12,…,k+n−1modn∪vr|r≡k+1+n−12,k+2+n−12,…,k+n−1modn,R3=ℛtkuk=VHn3−sr|r≡k+1modn,r≡k+n−2modn∪tr|r≡k+1modn,r≡k+n−1modn,R4=ℛukuk+1=VHn3−sr|r≡kmodn∪tr|r≡k+n+32−1modn∪ur|r≡k+n+32−1modn∪vr|r≡kmodn.The cardinality of each of the local resolving neighborhoodsR1¯,R2¯,R1,R2,R3, and R4 for each 1≤k≤n is 2n+1,2n+1,2n+6,2n+6,4n−1, and 4n−1, respectively. Since cardinality of R1¯ and R2¯ is 2n+1 which is less than the cardinality of all other resolving neighborhoods and ∪i=14Ri=VHn3, this implies that Rj¯∩∪i=14Ri≥Ri. Now, we construct a mapping ϑ:VHn3⟶0,1 as(19)ϑuk=ϑvk−1=ϑvk=12n+1.Since2n+1 is a very small cardinal number, there must exist a minimal resolving function ϑ′ such that ϑ′≤ϑ. Hence,(20)DimlfHn3≤∑i=14n12n+1=2nn+1.Lemma 2.
LetHn4 be a hexagonal planar circular network for n≥3 with n≡1mod2.(1)
Then, for1≤k≤n, ℛk=ℛtkuk=ℛskuk=2n+1, and ∪k=1nℛk=4n.(2)
ℛk<ℛsktk and ℛsktk∩∪k=1nℛk≥ℛk.(3)
ℛk<ℛsktk+1 and ℛsktk+1∩∪k=1nℛk≥ℛk with tn+1=t1.(4)
ℛk<ℛukvk and ℛukvk∩∪k=1nℛk≥ℛk.(5)
ℛk<ℛukvk−1 and ℛukvk−1∩∪k=1nℛk≥ℛk with v0=vn.Theorem 3.
IfHn4 is a hexagonal planar circular network for n≥3 with n≡1mod2, then DimlfHnm≤4n/2n+1.


### 3.2. LFMD of Rotationally Symmetric Hexagonal Planar Networks with 4 Faces

In this section, we find the sequences of upper bounds of local fractional metric dimension of rotationally symmetric hexagonal planar networks with 4 faces.Lemma 3.
LetHn6 be a hexagonal planar circular network for n≥3 with n≡1mod2.(1)
Then, for1≤k≤n, ℛk=ℛskuk=2n, and ∪k=1nℛk=4n.(2)
ℛk<ℛsktk and ℛsktk∩∪k=1nℛk≥ℛk.(3)
ℛk<ℛsktk+1 and ℛsktk+1∩∪k=1nℛk≥ℛk with tn+1=t1.(4)
ℛk<ℛukvk and ℛukvk∩∪k=1nℛk≥ℛk.(5)
ℛk<ℛukvk−1 and ℛukvk−1∩∪k=1nℛk≥ℛk with v0=vn.(6)
ℛk<ℛtkuk and ℛtkuk∩∪k=1nℛk≥ℛk.(7)
ℛk<ℛskvk and ℛskvk∩∪k=1nℛk≥ℛk.Proof.
(1)
The set of local resolving neighborhoods ofskuk is ℛskuk=VHn6−sr|r≡k+1+n−1/2,k+2+n−1/2,…,k+n−1modn∪tr|r≡kmodn,r≡k+1+n−1/2,k+2+n−1/2,…,k+n−1modn∪ur|r≡k+1,k+2,…,k+n−1/2modn∪vr|r≡k,k+1,…,k−1+n+1/2modn. Clearly, ∪k=1nℛk=VHn6. Therefore, ∪k=1nℛk=4n.(2)
The set of local resolving neighborhoods ofsktk is ℛsktk=VHn6−ur|r≡kmodn∪vr|r≡k−n−1/2modn,r≡k+n−1modn. Hence, ℛsktk=4n−3. Since ℛk=2n<ℛsktk=4n−3, ℛsktk∩∪k=1nℛk=4n−3≥ℛk.(3)
The set of local resolving neighborhoods ofsktk+1 with tk+1=t1 is ℛsktk+1=VHn6−ur|r≡k+n−1/2+1modn. Hence, ℛsktk+1=4n−1. Since ℛk=2n<ℛsktk+1=4n−1, ℛsktk+1∩∪k=1nℛk=4n−1≥ℛk.(4)
The set of local resolving neighborhoods ofukvk is ℛukvk=VHn6−sr|r≡kmodn∪tr|r≡k+1modn,r≡k+n+1/2modn. Hence, ℛukvk=4n−3. Since ℛk=2n<ℛukvk=4n−3, ℛukvk∩∪k=1nℛk=4n−3≥ℛk.(5)
The set of local resolving neighborhoods ofukvk−1 with vn=v0 is ℛukvk−1=VHn6−sr|r≡k+n−1/2modn. Hence, ℛukvk−1=4n−1. Since ℛk=2n<ℛukvk−1=4n−1, ℛukvk−1∩∪k=1nℛk=4n−1≥ℛk.(6)
The set of local resolving neighborhoods oftkuk is ℛtkuk=VHn6−sr|r≡k,k+1,…,k−1+n+1/2modn∪tr|r≡k+1,k+2,…,k+n−1/2modn∪ur|r≡k+1+n−1/2,k+2+n−1/2,…,k+n−1modn∪vr|r≡k+n−1/2,k+1+n−1/2,…,k+n−2modn. Hence, ℛtkuk=2n+1. Since ℛk=2n<ℛtkuk=2n+1, ℛtkuk∩∪k=1nℛk=2n+1≥ℛk.(7)
The set of local resolving neighborhoods ofskvk is ℛskvk=VHn6−sr|r≡k+1,k+2,…,k+n−1/2modn∪tr|r≡k+2,k+3,…,k+n+1/2modn∪vr|r≡k+1+n−1/2,k+2+n−1/2,…,k+n−1modn∪ur|r≡kmodn,r≡k+1+n−1/2,k+2+n−1/2,…,k+n−1modn. Hence, ℛskvk=2n+1. Since ℛk=2n<ℛskvk=2n+1, ℛskvk∩∪k=1nℛk=2n+1≥ℛk.Theorem 4.
IfHn6 is a hexagonal planar circular network for n≥3 with n≡1mod2, then DimlfHn6≤2.Proof.
Whenn≥3 with n≡1mod2, then by Lemma 3, the local resolving neighborhoods are(21)R1¯=ℛskuk=VHn6−sr|r≡k+1+n−12,k+2+n−12,…,k+n−1modn∪⋅tr|r≡kmodn,r≡k+1+n−12,k+2+n−12,…,k+n−1modn∪⋅ur|r≡k+1,k+2,…,k+n−12modn∪vr|r≡k,k+1,…,k−1+n+12modn,R1=ℛsktk=VHn6−ur|r≡kmodn∪vr|r≡k−n−12modn,r≡k+n−1modn,R2=ℛsktk+1=VHn6−ur|r≡k+n−12+1modn,R3=ℛukvk=VHn6−sr|r≡kmodn∪tr|r≡k+1modn,r≡k+n+12modn,R4=ℛukvk−1=VHn6−sr|r≡k+n−12modn,R5=ℛtkuk=VH46−sr|r≡k,k+1mod4∪tr|r≡k+1,k+2mod4∪⋅vr|r≡k+2mod4∪ur|r≡k+2,k+3mod4,R6=ℛskvk=VHn6−sr|r≡k+1,k+2,…,k+n−12modn∪⋅tr|r≡k+2,k+3,…,k+n+12modn∪vr|r≡k+1+n−12,k+2+n−12,…,k+n−1modn∪ur|r≡kmodn,r≡k+1+n−12,k+2+n−12,…,k+n−1modn.
The cardinality of each of the local resolving neighborhoodsR1¯,R1,R2,R3,R4,R5, and R6 for each 1≤k≤n is 2n,4n−3,4n−1,4n−3,4n−1,2n+1, and 2n+1, respectively. Since cardinality of R1¯ is 2n which is less than the cardinality of all other resolving neighborhoods and ∪i=16Ri=VHn6, this implies that Rj¯∩∪i=16Ri≥Ri. Now, we construct a mapping ϑ:VHn6⟶0,1 as(22)ϑuk=ϑsk=12n.
Since2n is a minimal cardinal number, there must exist a minimal resolving function ϑ′ such that ϑ′≤ϑ. Hence,(23)DimlfHn6≤∑i=14n12n=2.Lemma 4.
LetHn7 be a hexagonal planar circular network for n≥5 with n≡1mod2.(1)
Then, for1≤k≤n, ℛk=ℛsktk=ℛsktk+1=ℛtkuk=ℛtkuk+1=2n+3 with tn+1=t1,un+1=u1 and ∪k=1nℛk=4n.(2)
ℛk<ℛukvk and ℛukvk∩∪k=1nℛk≥ℛk.(3)
ℛk<ℛukvk−1 and ℛukvk−1∩∪k=1nℛk≥ℛk with vn=v0.(4)
ℛk<ℛtktk+1 and ℛtktk+1∩∪k=1nℛk≥ℛk, tn+1=t1.Proof.
(1)
The set of local resolving neighborhoods ofsktk,sktk+1,tkuk and tkuk+1 with un+1=u1,tn+1=t1 is ℛsktk=VHn7−sr|r≡k+1,k+2,…,k+n+1/2−1modn∪tr|r≡k+1,k+2,…,k+n+1/2−1modn∪ur|r≡k+2,k+3,…,k+n+3/2−1modn∪vr|r≡k+2,k+3,…,k+n+1/2−1modn, ℛsktk+1=VHn7−sr|r≡k+1+n−1/2,k+2+n−1/2,…,k+n−1modn∪tr|r≡kmodn,r≡k+2+n−1/2,k+3+n−1/2,…,k+n−1modn∪ur|r≡kmodn,r≡k+2+n−1/2,k+3+n−1/2,…,k+n−1modn∪vr|r≡k+2+n−1/2,k+3+n−1/2,…,k+n−1modn, ℛtkuk=VHn7−sr|r≡k+n−1/2,k+1+n−1/2,…,k+n−2modn∪tr|r≡k+n−1/2+1,k+2+n−1/2,…,k+n−1modn∪ur|r≡k+1+n−1/2,k+2+n−1/2,…,k+n−1modn∪vr|r≡k+1+n−1/2,k+2+n−1/2,…,k+n−2modn, and ℛtkuk+1=VHn7−sr|r≡k+1,k+2,…,k+n+1/2−1modn∪tr|r≡k+1,k+2,…,k+n+1/2−1modn∪ur|r≡k+2,k+3,…,k+n+3/2−1modn∪vr|r≡k+2,k+3,…,k+n+1/2−1modn. Clearly, ∪k=1nℛk=VHn7. Therefore, ∪k=1nℛk=4n.(2)
The set of local resolving neighborhoods ofukvk is ℛukvk=VHn7−sr|r≡k+1,k+2,…,k+n−1/2−1modn∪tr|r≡k+1,k+2,…,k+n−1/2−1modn∪ur|r≡k+2,k+3,…,k+n+1/2−1modn∪vr|r≡k+2,k+3,…,k+n+1/2−1modn. Hence, ℛukvk=2n+5. Since ℛk=2n+3<ℛukvk=2n+5, ℛukvk∩∪k=1nℛk=2n+3≥ℛk.(3)
The set of local resolving neighborhoods ofukvk−1 is ℛukvk−1=VHn7−sr|r≡k+n−1/2,k+1+n−1/2,k+2+n−1/2,…,k+n−3modn∪tr|r≡k+n−1/2,k+1+n−1/2,…,k+n−2modn∪ur|r≡k+1+n−1/2,k+2+n−1/2,…,k+n−2modn∪vr|r≡k+n−1/2,k+1+n−1/2,…,k+n−3modn. Hence, ℛukvk−1=2n+5. Since ℛk=2n+3<ℛukvk−1=2n+5, ℛukvk−1∩∪k=1nℛk=2n+3≥ℛk.(4)
The set of local resolving neighborhoods oftktk+1, with vn+1=v1, is ℛtktk+1=VHn7−sr|r≡kmodn∪tr|r≡k+n+3/2−1modn∪ur|r≡k+1modn∪vr|r≡kmodn,r≡k+1modn,r≡k+n+3/2−1modn. Hence, ℛtktk+1=4n−6. Since ℛk=2n+3<ℛtktk+1=4n−6, ℛukvk−1∩∪k=1nℛk=2n+3≥ℛk.Theorem 5.
IfHn7 is a hexagonal planar circular network, then(24)DimlfHn7≤2,n=3,4n2n+3,n≥5.Proof.
The local resolving neighborhoods areCase (a): forn=3,1≤k≤3, by Lemma 4,(25)R1¯=ℛtktk+1=VH36−sr|r≡kmod3∪tr|r≡k+n+32−1mod3∪⋅ur|r≡k+1mod3∪vr|r≡k,k+1,k+n+32−1mod3,R1=ℛsktk=VH36−sr|r≡k+1mod3∪tr|r≡k+1mod3∪ur|r≡k+2mod3,R2=ℛsktk+1=VH36−sr|r≡k+2mod3∪tr|r≡kmod3∪ur|r≡kmod3,R3=ℛtkuk=VH36−sr|r≡k+1mod3∪tr|r≡k+2mod3∪ur|r≡k+2mod3,R4=ℛtkuk+1=VH36−sr|r≡k+1mod3∪tr|r≡k+1mod3∪ur|r≡k+2mod3,R5=ℛukvk=VH36−sr|r≡k+1mod3∪tr|r≡k+1,k+2mod3∪⋅ur|r≡k+2mod3∪vr|r≡k+2mod3,R6=ℛukvk−1=VH36−tr|r≡k+1mod3.The cardinality of each of the local resolving neighborhoodsR1¯,R1,R2,R3,R4,R5, and R6 for each k=1,2,3 is 6, 9, 9, 9, 9, 11, and 11, respectively. Since cardinality of R1¯ is 6 which is less than the cardinality of all other resolving neighborhoods and ∪i=16Ri=VH36, this implies that Rj¯∩∪i=16Ri≥Ri. Now, we construct a mapping ϑ:VH36⟶0,1 as(26)ϑtk=ϑtk+1=16.Since 6 is a minimal cardinal number, there must exist a minimal resolving functionϑ′ such that ϑ′≤ϑ. Hence,(27)DimlfH36≤∑i=11216=2.Case (b): forn≥5,1≤k≤5, by Lemma 8, we have(28)R1¯=ℛsktk=VHn7−sr|r≡k+1,k+2,…,k+n+12−1modn∪⋅tr|r≡k+1,k+2,…,k+n+12−1modn∪ur|r≡k+2,k+3,…,k+n+32−1modn∪vr|r≡k+2,k+3,…,k+n+12−1modn,R2¯=ℛsktk+1=VHn7−sr|r≡k+1+n−12,k+2+n−12,…,k+n−1modn∪⋅tr|r≡kmodn,r≡k+2+n−12k+3+n−12,…,k+n−1modn∪⋅ur|r≡kmodn,r≡k+2+n−12,k+3+n−12,…,k+n−1modn∪⋅vr|r≡k+2+n−12,k+3+n−12,…,k+n−1modn,R3¯=ℛtkuk=VHn7−sr|r≡k+n−12,k+1+n−12,…,k+n−2modn∪⋅tr|r≡k+n−12+1,k+2+n−12,…,k+n−1modn∪⋅ur|r≡k+1+n−12,k+2+n−12,…,k+n−1modn∪⋅vr|r≡k+1+n−12,k+2+n−12,…,k+n−2modn,R4¯=ℛtkuk+1=VHn7−sr|r≡k+1,k+2,…,k+n+12−1modn∪⋅tr|r≡k+1,k+2,…,k+n+12−1modn∪ur|r≡k+2,k+3,…,k+n+32−1modn∪vr|r≡k+2,k+3,…,k+n+12−1modn,R1=ℛukvk=VHn7−sr|r≡k+1,k+2,…,k+n−12−1modn∪⋅tr|r≡k+1,k+2,…,k+n−12−1modn∪ur|r≡k+2,k+3,…,k+n+12−1modn∪vr|r≡k+2,k+3,…,k+n+12−1modn,R2=ℛukvk−1=VHn7−sr|r≡k+n−12,k+1+n−12,k+2+n−12,…,k+n−3modn∪tr|r≡k+n−12,k+1+n−12,…,k+n−2modn∪⋅ur|r≡k+1+n−12,k+2+n−12,…,k+n−2modn∪vr|r≡k+n−12,k+1+n−12,…,k+n−3modn,R3=ℛtktk+1=VHn7−sr|r≡kmodn∪tr|r≡k+n+32−1modn∪ur|r≡k+1modn∪vr|r≡kmodn,r≡k+1modn,r≡k+n+32−1modn.The cardinality of each of the local resolving neighborhoodsR1¯,R2¯,R3¯,R4¯,R1,R2, and R3 for each 1≤k≤n is 2n+3,2n+3,2n+3,2n+3,2n+5, and 4n−6, respectively. Since cardinality of Ri¯,1≤i≤4 is 2n+3 which is less than the cardinality of all other resolving neighborhoods and ∪i=13Ri=VHn7, this implies that Rj¯∩∪i=13Ri≥Ri. Now, we construct a mapping ϑ:VHn7⟶0,1 as(29)ϑtk=ϑsk=ϑtk+1=ϑuk=ϑuk+1=12n+3.Since2n+3 is a minimal cardinal number, there must exist a minimal resolving function ϑ′ such that ϑ′≤ϑ. Hence,(30)DimlfHn7≤∑i=14n12n+3=4n2n+3.Lemma 5.
LetHn8 be a hexagonal planar circular network for n≥5 with n≡1mod2.(1)
Then, for1≤k≤n, ℛk=ℛukvk=8 and ∪k=1nℛk=4n with un+1=u1.(2)
ℛk<ℛsktk and ℛsktk∩∪k=1nℛk≥ℛk.(3)
ℛk<ℛtkuk and ℛtkuk∩∪k=1nℛk≥ℛk.(4)
ℛk<ℛtktk+1 and ℛtktk+1∩∪k=1nℛk≥ℛk, with tn+1=t1.(5)
ℛk<ℛtkvk and ℛtkvk∩∪k=1nℛk≥ℛk.(6)
ℛk<ℛukvk−1 and ℛukvk−1∩∪k=1nℛk≥ℛk, with vn=v0.(7)
ℛk<ℛsktk+1 and ℛsktk+1∩∪k=1nℛk≥ℛk, with tk+1=t1.Theorem 6.
IfHn8 is a hexagonal planar circular network, then(31)DimlfHn8≤2,n=3,n2,n≥5.Lemma 6.
LetHn9 be a hexagonal planar circular network for n≥3 with n≡1mod2.(1)
Then, for1≤k≤n, ℛk=ℛtkvk=4, and ∪k=1nℛk=4n.(2)
ℛk<ℛtkuk and ℛtkuk∩∪k=1nℛk≥ℛk.(3)
ℛk<ℛukvk−1 and ℛukvk−1∩∪k=1nℛk≥ℛk with vn=u0.(4)
ℛk<ℛukvk and ℛukvk∩∪k=1nℛk≥ℛk.(5)
ℛk<ℛsktk+1 and ℛsktk+1∩∪k=1nℛk≥ℛk, with tn+1=t1.(6)
ℛk<ℛsktk and ℛsktk∩∪k=1nℛk≥ℛk.(7)
ℛk<ℛtkuk+1 and ℛtkuk+1∩∪k=1nℛk≥ℛk, with un+1=u1.Proof.
(1)
The set of local resolving neighborhoods oftkvk is ℛtkvk=VHn9−sr|r≡k+1,k+2,…,k+n−modn∪tr|r≡k+1,k+2,…,k+n−2modn∪ur|r≡k,k+1,k+2,…,k+n−1modn∪vr|r≡k+1,k+2,…,k+n−1modn. Clearly, ∪k=1nℛk=VHn9. Therefore, ∪k=1nℛk=4n.(2)
The set of local resolving neighborhoods ofsktk+1 is ℛsktk+1=VHn9−vr|r≡k+1modn. Hence, ℛsktk+1=4n−1. Since ℛk=4<ℛsktk+1=4n−1, ℛsktk+1∩∪k=1nℛk=4n−1≥ℛk.(3)
The set of local resolving neighborhoods ofsktk is ℛsktk=VHn9−vr|r≡kmodn. Hence, ℛsktk=4n−1. Since ℛk=4<ℛsktk=4n−1, ℛsktk∩∪k=1nℛk=4n−1≥ℛk.(4)
The set of local resolving neighborhoods oftkuk is ℛtkuk=VHn9−sr|r≡kmodn,r≡k+n−1modn∪tr|r≡kmodn. Hence, ℛtkuk=4n−1. Since ℛk=4<ℛtkuk=4n−1, ℛtkuk∩∪k=1nℛk=4n−1≥ℛk.(5)
The set of local resolving neighborhoods oftkuk+1 is ℛtkuk+1=VHn9−vr|r≡kmodn. Hence, ℛtkuk+1=4n−1. Since ℛk=4<ℛtkuk+1=4n−1, ℛtkuk+1∩∪k=1nℛk=4n−1≥ℛk.(6)
The set of local resolving neighborhoods ofukvk−1 is ℛukvk−1=VHn9−vr|r≡k+n−2modn,r≡k+n−1modn∪tr|r≡k+n−1modn. Hence, ℛukvk−1=4n−3. Since ℛk=4<ℛukvk−1=4n−3, ℛukvk−1∩∪k=1nℛk=4n−3≥ℛk.(7)
The set of local resolving neighborhoods ofukvk is ℛukvk=VHn9−vr|r≡kmodn,r≡k+n−1modn∪tr|r≡k+n−1modn. Hence, ℛukvk=4n−3. Since ℛk=4<ℛukvk=4n−3, ℛukvk∩∪k=1nℛk=4n−3≥ℛk.Theorem 7.
IfHn9 is a hexagonal planar circular network for n≥3 with n≡1mod2, then DimlfHn9≤n.Proof.
Whenn≥3 with n≡1mod2, by Lemma 6, the local resolving neighborhoods are(32)R1¯=ℛtkvk=VHn9−sr|r≡k+1,k+2,…,k+n−2modn∪⋅tr|r≡k+1,k+2,…,k+n−2modn∪ur|r≡k,k+1,k+2,…,k+n−1modn∪⋅vr|r≡k+1,k+2,…,k+n−1modn,R1=ℛsktk+1=VHn9−vr|r≡k+1modn,R2=ℛsktk=VHn9−vr|r≡kmodn,R3=ℛtkuk=VHn9−sr|r≡kmodn,r≡k+n−1modn∪tr|r≡kmodn,R4=ℛtkuk+1=VHn9−vr|r≡kmodn,R5=ℛukvk−1=VHn9−vr|r≡k+n−2modn,r≡k+n−1modn∪tr|r≡k+n−1modn,R6=ℛukvk=VHn9−vr|r≡kmodn,r≡k+n−1modn∪tr|r≡k+n−1modn.
The cardinality of each of the local resolving neighborhoodsR1¯,R2¯,R1,R2,R3,R4,R5, and R6 for each 1≤k≤n is 4, 4n−1,4n−1,4n−1,4n−1,4n−1,4n−3, and 4n−3, respectively. Since cardinality of R1¯ is 4 which is less than the cardinality of all other resolving neighborhoods and ∪i=16Ri=VHn9, this implies that Rj¯∩∪i=16Ri≥Ri. Now, we construct a mapping ϑ:VHn9⟶0,1 as(33)ϑtk=ϑvk=14.
Since 4 is a minimal cardinal number, there must exist a minimal resolving functionϑ′ such that ϑ′≤ϑ. Hence,(34)DimlfHn9≤∑i=14n14=n.Lemma 7.
LetHn10 be a hexagonal planar circular network for n≥3 with n≡1mod2.(1)
Then, for1≤k≤n, ℛk=ℛtktk+1=ℛukvk−1=ℛukvk=ℛsktk+1=ℛsktk=ℛukuk+1=2n+2, with tn+1=t1, vn=v0, un+1=u1, and ∪k=1nℛk=4n.(2)
ℛk<ℛtkuk and ℛtkuk∩∪k=1nℛk≥ℛk.Theorem 8.
IfHn10 is a hexagonal planar circular network for n≥3 with n≡1mod2, then DimlfHn10≤2n/n+1.Lemma 8.
LetHn11 be a hexagonal planar circular network for n≥3 with n≡1mod2.(1)
Then, for1≤k≤n, ℛk=ℛtkuk=2n, and ∪k=1nℛk=4n.(2)
ℛk<ℛukvk−1 and ℛukvk−1∩∪k=1nℛk≥ℛk with vn=v0.(3)
ℛk<ℛukvk and ℛukvk∩∪k=1nℛk≥ℛk.(4)
ℛk<ℛsktk+1 and ℛsktk+1∩∪k=1nℛk≥ℛk, with tk+1=t1.(5)
ℛk<ℛsktk and ℛsktk∩∪k=1nℛk≥ℛk.(6)
ℛk<ℛtkvk and ℛtkvk∩∪k=1nℛk≥ℛk.(7)
ℛk<ℛskuk+1 and ℛskuk+1∩∪k=1nℛk≥ℛk, with uk+1=u1.Theorem 9.
IfHn11 is a hexagonal planar circular network for n≥3 with n≡1mod2, then DimlfHn11≤2.


### 3.3. LFMD of Rotationally Symmetric Hexagonal Planar Networks with 5 Faces

In this section, we find the sequences of upper bounds of local fractional metric dimension of rotationally symmetric hexagonal planar networks with 5 faces.Lemma 9.
LetHn12 be a hexagonal planar circular network for n≥5 with n≡1mod2.(1)
Then for1≤k≤n, ℛk=ℛsktk=ℛsktk+1=ℛukvk=ℛukvk−1=ℛskuk=ℛskuk+1=2n+1, with un+1=u1,tn+1=t1,vn=v0, and ∪k=1nℛk=4n.(2)
ℛk<ℛtkuk and ℛtkuk∩∪k=1nℛk≥ℛk.(3)
ℛk<ℛukuk+1 and ℛukvk∩∪k=1nℛk≥ℛk, with un+1=u1.Proof.
(1)
The set of local resolving neighborhoods ofsktk,sktk+1,ukvk,ukvk−1,skuk, and skuk+1 with uk+1=u1,tk+1=t1 is ℛsktk=VHn12−sr|r≡k+1+n−1/2,k+2+n−1/2,…,k+n−2modn∪tr|r≡k+1+n−1/2,k+2+n−1/2,…,k+n−2modn∪ur|r≡kmodn,r≡k+1+n−1/2,k+2+n−1/2,…,k+n−1modn∪vr|r≡kmodn,r≡k+1+n−1/2,k+2+n−1/2,…,k+n−1modn, ℛsktk+1=VHn12−sr|r≡k+2,k+3,…,k+n−1/2modn∪tr|r≡k+3,k+4,…,k+n+1/2modn∪ur|r≡k+1,k+2,…,k+n+1/2modn∪vr|r≡k,k+1,…,k+n−1/2modn, ℛukvk=VHn12−sr|r≡k+1,k+2,…,k+n−1/2modn∪tr|r≡k+1,k+2,…,k+n−1/2modn∪ur|r≡k+1,k+2,…,k+n−1/2modn∪vr|r≡k+1,k+2,…,k+n−1/2modn, ℛukvk−1=VHn12−sr|r≡k+n−1/2,k+1+n−1/2,…,k+n−2modn∪tr|r≡k+1+n−1/2,k+2+n−1/2,…,k+n−1modn∪ur|r≡k+1+n−1/2,k+2+n/2,…,k+n−1modn∪vr|r≡k+n−1/2,k+1+n−1/2,…,k+n−2modn, ℛskuk=VHn12−sr|r≡k+1,k+2,…,k+n−1/2modn∪tr|r≡kmodn,r≡k+2,k+3,…,k+n−1/2modn∪ur|r≡k+1,k+2,…,k+n−1/2modn∪vr|r≡k+1,k+2,…,k+n−1/2modn, and ℛskuk+1=VHn12−sr|r≡k+1+n−1/2,k+2+n−1/2,…,k+n−1modn∪tr|r≡k+1modn,r≡k+2+n−1/2,k+3+n−1/2,…,k+n−1modn∪ur|r≡kmodn,r≡k+2+n−1/2,k+3+n−1/2,…,k+n−1modn∪vr|r≡k+1+n−1/2,k+2+n−1/2,…,k+n−1modn. Clearly, ∪k=1nℛk=VHn12. Therefore, ∪k=1nℛk=4n.(2)
The set of local resolving neighborhoods oftkuk is ℛtkuk=VHn12−sr|r≡kmodn,r≡k+n−1modn∪tr|r≡k+1modn,r≡k+n−1modn. Hence, ℛtkuk=4n−1. Since ℛk=2n+1<ℛtkuk=4n−1, ℛtkuk∩∪k=1nℛk=4n−1≥ℛk.(3)
The set of local resolving neighborhoods ofukuk+1 with un+1=u1 is ℛukuk+1=VHn12−sr|r≡kmodn∪tr|r≡k+n+1/2modn∪ur|r≡k+n+1/2modn∪vr|r≡kmodn. Hence, ℛukuk+1=4n−1. Since ℛk=2n+1<ℛukuk+1=4n−1, ℛukuk+1∩∪k=1nℛk=4n−1≥ℛk.Theorem 10.
IfHn12 is a hexagonal planar circular network with n≡1mod2, then(35)DimlfHn12≤32,n=3,2nn+1,n≥5.Proof.
The local resolving neighborhoods areCase (a): forn=3,1≤k≤3, by Lemma 9, we have(36)R1¯=ℛsktk=VH312−ur|r≡k,k+2mod3∪vr|r≡k,k+2mod3,R2¯=ℛsktk+1=VH312−ur|r≡k+1,k+2mod3∪vr|r≡k,k+1mod3,R3¯=ℛukvk=VH312−sr|r≡k+1mod3∪tr|r≡k+1mod3∪⋅ur|r≡k+1mod3∪vr|r≡k+1mod3,R4¯=ℛukvk−1=VH312−sr|r≡k+1mod3∪tr|r≡k+2mod3∪⋅ur|r≡k+2mod3∪vr|r≡k+1mod3,R5¯=ℛskuk=VH312−sr|r≡k+1mod3∪tr|r≡kmod3∪ur|r≡k+1mod3∪vr|r≡k+1mod3,R6¯=ℛskuk+1=VH312−sr|r≡k+2mod3∪tr|r≡k+1mod3∪⋅ur|r≡kmod3∪vr|r≡k+2mod3,R1=ℛtkuk=VH312−sr|r≡k,k+2mod3∪tr|r≡k+1,k+2mod3,R2=ℛukuk+1=VH312−sr|r≡kmod3∪tr|r≡k+2mod3∪⋅ur|r≡k+2mod3∪vr|r≡kmod3.The cardinality of each of the local resolving neighborhoodsR1¯,R2¯,R3¯,R4¯,R5¯,R6¯,R1, and R2 for each k=1,2,3 is 8, 8, 8, 8, 8, 8, and 8, respectively. Now, we construct a mapping ϑ:VH312⟶0,1 as(37)ϑsk=ϑuk=ϑvk=ϑtk=ϑuk+1=ϑtk+1=ϑvk−1=18.Since 8 is a minimal cardinal number, there must exist a minimal resolving functionϑ′ such that ϑ′≤ϑ. Hence,(38)DimlfH312≤∑i=11218=32.Case (b): forn≥5,1≤k≤n, by Lemma 9, we have(39)R1¯=ℛsktk=VHn12−sr|r≡k+1+n−12,k+2+n−12,…,k+n−2modn∪⋅tr|r≡k+1+n−12,k+2+n−12,…,k+n−2modn∪⋅ur|r≡kmodn,r≡k+1+n−12,k+2+n−12,…,k+n−1modn∪⋅vr|r≡kmodn,r≡k+1+n−12,k+2+n−12,…,k+n−1modn,R2¯=ℛsktk+1=VHn12−sr|r≡k+2,k+3,…,k+n−12modn∪⋅tr|r≡k+3,k+4,…,k+n+12modn∪ur|r≡k+1,k+2,…,k+n+12modn∪vr|r≡k,k+1,…,k+n−12modn,R3¯=ℛukvk=VHn12−sr|r≡k+1,k+2,…,k+n−12modn∪⋅tr|r≡k+1,k+2,…,k+n−12modn∪ur|r≡k+1,k+2,…,k+n−12modn∪vr|r≡k+1,k+2,…,k+n−12modn,R4¯=ℛukvk−1=VHn12−sr|r≡k+n−12,k+1+n−12,…,k+n−2modn∪⋅tr|r≡k+1+n−12,k+2+n−12,…,k+n−1modn∪⋅ur|r≡k+1+n−12,k+2+n2,…,k+n−1modn∪⋅vr|r≡k+n−12,k+1+n−12,…,k+n−2modn,R5¯=ℛskuk=VHn12−sr|r≡k+1,k+2,…,k+n−12modn∪⋅tr|r≡kmodn,r≡k+2,k+3,…,k+n−12modn∪ur|r≡k+1,k+2,…,k+n−12modn∪vr|r≡k+1,k+2,…,k+n−12modn,R6¯=ℛskuk+1=VHn12−sr|r≡k+1+n−12,k+2+n−12,…,k+n−1modn∪⋅tr|r≡k+1modn,r≡k+2+n−12,k+3+n−12,…,k+n−1modn∪⋅ur|r≡kmodn,r≡k+2+n−12,k+3+n−12,…,k+n−1modn∪⋅vr|r≡k+1+n−12,k+2+n−12,…,k+n−1modn,R1=ℛtkuk=VHn12−sr|r≡kmodn,r≡k+n−1modn∪⋅tr|r≡k+1modn,r≡k+n−1modn,R2=ℛukuk+1=VHn12−sr|r≡kmodn∪tr|r≡k+n+12modn∪⋅ur|r≡k+n+12modn∪vr|r≡kmodn.The cardinality of each of the local resolving neighborhoodsR1¯,R2¯,R3¯,R4¯,R5¯,R6¯,R1, and R2 for each 1≤k≤n is 2n+1,2n+1,2n+1,2n+1,2n+1,2n+1,4n−1, and 4n−1, respectively. Since cardinality of Ri¯,1≤i≤6 is 2n+1 which is less than the cardinality of all other resolving neighborhoods and ∪i=12Ri=VHn12, this implies that Rj¯∩∪i=12Ri≥Ri. Now, we construct a mapping ϑ:VHn12⟶0,1 as(40)ϑsk=ϑuk=ϑvk=ϑtk=ϑuk+1=ϑtk+1=ϑvk−1=12n+1.Since2n+1 is a minimal cardinal number, there must exist a minimal resolving function ϑ′ such that ϑ′≤ϑ. Hence,(41)DimlfHn12≤∑i=14n12n+1=2nn+1.Lemma 10.
LetHn13 be a hexagonal planar circular network for n≥5 with n≡1mod2.(1)
Then, for1≤k≤n, ℛk=ℛtkuk=ℛtkuk+1=2n+1, with u1=un+1 and ∪k=1nℛk=4n.(2)
ℛk<ℛsktk and ℛkktk∩∪k=1nℛk≥ℛk.(3)
ℛk<ℛukvk and ℛukvk∩∪k=1nℛk≥ℛk.(4)
ℛk<ℛtktk+1 and ℛtktk+1∩∪k=1nℛk≥ℛk, with tn+1=t1.(5)
ℛk<ℛukuk+1 and ℛukuk+1∩∪k=1nℛk≥ℛk, with un+1=u1.(6)
ℛk<ℛsktk+1 and ℛsktk+1∩∪k=1nℛk≥ℛk, with t1=tn+1.(7)
ℛk<ℛukvk−1 and ℛukvk−1∩∪k=1nℛk≥ℛk, with v0=vn.Theorem 11.
IfHn13 is a hexagonal planar circular network with n≡1mod2, then(42)DimlfHn13≤2,n=3,2nn+1,n≥5.Lemma 11.
LetHn14 be a hexagonal planar circular network for n≥3 with n≡1mod2.(1)
Then, for1≤k≤n, ℛk=ℛtkuk=ℛskuk+1=ℛskvk=ℛtkvk=2n, with u1=un+1 and ∪k=1nℛk=4n.(2)
ℛk<ℛukvk and ℛukvk∩∪k=1nℛk≥ℛk.(3)
ℛk<ℛsktk and ℛsktk∩∪k=1nℛk≥ℛk.(4)
ℛk<ℛsktk+1 and ℛsktk+1∩∪k=1nℛk≥ℛk, with t1=tn+1.(5)
ℛk<ℛukvk−1 and ℛukvk−1∩∪k=1nℛk≥ℛk, with v0=vn.Theorem 12.
IfHn14 is a hexagonal planar circular network for n≥3 with n≡1mod2, then DimlfHn14≤2.Lemma 12.
LetHn15 be a hexagonal planar circular network for n≥3 with n≡1mod2.(1)
Then, for1≤k≤n, ℛk=ℛskuk+1=ℛtkvk=4 and ∪k=1nℛk=4n with un+1=u1.(2)
ℛk<ℛsktk and ℛsktk∩∪k=1nℛk≥ℛk.(3)
ℛk<ℛtkuk and ℛtkuk∩∪k=1nℛk≥ℛk.(4)
ℛk<ℛukvk and ℛukvk∩∪k=1nℛk≥ℛk.(5)
ℛk<ℛtkuk+1 and ℛtkuk+1∩∪k=1nℛk≥ℛk, with un+1=u1.(6)
ℛk<ℛsktk+1 and ℛsktk+1∩∪k=1nℛk≥ℛk, with t1=tn+1.(7)
ℛk<ℛukvk−1 and ℛukvk−1∩∪k=1nℛk≥ℛk with v0=vn.Proof.
(1)
The set of local resolving neighborhoods ofskuk+1 and tkvk with un+1=u1 is ℛskuk+1=VHn15−sr|r≡k+1,k+2,…,k+n−1modn∪tr|r≡k,k+1,…,k+n−1modn∪ur|r≡kmodn,r≡k+2,k+3,…,k+n−1modn∪vr|r≡k+2,k+3,…,k+n−1modn and ℛtkvk=VHn15−sr|r≡k+1,k+2,…,k+n−2modn∪tr|r≡k+1,k+2,…,k+n−1modnur|r≡k,k+1,…,k+n−1modnvr|r≡k+1,k+2,…,k+n−1modn. Clearly, ∪k=1nℛk=VHn15. Therefore, ∪k=1nℛk=4n.(2)
The set of local resolving neighborhoods ofsktk is ℛsktk=VHn15−ur|r≡k+1modn∪vr|r≡k+1modn. Hence, ℛsktk=22n−1. Since ℛk=4<ℛsktk=22n−1, ℛsktk∩∪k=1nℛk=22n−1≥ℛk.(3)
The set of local resolving neighborhoods oftkuk is ℛtkuk=VHn15−sr|r≡k+n−1modn∪vr|r≡kmodn. Hence, ℛtkuk=22n−1. Since ℛk=4<ℛtkuk=22n−1, ℛtkuk∩∪k=1nℛk=22n−1≥ℛk.(4)
The set of local resolving neighborhoods ofukvk is ℛukvk=VHn15−sr|r≡kmodn∪tr|r≡kmodn. Hence, ℛukvk=22n−1. Since ℛk=4<ℛukvk=22n−1, ℛukvk∩∪k=1nℛk=22n−1≥ℛk.(5)
The set of local resolving neighborhoods oftkuk+1, with un+1=u1 is ℛtkuk+1=VHn15−sr|r≡kmodn∪vr|r≡kmodn. Hence, ℛtkuk+1=22n−1. Since ℛk=4<ℛtkuk=22n−1, ℛtkuk∩∪k=1nℛk=22n−1≥ℛk.(6)
The set of local resolving neighborhoods ofsktk+1 with tn+1=t1 is ℛsktk+1=VHn15−ur|r≡k+1modn∪vr|r≡kmodn. Hence, ℛsktk+1=22n−1. Since ℛk=4<ℛsktk=22n−1, ℛsktk∩∪k=1nℛk=22n−1≥ℛk.(7)
The set of local resolving neighborhoods ofukvk−1 with vn−1=v0 is ℛukvk+1=VHn15−sr|r≡k+n−2modn∪tr|r≡k+n−1modn. Hence, ℛukvk+1=22n−1. Since ℛk=4<ℛukvk−1=22n−1, ℛukvk∩∪k=1nℛk=22n−1≥ℛk.Theorem 13.
IfHn15 is a hexagonal planar circular network for n≥3 with n≡1mod2, then DimlfHn15≤n.Proof.
Whenn≥3 with n≡1mod2, by Lemma 12, the local resolving neighborhoods are(43)R1¯=ℛskuk+1=VHn15−sr|r≡k+1,k+2,…,k+n−1modn∪tr|r≡k,k+1,…,k+n−1modn∪ur|r≡kmodn,r≡k+2,k+3,…,k+n−1modn∪vr|r≡k+2,k+3,…,k+n−1modn,R2¯=ℛtkvk=VHn15−sr|r≡k+1,k+2,…,k+n−2modn∪tr|r≡k+1,k+2,…,k+n−1modn∪ur|r≡k,k+1,…,k+n−1modn∪vr|r≡k+1,k+2,…,k+n−1modn,R1=ℛsktk=VHn15−ur|r≡k+1modn∪vr|r≡k+1modn,R2=ℛtkuk=VHn15−sr|r≡k+n−1modn∪vr|r≡kmodn,R3=ℛukvk=VHn15−sr|r≡kmodn∪tr|r≡kmodn,R4=ℛtkuk+1=VHn15−sr|r≡kmodn∪vr|r≡kmodn,R5=ℛsktk+1=VHn15−ur|r≡k+1modn∪vr|r≡kmodn,R6=ℛukvk−1=VHn15−sr|r≡k+n−2modn∪tr|r≡k+n−1modn.
The cardinality of each of the local resolving neighborhoodsR1¯,R2¯,R1,R2,R3,R4,R5, and R6 for each 1≤k≤n is 4, 4, 4n−2,4n−2,4n−2,4n−2,4n−2,4n−2, and 4n−2, respectively. Since cardinality of Ri¯,1≤i≤2 is 4 which is less than the cardinality of all other resolving neighborhoods and ∪i=16Ri=VHn15, this implies that Rj¯∩∪i=16Ri≥Ri. Now, we construct a mapping ϑ:VHn15⟶0,1 as(44)ϑsk=ϑuk+1=ϑvk=ϑtk=14.
Since 4 is a minimal cardinal number, there must exist a minimal resolving functionϑ′ such that ϑ′≤ϑ. Hence,(45)DimlfHn15≤∑i=14n14=n.Lemma 13.
LetHn16 be a hexagonal planar circular network for n≥3 with n≡1mod2.(1)
Then, for1≤k≤n, ℛk=ℛtkuk=ℛskvk=4 and ∪k=1nℛk=4n.(2)
ℛk<ℛsktk and ℛsktk∩∪k=1nℛk≥ℛk.(3)
ℛk<ℛukvk and ℛukvk∩∪k=1nℛk≥ℛk.(4)
ℛk<ℛskuk+1 and ℛskuk+1∩∪k=1nℛk≥ℛk, with un+1=u1.(5)
ℛk<ℛskuk and ℛskuk∩∪k=1nℛk≥ℛk.(6)
ℛk<ℛsktk+1 and ℛsktk+1∩∪k=1nℛk≥ℛk, with tn+1=t1.(7)
ℛk<ℛukvk−1 and ℛukvk−1∩∪k=1nℛk≥ℛk, with vn=v0.Theorem 14.
IfHn16 is a hexagonal planar circular network for n≥3 with n≡1mod2, then DimlfHn16≤n.Lemma 14.
LetHn17 be a hexagonal planar circular network for n≥3 with n≡1mod2.(1)
Then, for1≤k≤n, ℛk=ℛsktk=ℛsktk+1=ℛtkuk=ℛukvk=ℛukvk−1=ℛtk+1uk=2n+1, with t1=tn+1,u1=un+1,v0=vn, and ∪k=1nℛk=4n.(2)
ℛk<ℛtktk+1 and ℛtktk+1∩∪k=1nℛk≥ℛk with tn+1=t1.(3)
ℛk<ℛvktk+1 and ℛvktk+1∩∪k=1nℛk≥ℛk with tn+1=t1.Theorem 15.
IfHn17 is a hexagonal planar circular network for n≥3 with n≡1mod2, then DimlfHn17≤2n/n+1.


## 3.1. LFMD of Rotationally Symmetric Hexagonal Planar Networks with 2 and 3 Faces

In this section, we find the sequences of upper bounds of local fractional metric dimension of rotationally symmetric hexagonal planar networks with 2 and 3 faces.Javaid et. al. [37] proved that if a network is a bipartite, then its local fractional metric should be one. Since our proposed networks Hnm,m=1,2,5, for each n are bipartite, we have the following result.Theorem 1.
The local fractional metric dimension of rotationally symmetric hexagonal planar networkHnm,m=1,2,5, is 1. That is, DimlfHnm=1.Lemma 1.
LetHn3 be a hexagonal planar circular network for n≥5 with n≡1mod2.(1)
Then, for1≤k≤n, ℛk=ℛukvk=ℛukvk−1=2n+1, with v0=vn and ∪k=1nℛk=4n.(2)
ℛk<ℛsktk and ℛsktk∩∪k=1nℛk≥ℛk.(3)
ℛk<ℛsktk+1 and ℛsktk+1∩∪k=1nℛk≥ℛk with tn+1=t1.(4)
ℛk<ℛtkuk and ℛtkuk∩∪k=1nℛk≥ℛk.(5)
ℛk<ℛukuk+1 and ℛukuk+1∩∪k=1nℛk≥ℛk with un+1=u1.Proof.
(1)
The set of local resolving neighborhoods ofukvk and ukvk−1 is ℛukvk=VHn3−sr|r≡k+1,k+2,…,k−1+n+1/2modn∪tr|r≡k+1,k+2,…,k−1+n+1/2modn∪ur|r≡k+1,k+2,…,k−1+n+1/2modn∪vr|r≡k+1,k+2,…,k−1+n+1/2modn and ℛukvk−1=VHn3−sr|r≡k+n−1/2,k+1+n−1/2,…,k+n−2modn∪tr|r≡k+1+n−1/2,k+2+n−1/2,…,k+n−1modn∪ur|r≡k+1+n−1/2,k+2+n−1/2,…,k+n−1modn∪vr|r≡k+n−1/2,k+1+n−1/2,…,k+n−2modn. Clearly, ∪k=1nℛk=VHn3. Therefore, ∪k=1nℛk=4n.(2)
The set of local resolving neighborhoods ofsktk is ℛsktk=VHn3−sr|r≡k+3,k+4,…,k+n+1/2−1modn,n≥7∪tr|r≡k+3,k+4,…,k+n+1/2−1modn,n≥7∪ur|r≡k+1,k+2,…,k+n+1/2−1modn,n≥5∪vr|r≡k+1,k+2,…,k+n+1/2−1modn,n≥5. Hence, ℛsktk=2n+6. Since ℛk=2n+1<ℛsktk=2n+6, ℛsktk∩∪k=1nℛk=2n+6≥ℛk.(3)
The set of local resolving neighborhoods ofsktk+1, where tn+1=t1, is ℛsktk+1=VHn3−sr|r≡k+1+n−1/2,k+2+n−1/2,…,k+n−3modn,n≥7∪tr|r≡k+2+n−1/2,k+3+n−1/2,…,k+n−2modn,n≥7∪ur|r≡kmodn,r≡k+2+n−1/2,k+3+n−1/2,…,k+n−1modn∪vr|r≡k+1+n−1/2,k+2+n−1/2,…,k+n−1modn Hence, ℛsktk+1=2n+6. Since ℛk=2n+1<ℛsktk+1=2n+6, ℛsktk+1∩∪k=1nℛk=2n+6≥ℛk.(4)
The set of local resolving neighborhoods oftkuk is ℛtkuk=VHn3−sr|r≡k+1modn,r≡k+n−2modn∪tr|r≡k+1modn,r≡k+n−1modn. Hence, ℛtkuk=4n−1. Since ℛk=2n+1<ℛtkuk=4n−1, ℛtkuk∩∪k=1nℛk=4n−1≥ℛk.(5)
The set of local resolving neighborhoods ofukuk+1, where un+1=u1, is ℛukuk+1=VHn3−sr|r≡kmodn∪tr|r≡k+n+3/2−1modn∪ur|r≡k+n+3/2−1modn∪vr|r≡kmodn. Hence, ℛukuk+1=4n−1. Since ℛk=2n+1<ℛukuk+1=4n−1, ℛukuk+1∩∪k=1nℛk=4n−1≥ℛk.Theorem 2.
IfHn3 is a hexagonal planar circular network for n≥3 with n≡1mod2, then DimlfHnm≤2n/n+1.Proof.
The local resolving neighborhoods areCase (a): forn=3,1≤k≤3, by Lemma 1, we have(9)R1¯=ℛukvk=VH33−sr|r≡k+1mod3∪tr|r≡k+1mod3∪ur|r≡k+1mod3∪vr|r≡k+1mod3,R2¯=ℛukvk−1=VH33−sr|r≡k+1mod3∪tr|r≡k+2mod3∪ur|r≡k+2mod3∪vr|r≡k+1mod3,R3¯=ℛukuk+1=VH33−sr|r≡kmod3∪tr|r≡k+2mod3∪ur|r≡k+2mod3∪vr|r≡kmod3,R1=ℛsktk=VH33−ur|r≡k+1mod3∪vr|r≡k+1mod3,R2=ℛsktk+1=VH33−ur|r≡kmod3∪vr|r≡k+1mod3,R3=ℛtkuk=VH33−sr|r≡k+1mod3∪tr|r≡k+1mod3,r≡k+2mod3.The cardinality of each of the local resolving neighborhoodsR1¯,R2¯,R3¯,R1,R2, and R3 for each k=1,2,3 is 8, 8, 8, 10, 10, and 9, respectively. Since cardinality of R1¯, R2¯, and R3¯ is 8 which is less than the cardinality of all other resolving neighborhoods and ∪i=13Ri=VH33, this implies that Rj¯∩∪i=13Ri≥Ri. Now, we construct a mapping ϑ:VH33⟶0,1 as(10)ϑuk=ϑvk−1=ϑvk=ϑuk+1=18.Since 8 is a very small cardinal number, there must exist a minimal resolving functionϑ′ such that ϑ′≤ϑ. Hence,(11)DimlfH33≤∑i=11218=32.Case (b): forn=5,1≤k≤5, by Lemma 2, we have(12)R1¯=ℛukvk=VH53−sr|r≡k+1,k+2mod5∪tr|r≡k+1,k+2mod5∪ur|r≡k+1,k+2mod5∪vr|r≡k+1,k+2mod5,R2¯=ℛukvk−1=VH53−sr|r≡k+2,k+3mod5∪tr|r≡k+3,k+4mod5∪ur|r≡k+3,k+4mod5∪vr|r≡k+2,k+3mod5,R1=ℛsktk=VH53−ur|r≡k+1,k+2mod5∪vr|r≡k+1,k+2mod5,R2=ℛsktk+1=VH53−ur|r≡kmod5,r≡k+4mod5∪vr|r≡k+3,k+4mod5,R3=ℛtkuk=VH53−sr|r≡k+1mod5,r≡k+3mod5∪tr|r≡k+1mod5,r≡k+4mod5,R4=ℛukuk+1=VH53−sr|r≡kmod5,∪tr|r≡k+3mod5∪ur|r≡k+3mod5∪vr|r≡kmod5.The cardinality of each of the local resolving neighborhoodsR1¯,R2¯,R1,R2,R3, and R4 for each 1≤k≤5 is 12, 12, 16, 16, 16, and 16, respectively. Since cardinality of R1¯ and R2¯ is 12 which is less than the cardinality of all other resolving neighborhoods and ∪i=14Ri=VH53, this implies that Rj¯∩∪i=14Ri≥Ri. Now, we construct a mapping ϑ:VH53⟶0,1 as(13)ϑuk=ϑvk−1=ϑvk=112.Since 12 is a very small cardinal number, there must exist a minimal resolving functionϑ′ such that ϑ′≤ϑ. Hence,(14)DimlfH53≤∑i=120112=53.Case (c): forn=7,1≤k≤7, by Lemma 2, we have(15)R1¯=ℛukvk=VH73−sr|r≡k+1,k+2,k+3mod7∪tr|r≡k+1,k+2,k+3mod7∪ur|r≡k+1,k+2,k+3mod7∪vr|r≡k+1,k+2,k+3mod7,R2¯=ℛukvk−1=VH73−sr|r≡k+3,k+4,k+5mod7∪tr|r≡k+4,k+5,k+6mod7∪ur|r≡k+4,k+5,k+6mod7∪vr|r≡k+3,k+4,k+5mod7,R1=ℛsktk=VH73−sr|r≡k+3mod7∪tr|r≡k+3mod7∪ur|r≡k+1,k+2,k+3mod7∪vr|r≡k+1,k+2,k+3mod7,R2=ℛsktk+1=VH73−sr|r≡k+3mod7∪tr|r≡k+5mod7∪ur|r≡kmod7,r≡k+5,k+6mod7∪vr|r≡k+4,k+5,k+6mod7.R3=ℛtkuk=VH73−sr|r≡k+1mod7,r≡k+5mod7∪tr|r≡k+1mod7,r≡k+6mod7,R4=ℛukuk+1=VH73−sr|r≡kmod7,∪tr|r≡k+4mod7∪ur|r≡k+4mod7∪vr|r≡kmod7.The cardinality of each of the local resolving neighborhoodsR1¯,R2¯,R1,R2,R3, and R4 for each 1≤k≤7 is 16, 16, 20, 20, 24, and 24, respectively. Since cardinality of R1¯ and R2¯ is 16 which is less than the cardinality of all other resolving neighborhoods and ∪i=14Ri=VH73, this implies that Rj¯∩∪i=14Ri≥Ri. Now, we construct a mapping ϑ:VH73⟶0,1 as(16)ϑuk=ϑvk−1=ϑvk=116.Since 16 is a very small cardinal number, there must exist a minimal resolving functionϑ′ such that ϑ′≤ϑ. Hence,(17)DimlfH73≤∑i=128116=74.Case (d): forn≥9,1≤k≤n, by Lemma 2, we have(18)R1¯=ℛukvk=VHn3−sr|r≡k+1,k+2,…,k−1+n+12modn∪tr|r≡k+1,k+2,…,k−1+n+12modn∪ur|r≡k+1,k+2,…,k−1+n+12modn∪vr|r≡k+1,k+2,…,k−1+n+12modn,R2¯=ℛukvk−1=VHn3−sr|r≡k+n−12,k+1+n−12,…,k+n−2modn∪tr|r≡k+1+n−12,k+2+n−12,…,k+n−1modn∪ur|r≡k+1+n−12,k+2+n−12,…,k+n−1modn∪vr|r≡k+n−12,k+1+n−12,…,k+n−2modn,R1=ℛsktk=VHn3−sr|r≡k+3,k+4,…,k+n+12−1modn,n≥7∪tr|r≡k+3,k+4,…,k+n+12−1modn,n≥7∪ur|r≡k+1,k+2,…,k+n+12−1modn,n≥5∪vr|r≡k+1,k+2,…,k+n+12−1modn,n≥5,R2=ℛsktk+1=VHn3−sr|r≡k+1+n−12,k+2+n−12,…,k+n−3modn,n≥7∪tr|r≡k+2+n−12,k+3+n−12,…,k+n−2modn,n≥7∪ur|r≡kmodn,r≡k+2+n−12,k+3+n−12,…,k+n−1modn∪vr|r≡k+1+n−12,k+2+n−12,…,k+n−1modn,R3=ℛtkuk=VHn3−sr|r≡k+1modn,r≡k+n−2modn∪tr|r≡k+1modn,r≡k+n−1modn,R4=ℛukuk+1=VHn3−sr|r≡kmodn∪tr|r≡k+n+32−1modn∪ur|r≡k+n+32−1modn∪vr|r≡kmodn.The cardinality of each of the local resolving neighborhoodsR1¯,R2¯,R1,R2,R3, and R4 for each 1≤k≤n is 2n+1,2n+1,2n+6,2n+6,4n−1, and 4n−1, respectively. Since cardinality of R1¯ and R2¯ is 2n+1 which is less than the cardinality of all other resolving neighborhoods and ∪i=14Ri=VHn3, this implies that Rj¯∩∪i=14Ri≥Ri. Now, we construct a mapping ϑ:VHn3⟶0,1 as(19)ϑuk=ϑvk−1=ϑvk=12n+1.Since2n+1 is a very small cardinal number, there must exist a minimal resolving function ϑ′ such that ϑ′≤ϑ. Hence,(20)DimlfHn3≤∑i=14n12n+1=2nn+1.Lemma 2.
LetHn4 be a hexagonal planar circular network for n≥3 with n≡1mod2.(1)
Then, for1≤k≤n, ℛk=ℛtkuk=ℛskuk=2n+1, and ∪k=1nℛk=4n.(2)
ℛk<ℛsktk and ℛsktk∩∪k=1nℛk≥ℛk.(3)
ℛk<ℛsktk+1 and ℛsktk+1∩∪k=1nℛk≥ℛk with tn+1=t1.(4)
ℛk<ℛukvk and ℛukvk∩∪k=1nℛk≥ℛk.(5)
ℛk<ℛukvk−1 and ℛukvk−1∩∪k=1nℛk≥ℛk with v0=vn.Theorem 3.
IfHn4 is a hexagonal planar circular network for n≥3 with n≡1mod2, then DimlfHnm≤4n/2n+1.


## 3.2. LFMD of Rotationally Symmetric Hexagonal Planar Networks with 4 Faces

In this section, we find the sequences of upper bounds of local fractional metric dimension of rotationally symmetric hexagonal planar networks with 4 faces.Lemma 3.
LetHn6 be a hexagonal planar circular network for n≥3 with n≡1mod2.(1)
Then, for1≤k≤n, ℛk=ℛskuk=2n, and ∪k=1nℛk=4n.(2)
ℛk<ℛsktk and ℛsktk∩∪k=1nℛk≥ℛk.(3)
ℛk<ℛsktk+1 and ℛsktk+1∩∪k=1nℛk≥ℛk with tn+1=t1.(4)
ℛk<ℛukvk and ℛukvk∩∪k=1nℛk≥ℛk.(5)
ℛk<ℛukvk−1 and ℛukvk−1∩∪k=1nℛk≥ℛk with v0=vn.(6)
ℛk<ℛtkuk and ℛtkuk∩∪k=1nℛk≥ℛk.(7)
ℛk<ℛskvk and ℛskvk∩∪k=1nℛk≥ℛk.Proof.
(1)
The set of local resolving neighborhoods ofskuk is ℛskuk=VHn6−sr|r≡k+1+n−1/2,k+2+n−1/2,…,k+n−1modn∪tr|r≡kmodn,r≡k+1+n−1/2,k+2+n−1/2,…,k+n−1modn∪ur|r≡k+1,k+2,…,k+n−1/2modn∪vr|r≡k,k+1,…,k−1+n+1/2modn. Clearly, ∪k=1nℛk=VHn6. Therefore, ∪k=1nℛk=4n.(2)
The set of local resolving neighborhoods ofsktk is ℛsktk=VHn6−ur|r≡kmodn∪vr|r≡k−n−1/2modn,r≡k+n−1modn. Hence, ℛsktk=4n−3. Since ℛk=2n<ℛsktk=4n−3, ℛsktk∩∪k=1nℛk=4n−3≥ℛk.(3)
The set of local resolving neighborhoods ofsktk+1 with tk+1=t1 is ℛsktk+1=VHn6−ur|r≡k+n−1/2+1modn. Hence, ℛsktk+1=4n−1. Since ℛk=2n<ℛsktk+1=4n−1, ℛsktk+1∩∪k=1nℛk=4n−1≥ℛk.(4)
The set of local resolving neighborhoods ofukvk is ℛukvk=VHn6−sr|r≡kmodn∪tr|r≡k+1modn,r≡k+n+1/2modn. Hence, ℛukvk=4n−3. Since ℛk=2n<ℛukvk=4n−3, ℛukvk∩∪k=1nℛk=4n−3≥ℛk.(5)
The set of local resolving neighborhoods ofukvk−1 with vn=v0 is ℛukvk−1=VHn6−sr|r≡k+n−1/2modn. Hence, ℛukvk−1=4n−1. Since ℛk=2n<ℛukvk−1=4n−1, ℛukvk−1∩∪k=1nℛk=4n−1≥ℛk.(6)
The set of local resolving neighborhoods oftkuk is ℛtkuk=VHn6−sr|r≡k,k+1,…,k−1+n+1/2modn∪tr|r≡k+1,k+2,…,k+n−1/2modn∪ur|r≡k+1+n−1/2,k+2+n−1/2,…,k+n−1modn∪vr|r≡k+n−1/2,k+1+n−1/2,…,k+n−2modn. Hence, ℛtkuk=2n+1. Since ℛk=2n<ℛtkuk=2n+1, ℛtkuk∩∪k=1nℛk=2n+1≥ℛk.(7)
The set of local resolving neighborhoods ofskvk is ℛskvk=VHn6−sr|r≡k+1,k+2,…,k+n−1/2modn∪tr|r≡k+2,k+3,…,k+n+1/2modn∪vr|r≡k+1+n−1/2,k+2+n−1/2,…,k+n−1modn∪ur|r≡kmodn,r≡k+1+n−1/2,k+2+n−1/2,…,k+n−1modn. Hence, ℛskvk=2n+1. Since ℛk=2n<ℛskvk=2n+1, ℛskvk∩∪k=1nℛk=2n+1≥ℛk.Theorem 4.
IfHn6 is a hexagonal planar circular network for n≥3 with n≡1mod2, then DimlfHn6≤2.Proof.
Whenn≥3 with n≡1mod2, then by Lemma 3, the local resolving neighborhoods are(21)R1¯=ℛskuk=VHn6−sr|r≡k+1+n−12,k+2+n−12,…,k+n−1modn∪⋅tr|r≡kmodn,r≡k+1+n−12,k+2+n−12,…,k+n−1modn∪⋅ur|r≡k+1,k+2,…,k+n−12modn∪vr|r≡k,k+1,…,k−1+n+12modn,R1=ℛsktk=VHn6−ur|r≡kmodn∪vr|r≡k−n−12modn,r≡k+n−1modn,R2=ℛsktk+1=VHn6−ur|r≡k+n−12+1modn,R3=ℛukvk=VHn6−sr|r≡kmodn∪tr|r≡k+1modn,r≡k+n+12modn,R4=ℛukvk−1=VHn6−sr|r≡k+n−12modn,R5=ℛtkuk=VH46−sr|r≡k,k+1mod4∪tr|r≡k+1,k+2mod4∪⋅vr|r≡k+2mod4∪ur|r≡k+2,k+3mod4,R6=ℛskvk=VHn6−sr|r≡k+1,k+2,…,k+n−12modn∪⋅tr|r≡k+2,k+3,…,k+n+12modn∪vr|r≡k+1+n−12,k+2+n−12,…,k+n−1modn∪ur|r≡kmodn,r≡k+1+n−12,k+2+n−12,…,k+n−1modn.
The cardinality of each of the local resolving neighborhoodsR1¯,R1,R2,R3,R4,R5, and R6 for each 1≤k≤n is 2n,4n−3,4n−1,4n−3,4n−1,2n+1, and 2n+1, respectively. Since cardinality of R1¯ is 2n which is less than the cardinality of all other resolving neighborhoods and ∪i=16Ri=VHn6, this implies that Rj¯∩∪i=16Ri≥Ri. Now, we construct a mapping ϑ:VHn6⟶0,1 as(22)ϑuk=ϑsk=12n.
Since2n is a minimal cardinal number, there must exist a minimal resolving function ϑ′ such that ϑ′≤ϑ. Hence,(23)DimlfHn6≤∑i=14n12n=2.Lemma 4.
LetHn7 be a hexagonal planar circular network for n≥5 with n≡1mod2.(1)
Then, for1≤k≤n, ℛk=ℛsktk=ℛsktk+1=ℛtkuk=ℛtkuk+1=2n+3 with tn+1=t1,un+1=u1 and ∪k=1nℛk=4n.(2)
ℛk<ℛukvk and ℛukvk∩∪k=1nℛk≥ℛk.(3)
ℛk<ℛukvk−1 and ℛukvk−1∩∪k=1nℛk≥ℛk with vn=v0.(4)
ℛk<ℛtktk+1 and ℛtktk+1∩∪k=1nℛk≥ℛk, tn+1=t1.Proof.
(1)
The set of local resolving neighborhoods ofsktk,sktk+1,tkuk and tkuk+1 with un+1=u1,tn+1=t1 is ℛsktk=VHn7−sr|r≡k+1,k+2,…,k+n+1/2−1modn∪tr|r≡k+1,k+2,…,k+n+1/2−1modn∪ur|r≡k+2,k+3,…,k+n+3/2−1modn∪vr|r≡k+2,k+3,…,k+n+1/2−1modn, ℛsktk+1=VHn7−sr|r≡k+1+n−1/2,k+2+n−1/2,…,k+n−1modn∪tr|r≡kmodn,r≡k+2+n−1/2,k+3+n−1/2,…,k+n−1modn∪ur|r≡kmodn,r≡k+2+n−1/2,k+3+n−1/2,…,k+n−1modn∪vr|r≡k+2+n−1/2,k+3+n−1/2,…,k+n−1modn, ℛtkuk=VHn7−sr|r≡k+n−1/2,k+1+n−1/2,…,k+n−2modn∪tr|r≡k+n−1/2+1,k+2+n−1/2,…,k+n−1modn∪ur|r≡k+1+n−1/2,k+2+n−1/2,…,k+n−1modn∪vr|r≡k+1+n−1/2,k+2+n−1/2,…,k+n−2modn, and ℛtkuk+1=VHn7−sr|r≡k+1,k+2,…,k+n+1/2−1modn∪tr|r≡k+1,k+2,…,k+n+1/2−1modn∪ur|r≡k+2,k+3,…,k+n+3/2−1modn∪vr|r≡k+2,k+3,…,k+n+1/2−1modn. Clearly, ∪k=1nℛk=VHn7. Therefore, ∪k=1nℛk=4n.(2)
The set of local resolving neighborhoods ofukvk is ℛukvk=VHn7−sr|r≡k+1,k+2,…,k+n−1/2−1modn∪tr|r≡k+1,k+2,…,k+n−1/2−1modn∪ur|r≡k+2,k+3,…,k+n+1/2−1modn∪vr|r≡k+2,k+3,…,k+n+1/2−1modn. Hence, ℛukvk=2n+5. Since ℛk=2n+3<ℛukvk=2n+5, ℛukvk∩∪k=1nℛk=2n+3≥ℛk.(3)
The set of local resolving neighborhoods ofukvk−1 is ℛukvk−1=VHn7−sr|r≡k+n−1/2,k+1+n−1/2,k+2+n−1/2,…,k+n−3modn∪tr|r≡k+n−1/2,k+1+n−1/2,…,k+n−2modn∪ur|r≡k+1+n−1/2,k+2+n−1/2,…,k+n−2modn∪vr|r≡k+n−1/2,k+1+n−1/2,…,k+n−3modn. Hence, ℛukvk−1=2n+5. Since ℛk=2n+3<ℛukvk−1=2n+5, ℛukvk−1∩∪k=1nℛk=2n+3≥ℛk.(4)
The set of local resolving neighborhoods oftktk+1, with vn+1=v1, is ℛtktk+1=VHn7−sr|r≡kmodn∪tr|r≡k+n+3/2−1modn∪ur|r≡k+1modn∪vr|r≡kmodn,r≡k+1modn,r≡k+n+3/2−1modn. Hence, ℛtktk+1=4n−6. Since ℛk=2n+3<ℛtktk+1=4n−6, ℛukvk−1∩∪k=1nℛk=2n+3≥ℛk.Theorem 5.
IfHn7 is a hexagonal planar circular network, then(24)DimlfHn7≤2,n=3,4n2n+3,n≥5.Proof.
The local resolving neighborhoods areCase (a): forn=3,1≤k≤3, by Lemma 4,(25)R1¯=ℛtktk+1=VH36−sr|r≡kmod3∪tr|r≡k+n+32−1mod3∪⋅ur|r≡k+1mod3∪vr|r≡k,k+1,k+n+32−1mod3,R1=ℛsktk=VH36−sr|r≡k+1mod3∪tr|r≡k+1mod3∪ur|r≡k+2mod3,R2=ℛsktk+1=VH36−sr|r≡k+2mod3∪tr|r≡kmod3∪ur|r≡kmod3,R3=ℛtkuk=VH36−sr|r≡k+1mod3∪tr|r≡k+2mod3∪ur|r≡k+2mod3,R4=ℛtkuk+1=VH36−sr|r≡k+1mod3∪tr|r≡k+1mod3∪ur|r≡k+2mod3,R5=ℛukvk=VH36−sr|r≡k+1mod3∪tr|r≡k+1,k+2mod3∪⋅ur|r≡k+2mod3∪vr|r≡k+2mod3,R6=ℛukvk−1=VH36−tr|r≡k+1mod3.The cardinality of each of the local resolving neighborhoodsR1¯,R1,R2,R3,R4,R5, and R6 for each k=1,2,3 is 6, 9, 9, 9, 9, 11, and 11, respectively. Since cardinality of R1¯ is 6 which is less than the cardinality of all other resolving neighborhoods and ∪i=16Ri=VH36, this implies that Rj¯∩∪i=16Ri≥Ri. Now, we construct a mapping ϑ:VH36⟶0,1 as(26)ϑtk=ϑtk+1=16.Since 6 is a minimal cardinal number, there must exist a minimal resolving functionϑ′ such that ϑ′≤ϑ. Hence,(27)DimlfH36≤∑i=11216=2.Case (b): forn≥5,1≤k≤5, by Lemma 8, we have(28)R1¯=ℛsktk=VHn7−sr|r≡k+1,k+2,…,k+n+12−1modn∪⋅tr|r≡k+1,k+2,…,k+n+12−1modn∪ur|r≡k+2,k+3,…,k+n+32−1modn∪vr|r≡k+2,k+3,…,k+n+12−1modn,R2¯=ℛsktk+1=VHn7−sr|r≡k+1+n−12,k+2+n−12,…,k+n−1modn∪⋅tr|r≡kmodn,r≡k+2+n−12k+3+n−12,…,k+n−1modn∪⋅ur|r≡kmodn,r≡k+2+n−12,k+3+n−12,…,k+n−1modn∪⋅vr|r≡k+2+n−12,k+3+n−12,…,k+n−1modn,R3¯=ℛtkuk=VHn7−sr|r≡k+n−12,k+1+n−12,…,k+n−2modn∪⋅tr|r≡k+n−12+1,k+2+n−12,…,k+n−1modn∪⋅ur|r≡k+1+n−12,k+2+n−12,…,k+n−1modn∪⋅vr|r≡k+1+n−12,k+2+n−12,…,k+n−2modn,R4¯=ℛtkuk+1=VHn7−sr|r≡k+1,k+2,…,k+n+12−1modn∪⋅tr|r≡k+1,k+2,…,k+n+12−1modn∪ur|r≡k+2,k+3,…,k+n+32−1modn∪vr|r≡k+2,k+3,…,k+n+12−1modn,R1=ℛukvk=VHn7−sr|r≡k+1,k+2,…,k+n−12−1modn∪⋅tr|r≡k+1,k+2,…,k+n−12−1modn∪ur|r≡k+2,k+3,…,k+n+12−1modn∪vr|r≡k+2,k+3,…,k+n+12−1modn,R2=ℛukvk−1=VHn7−sr|r≡k+n−12,k+1+n−12,k+2+n−12,…,k+n−3modn∪tr|r≡k+n−12,k+1+n−12,…,k+n−2modn∪⋅ur|r≡k+1+n−12,k+2+n−12,…,k+n−2modn∪vr|r≡k+n−12,k+1+n−12,…,k+n−3modn,R3=ℛtktk+1=VHn7−sr|r≡kmodn∪tr|r≡k+n+32−1modn∪ur|r≡k+1modn∪vr|r≡kmodn,r≡k+1modn,r≡k+n+32−1modn.The cardinality of each of the local resolving neighborhoodsR1¯,R2¯,R3¯,R4¯,R1,R2, and R3 for each 1≤k≤n is 2n+3,2n+3,2n+3,2n+3,2n+5, and 4n−6, respectively. Since cardinality of Ri¯,1≤i≤4 is 2n+3 which is less than the cardinality of all other resolving neighborhoods and ∪i=13Ri=VHn7, this implies that Rj¯∩∪i=13Ri≥Ri. Now, we construct a mapping ϑ:VHn7⟶0,1 as(29)ϑtk=ϑsk=ϑtk+1=ϑuk=ϑuk+1=12n+3.Since2n+3 is a minimal cardinal number, there must exist a minimal resolving function ϑ′ such that ϑ′≤ϑ. Hence,(30)DimlfHn7≤∑i=14n12n+3=4n2n+3.Lemma 5.
LetHn8 be a hexagonal planar circular network for n≥5 with n≡1mod2.(1)
Then, for1≤k≤n, ℛk=ℛukvk=8 and ∪k=1nℛk=4n with un+1=u1.(2)
ℛk<ℛsktk and ℛsktk∩∪k=1nℛk≥ℛk.(3)
ℛk<ℛtkuk and ℛtkuk∩∪k=1nℛk≥ℛk.(4)
ℛk<ℛtktk+1 and ℛtktk+1∩∪k=1nℛk≥ℛk, with tn+1=t1.(5)
ℛk<ℛtkvk and ℛtkvk∩∪k=1nℛk≥ℛk.(6)
ℛk<ℛukvk−1 and ℛukvk−1∩∪k=1nℛk≥ℛk, with vn=v0.(7)
ℛk<ℛsktk+1 and ℛsktk+1∩∪k=1nℛk≥ℛk, with tk+1=t1.Theorem 6.
IfHn8 is a hexagonal planar circular network, then(31)DimlfHn8≤2,n=3,n2,n≥5.Lemma 6.
LetHn9 be a hexagonal planar circular network for n≥3 with n≡1mod2.(1)
Then, for1≤k≤n, ℛk=ℛtkvk=4, and ∪k=1nℛk=4n.(2)
ℛk<ℛtkuk and ℛtkuk∩∪k=1nℛk≥ℛk.(3)
ℛk<ℛukvk−1 and ℛukvk−1∩∪k=1nℛk≥ℛk with vn=u0.(4)
ℛk<ℛukvk and ℛukvk∩∪k=1nℛk≥ℛk.(5)
ℛk<ℛsktk+1 and ℛsktk+1∩∪k=1nℛk≥ℛk, with tn+1=t1.(6)
ℛk<ℛsktk and ℛsktk∩∪k=1nℛk≥ℛk.(7)
ℛk<ℛtkuk+1 and ℛtkuk+1∩∪k=1nℛk≥ℛk, with un+1=u1.Proof.
(1)
The set of local resolving neighborhoods oftkvk is ℛtkvk=VHn9−sr|r≡k+1,k+2,…,k+n−modn∪tr|r≡k+1,k+2,…,k+n−2modn∪ur|r≡k,k+1,k+2,…,k+n−1modn∪vr|r≡k+1,k+2,…,k+n−1modn. Clearly, ∪k=1nℛk=VHn9. Therefore, ∪k=1nℛk=4n.(2)
The set of local resolving neighborhoods ofsktk+1 is ℛsktk+1=VHn9−vr|r≡k+1modn. Hence, ℛsktk+1=4n−1. Since ℛk=4<ℛsktk+1=4n−1, ℛsktk+1∩∪k=1nℛk=4n−1≥ℛk.(3)
The set of local resolving neighborhoods ofsktk is ℛsktk=VHn9−vr|r≡kmodn. Hence, ℛsktk=4n−1. Since ℛk=4<ℛsktk=4n−1, ℛsktk∩∪k=1nℛk=4n−1≥ℛk.(4)
The set of local resolving neighborhoods oftkuk is ℛtkuk=VHn9−sr|r≡kmodn,r≡k+n−1modn∪tr|r≡kmodn. Hence, ℛtkuk=4n−1. Since ℛk=4<ℛtkuk=4n−1, ℛtkuk∩∪k=1nℛk=4n−1≥ℛk.(5)
The set of local resolving neighborhoods oftkuk+1 is ℛtkuk+1=VHn9−vr|r≡kmodn. Hence, ℛtkuk+1=4n−1. Since ℛk=4<ℛtkuk+1=4n−1, ℛtkuk+1∩∪k=1nℛk=4n−1≥ℛk.(6)
The set of local resolving neighborhoods ofukvk−1 is ℛukvk−1=VHn9−vr|r≡k+n−2modn,r≡k+n−1modn∪tr|r≡k+n−1modn. Hence, ℛukvk−1=4n−3. Since ℛk=4<ℛukvk−1=4n−3, ℛukvk−1∩∪k=1nℛk=4n−3≥ℛk.(7)
The set of local resolving neighborhoods ofukvk is ℛukvk=VHn9−vr|r≡kmodn,r≡k+n−1modn∪tr|r≡k+n−1modn. Hence, ℛukvk=4n−3. Since ℛk=4<ℛukvk=4n−3, ℛukvk∩∪k=1nℛk=4n−3≥ℛk.Theorem 7.
IfHn9 is a hexagonal planar circular network for n≥3 with n≡1mod2, then DimlfHn9≤n.Proof.
Whenn≥3 with n≡1mod2, by Lemma 6, the local resolving neighborhoods are(32)R1¯=ℛtkvk=VHn9−sr|r≡k+1,k+2,…,k+n−2modn∪⋅tr|r≡k+1,k+2,…,k+n−2modn∪ur|r≡k,k+1,k+2,…,k+n−1modn∪⋅vr|r≡k+1,k+2,…,k+n−1modn,R1=ℛsktk+1=VHn9−vr|r≡k+1modn,R2=ℛsktk=VHn9−vr|r≡kmodn,R3=ℛtkuk=VHn9−sr|r≡kmodn,r≡k+n−1modn∪tr|r≡kmodn,R4=ℛtkuk+1=VHn9−vr|r≡kmodn,R5=ℛukvk−1=VHn9−vr|r≡k+n−2modn,r≡k+n−1modn∪tr|r≡k+n−1modn,R6=ℛukvk=VHn9−vr|r≡kmodn,r≡k+n−1modn∪tr|r≡k+n−1modn.
The cardinality of each of the local resolving neighborhoodsR1¯,R2¯,R1,R2,R3,R4,R5, and R6 for each 1≤k≤n is 4, 4n−1,4n−1,4n−1,4n−1,4n−1,4n−3, and 4n−3, respectively. Since cardinality of R1¯ is 4 which is less than the cardinality of all other resolving neighborhoods and ∪i=16Ri=VHn9, this implies that Rj¯∩∪i=16Ri≥Ri. Now, we construct a mapping ϑ:VHn9⟶0,1 as(33)ϑtk=ϑvk=14.
Since 4 is a minimal cardinal number, there must exist a minimal resolving functionϑ′ such that ϑ′≤ϑ. Hence,(34)DimlfHn9≤∑i=14n14=n.Lemma 7.
LetHn10 be a hexagonal planar circular network for n≥3 with n≡1mod2.(1)
Then, for1≤k≤n, ℛk=ℛtktk+1=ℛukvk−1=ℛukvk=ℛsktk+1=ℛsktk=ℛukuk+1=2n+2, with tn+1=t1, vn=v0, un+1=u1, and ∪k=1nℛk=4n.(2)
ℛk<ℛtkuk and ℛtkuk∩∪k=1nℛk≥ℛk.Theorem 8.
IfHn10 is a hexagonal planar circular network for n≥3 with n≡1mod2, then DimlfHn10≤2n/n+1.Lemma 8.
LetHn11 be a hexagonal planar circular network for n≥3 with n≡1mod2.(1)
Then, for1≤k≤n, ℛk=ℛtkuk=2n, and ∪k=1nℛk=4n.(2)
ℛk<ℛukvk−1 and ℛukvk−1∩∪k=1nℛk≥ℛk with vn=v0.(3)
ℛk<ℛukvk and ℛukvk∩∪k=1nℛk≥ℛk.(4)
ℛk<ℛsktk+1 and ℛsktk+1∩∪k=1nℛk≥ℛk, with tk+1=t1.(5)
ℛk<ℛsktk and ℛsktk∩∪k=1nℛk≥ℛk.(6)
ℛk<ℛtkvk and ℛtkvk∩∪k=1nℛk≥ℛk.(7)
ℛk<ℛskuk+1 and ℛskuk+1∩∪k=1nℛk≥ℛk, with uk+1=u1.Theorem 9.
IfHn11 is a hexagonal planar circular network for n≥3 with n≡1mod2, then DimlfHn11≤2.


## 3.3. LFMD of Rotationally Symmetric Hexagonal Planar Networks with 5 Faces

In this section, we find the sequences of upper bounds of local fractional metric dimension of rotationally symmetric hexagonal planar networks with 5 faces.Lemma 9.
LetHn12 be a hexagonal planar circular network for n≥5 with n≡1mod2.(1)
Then for1≤k≤n, ℛk=ℛsktk=ℛsktk+1=ℛukvk=ℛukvk−1=ℛskuk=ℛskuk+1=2n+1, with un+1=u1,tn+1=t1,vn=v0, and ∪k=1nℛk=4n.(2)
ℛk<ℛtkuk and ℛtkuk∩∪k=1nℛk≥ℛk.(3)
ℛk<ℛukuk+1 and ℛukvk∩∪k=1nℛk≥ℛk, with un+1=u1.Proof.
(1)
The set of local resolving neighborhoods ofsktk,sktk+1,ukvk,ukvk−1,skuk, and skuk+1 with uk+1=u1,tk+1=t1 is ℛsktk=VHn12−sr|r≡k+1+n−1/2,k+2+n−1/2,…,k+n−2modn∪tr|r≡k+1+n−1/2,k+2+n−1/2,…,k+n−2modn∪ur|r≡kmodn,r≡k+1+n−1/2,k+2+n−1/2,…,k+n−1modn∪vr|r≡kmodn,r≡k+1+n−1/2,k+2+n−1/2,…,k+n−1modn, ℛsktk+1=VHn12−sr|r≡k+2,k+3,…,k+n−1/2modn∪tr|r≡k+3,k+4,…,k+n+1/2modn∪ur|r≡k+1,k+2,…,k+n+1/2modn∪vr|r≡k,k+1,…,k+n−1/2modn, ℛukvk=VHn12−sr|r≡k+1,k+2,…,k+n−1/2modn∪tr|r≡k+1,k+2,…,k+n−1/2modn∪ur|r≡k+1,k+2,…,k+n−1/2modn∪vr|r≡k+1,k+2,…,k+n−1/2modn, ℛukvk−1=VHn12−sr|r≡k+n−1/2,k+1+n−1/2,…,k+n−2modn∪tr|r≡k+1+n−1/2,k+2+n−1/2,…,k+n−1modn∪ur|r≡k+1+n−1/2,k+2+n/2,…,k+n−1modn∪vr|r≡k+n−1/2,k+1+n−1/2,…,k+n−2modn, ℛskuk=VHn12−sr|r≡k+1,k+2,…,k+n−1/2modn∪tr|r≡kmodn,r≡k+2,k+3,…,k+n−1/2modn∪ur|r≡k+1,k+2,…,k+n−1/2modn∪vr|r≡k+1,k+2,…,k+n−1/2modn, and ℛskuk+1=VHn12−sr|r≡k+1+n−1/2,k+2+n−1/2,…,k+n−1modn∪tr|r≡k+1modn,r≡k+2+n−1/2,k+3+n−1/2,…,k+n−1modn∪ur|r≡kmodn,r≡k+2+n−1/2,k+3+n−1/2,…,k+n−1modn∪vr|r≡k+1+n−1/2,k+2+n−1/2,…,k+n−1modn. Clearly, ∪k=1nℛk=VHn12. Therefore, ∪k=1nℛk=4n.(2)
The set of local resolving neighborhoods oftkuk is ℛtkuk=VHn12−sr|r≡kmodn,r≡k+n−1modn∪tr|r≡k+1modn,r≡k+n−1modn. Hence, ℛtkuk=4n−1. Since ℛk=2n+1<ℛtkuk=4n−1, ℛtkuk∩∪k=1nℛk=4n−1≥ℛk.(3)
The set of local resolving neighborhoods ofukuk+1 with un+1=u1 is ℛukuk+1=VHn12−sr|r≡kmodn∪tr|r≡k+n+1/2modn∪ur|r≡k+n+1/2modn∪vr|r≡kmodn. Hence, ℛukuk+1=4n−1. Since ℛk=2n+1<ℛukuk+1=4n−1, ℛukuk+1∩∪k=1nℛk=4n−1≥ℛk.Theorem 10.
IfHn12 is a hexagonal planar circular network with n≡1mod2, then(35)DimlfHn12≤32,n=3,2nn+1,n≥5.Proof.
The local resolving neighborhoods areCase (a): forn=3,1≤k≤3, by Lemma 9, we have(36)R1¯=ℛsktk=VH312−ur|r≡k,k+2mod3∪vr|r≡k,k+2mod3,R2¯=ℛsktk+1=VH312−ur|r≡k+1,k+2mod3∪vr|r≡k,k+1mod3,R3¯=ℛukvk=VH312−sr|r≡k+1mod3∪tr|r≡k+1mod3∪⋅ur|r≡k+1mod3∪vr|r≡k+1mod3,R4¯=ℛukvk−1=VH312−sr|r≡k+1mod3∪tr|r≡k+2mod3∪⋅ur|r≡k+2mod3∪vr|r≡k+1mod3,R5¯=ℛskuk=VH312−sr|r≡k+1mod3∪tr|r≡kmod3∪ur|r≡k+1mod3∪vr|r≡k+1mod3,R6¯=ℛskuk+1=VH312−sr|r≡k+2mod3∪tr|r≡k+1mod3∪⋅ur|r≡kmod3∪vr|r≡k+2mod3,R1=ℛtkuk=VH312−sr|r≡k,k+2mod3∪tr|r≡k+1,k+2mod3,R2=ℛukuk+1=VH312−sr|r≡kmod3∪tr|r≡k+2mod3∪⋅ur|r≡k+2mod3∪vr|r≡kmod3.The cardinality of each of the local resolving neighborhoodsR1¯,R2¯,R3¯,R4¯,R5¯,R6¯,R1, and R2 for each k=1,2,3 is 8, 8, 8, 8, 8, 8, and 8, respectively. Now, we construct a mapping ϑ:VH312⟶0,1 as(37)ϑsk=ϑuk=ϑvk=ϑtk=ϑuk+1=ϑtk+1=ϑvk−1=18.Since 8 is a minimal cardinal number, there must exist a minimal resolving functionϑ′ such that ϑ′≤ϑ. Hence,(38)DimlfH312≤∑i=11218=32.Case (b): forn≥5,1≤k≤n, by Lemma 9, we have(39)R1¯=ℛsktk=VHn12−sr|r≡k+1+n−12,k+2+n−12,…,k+n−2modn∪⋅tr|r≡k+1+n−12,k+2+n−12,…,k+n−2modn∪⋅ur|r≡kmodn,r≡k+1+n−12,k+2+n−12,…,k+n−1modn∪⋅vr|r≡kmodn,r≡k+1+n−12,k+2+n−12,…,k+n−1modn,R2¯=ℛsktk+1=VHn12−sr|r≡k+2,k+3,…,k+n−12modn∪⋅tr|r≡k+3,k+4,…,k+n+12modn∪ur|r≡k+1,k+2,…,k+n+12modn∪vr|r≡k,k+1,…,k+n−12modn,R3¯=ℛukvk=VHn12−sr|r≡k+1,k+2,…,k+n−12modn∪⋅tr|r≡k+1,k+2,…,k+n−12modn∪ur|r≡k+1,k+2,…,k+n−12modn∪vr|r≡k+1,k+2,…,k+n−12modn,R4¯=ℛukvk−1=VHn12−sr|r≡k+n−12,k+1+n−12,…,k+n−2modn∪⋅tr|r≡k+1+n−12,k+2+n−12,…,k+n−1modn∪⋅ur|r≡k+1+n−12,k+2+n2,…,k+n−1modn∪⋅vr|r≡k+n−12,k+1+n−12,…,k+n−2modn,R5¯=ℛskuk=VHn12−sr|r≡k+1,k+2,…,k+n−12modn∪⋅tr|r≡kmodn,r≡k+2,k+3,…,k+n−12modn∪ur|r≡k+1,k+2,…,k+n−12modn∪vr|r≡k+1,k+2,…,k+n−12modn,R6¯=ℛskuk+1=VHn12−sr|r≡k+1+n−12,k+2+n−12,…,k+n−1modn∪⋅tr|r≡k+1modn,r≡k+2+n−12,k+3+n−12,…,k+n−1modn∪⋅ur|r≡kmodn,r≡k+2+n−12,k+3+n−12,…,k+n−1modn∪⋅vr|r≡k+1+n−12,k+2+n−12,…,k+n−1modn,R1=ℛtkuk=VHn12−sr|r≡kmodn,r≡k+n−1modn∪⋅tr|r≡k+1modn,r≡k+n−1modn,R2=ℛukuk+1=VHn12−sr|r≡kmodn∪tr|r≡k+n+12modn∪⋅ur|r≡k+n+12modn∪vr|r≡kmodn.The cardinality of each of the local resolving neighborhoodsR1¯,R2¯,R3¯,R4¯,R5¯,R6¯,R1, and R2 for each 1≤k≤n is 2n+1,2n+1,2n+1,2n+1,2n+1,2n+1,4n−1, and 4n−1, respectively. Since cardinality of Ri¯,1≤i≤6 is 2n+1 which is less than the cardinality of all other resolving neighborhoods and ∪i=12Ri=VHn12, this implies that Rj¯∩∪i=12Ri≥Ri. Now, we construct a mapping ϑ:VHn12⟶0,1 as(40)ϑsk=ϑuk=ϑvk=ϑtk=ϑuk+1=ϑtk+1=ϑvk−1=12n+1.Since2n+1 is a minimal cardinal number, there must exist a minimal resolving function ϑ′ such that ϑ′≤ϑ. Hence,(41)DimlfHn12≤∑i=14n12n+1=2nn+1.Lemma 10.
LetHn13 be a hexagonal planar circular network for n≥5 with n≡1mod2.(1)
Then, for1≤k≤n, ℛk=ℛtkuk=ℛtkuk+1=2n+1, with u1=un+1 and ∪k=1nℛk=4n.(2)
ℛk<ℛsktk and ℛkktk∩∪k=1nℛk≥ℛk.(3)
ℛk<ℛukvk and ℛukvk∩∪k=1nℛk≥ℛk.(4)
ℛk<ℛtktk+1 and ℛtktk+1∩∪k=1nℛk≥ℛk, with tn+1=t1.(5)
ℛk<ℛukuk+1 and ℛukuk+1∩∪k=1nℛk≥ℛk, with un+1=u1.(6)
ℛk<ℛsktk+1 and ℛsktk+1∩∪k=1nℛk≥ℛk, with t1=tn+1.(7)
ℛk<ℛukvk−1 and ℛukvk−1∩∪k=1nℛk≥ℛk, with v0=vn.Theorem 11.
IfHn13 is a hexagonal planar circular network with n≡1mod2, then(42)DimlfHn13≤2,n=3,2nn+1,n≥5.Lemma 11.
LetHn14 be a hexagonal planar circular network for n≥3 with n≡1mod2.(1)
Then, for1≤k≤n, ℛk=ℛtkuk=ℛskuk+1=ℛskvk=ℛtkvk=2n, with u1=un+1 and ∪k=1nℛk=4n.(2)
ℛk<ℛukvk and ℛukvk∩∪k=1nℛk≥ℛk.(3)
ℛk<ℛsktk and ℛsktk∩∪k=1nℛk≥ℛk.(4)
ℛk<ℛsktk+1 and ℛsktk+1∩∪k=1nℛk≥ℛk, with t1=tn+1.(5)
ℛk<ℛukvk−1 and ℛukvk−1∩∪k=1nℛk≥ℛk, with v0=vn.Theorem 12.
IfHn14 is a hexagonal planar circular network for n≥3 with n≡1mod2, then DimlfHn14≤2.Lemma 12.
LetHn15 be a hexagonal planar circular network for n≥3 with n≡1mod2.(1)
Then, for1≤k≤n, ℛk=ℛskuk+1=ℛtkvk=4 and ∪k=1nℛk=4n with un+1=u1.(2)
ℛk<ℛsktk and ℛsktk∩∪k=1nℛk≥ℛk.(3)
ℛk<ℛtkuk and ℛtkuk∩∪k=1nℛk≥ℛk.(4)
ℛk<ℛukvk and ℛukvk∩∪k=1nℛk≥ℛk.(5)
ℛk<ℛtkuk+1 and ℛtkuk+1∩∪k=1nℛk≥ℛk, with un+1=u1.(6)
ℛk<ℛsktk+1 and ℛsktk+1∩∪k=1nℛk≥ℛk, with t1=tn+1.(7)
ℛk<ℛukvk−1 and ℛukvk−1∩∪k=1nℛk≥ℛk with v0=vn.Proof.
(1)
The set of local resolving neighborhoods ofskuk+1 and tkvk with un+1=u1 is ℛskuk+1=VHn15−sr|r≡k+1,k+2,…,k+n−1modn∪tr|r≡k,k+1,…,k+n−1modn∪ur|r≡kmodn,r≡k+2,k+3,…,k+n−1modn∪vr|r≡k+2,k+3,…,k+n−1modn and ℛtkvk=VHn15−sr|r≡k+1,k+2,…,k+n−2modn∪tr|r≡k+1,k+2,…,k+n−1modnur|r≡k,k+1,…,k+n−1modnvr|r≡k+1,k+2,…,k+n−1modn. Clearly, ∪k=1nℛk=VHn15. Therefore, ∪k=1nℛk=4n.(2)
The set of local resolving neighborhoods ofsktk is ℛsktk=VHn15−ur|r≡k+1modn∪vr|r≡k+1modn. Hence, ℛsktk=22n−1. Since ℛk=4<ℛsktk=22n−1, ℛsktk∩∪k=1nℛk=22n−1≥ℛk.(3)
The set of local resolving neighborhoods oftkuk is ℛtkuk=VHn15−sr|r≡k+n−1modn∪vr|r≡kmodn. Hence, ℛtkuk=22n−1. Since ℛk=4<ℛtkuk=22n−1, ℛtkuk∩∪k=1nℛk=22n−1≥ℛk.(4)
The set of local resolving neighborhoods ofukvk is ℛukvk=VHn15−sr|r≡kmodn∪tr|r≡kmodn. Hence, ℛukvk=22n−1. Since ℛk=4<ℛukvk=22n−1, ℛukvk∩∪k=1nℛk=22n−1≥ℛk.(5)
The set of local resolving neighborhoods oftkuk+1, with un+1=u1 is ℛtkuk+1=VHn15−sr|r≡kmodn∪vr|r≡kmodn. Hence, ℛtkuk+1=22n−1. Since ℛk=4<ℛtkuk=22n−1, ℛtkuk∩∪k=1nℛk=22n−1≥ℛk.(6)
The set of local resolving neighborhoods ofsktk+1 with tn+1=t1 is ℛsktk+1=VHn15−ur|r≡k+1modn∪vr|r≡kmodn. Hence, ℛsktk+1=22n−1. Since ℛk=4<ℛsktk=22n−1, ℛsktk∩∪k=1nℛk=22n−1≥ℛk.(7)
The set of local resolving neighborhoods ofukvk−1 with vn−1=v0 is ℛukvk+1=VHn15−sr|r≡k+n−2modn∪tr|r≡k+n−1modn. Hence, ℛukvk+1=22n−1. Since ℛk=4<ℛukvk−1=22n−1, ℛukvk∩∪k=1nℛk=22n−1≥ℛk.Theorem 13.
IfHn15 is a hexagonal planar circular network for n≥3 with n≡1mod2, then DimlfHn15≤n.Proof.
Whenn≥3 with n≡1mod2, by Lemma 12, the local resolving neighborhoods are(43)R1¯=ℛskuk+1=VHn15−sr|r≡k+1,k+2,…,k+n−1modn∪tr|r≡k,k+1,…,k+n−1modn∪ur|r≡kmodn,r≡k+2,k+3,…,k+n−1modn∪vr|r≡k+2,k+3,…,k+n−1modn,R2¯=ℛtkvk=VHn15−sr|r≡k+1,k+2,…,k+n−2modn∪tr|r≡k+1,k+2,…,k+n−1modn∪ur|r≡k,k+1,…,k+n−1modn∪vr|r≡k+1,k+2,…,k+n−1modn,R1=ℛsktk=VHn15−ur|r≡k+1modn∪vr|r≡k+1modn,R2=ℛtkuk=VHn15−sr|r≡k+n−1modn∪vr|r≡kmodn,R3=ℛukvk=VHn15−sr|r≡kmodn∪tr|r≡kmodn,R4=ℛtkuk+1=VHn15−sr|r≡kmodn∪vr|r≡kmodn,R5=ℛsktk+1=VHn15−ur|r≡k+1modn∪vr|r≡kmodn,R6=ℛukvk−1=VHn15−sr|r≡k+n−2modn∪tr|r≡k+n−1modn.
The cardinality of each of the local resolving neighborhoodsR1¯,R2¯,R1,R2,R3,R4,R5, and R6 for each 1≤k≤n is 4, 4, 4n−2,4n−2,4n−2,4n−2,4n−2,4n−2, and 4n−2, respectively. Since cardinality of Ri¯,1≤i≤2 is 4 which is less than the cardinality of all other resolving neighborhoods and ∪i=16Ri=VHn15, this implies that Rj¯∩∪i=16Ri≥Ri. Now, we construct a mapping ϑ:VHn15⟶0,1 as(44)ϑsk=ϑuk+1=ϑvk=ϑtk=14.
Since 4 is a minimal cardinal number, there must exist a minimal resolving functionϑ′ such that ϑ′≤ϑ. Hence,(45)DimlfHn15≤∑i=14n14=n.Lemma 13.
LetHn16 be a hexagonal planar circular network for n≥3 with n≡1mod2.(1)
Then, for1≤k≤n, ℛk=ℛtkuk=ℛskvk=4 and ∪k=1nℛk=4n.(2)
ℛk<ℛsktk and ℛsktk∩∪k=1nℛk≥ℛk.(3)
ℛk<ℛukvk and ℛukvk∩∪k=1nℛk≥ℛk.(4)
ℛk<ℛskuk+1 and ℛskuk+1∩∪k=1nℛk≥ℛk, with un+1=u1.(5)
ℛk<ℛskuk and ℛskuk∩∪k=1nℛk≥ℛk.(6)
ℛk<ℛsktk+1 and ℛsktk+1∩∪k=1nℛk≥ℛk, with tn+1=t1.(7)
ℛk<ℛukvk−1 and ℛukvk−1∩∪k=1nℛk≥ℛk, with vn=v0.Theorem 14.
IfHn16 is a hexagonal planar circular network for n≥3 with n≡1mod2, then DimlfHn16≤n.Lemma 14.
LetHn17 be a hexagonal planar circular network for n≥3 with n≡1mod2.(1)
Then, for1≤k≤n, ℛk=ℛsktk=ℛsktk+1=ℛtkuk=ℛukvk=ℛukvk−1=ℛtk+1uk=2n+1, with t1=tn+1,u1=un+1,v0=vn, and ∪k=1nℛk=4n.(2)
ℛk<ℛtktk+1 and ℛtktk+1∩∪k=1nℛk≥ℛk with tn+1=t1.(3)
ℛk<ℛvktk+1 and ℛvktk+1∩∪k=1nℛk≥ℛk with tn+1=t1.Theorem 15.
IfHn17 is a hexagonal planar circular network for n≥3 with n≡1mod2, then DimlfHn17≤2n/n+1.


## 4. Table of Sequences of LFMD of Rotationally Symmetric Hexagonal Planar Networks and Their Comparison via Graphs

In this section, we propose the table of sequences of local fractional metric dimension of rotationally symmetric hexagonal planar networks and their comparison via plotting the graphs of these sequences (Table1).Table 1
Sequences of local fractional metric dimension of rotationally symmetric hexagonal planar networks.
NetworkUpper bound ofDimlfHn1,Hn2,Hn51Hn3,Hn10Hn172n/n+1Hn44n/2n+1Hn6,Hn11,Hn142Hn72,n=3;4n/2n+3,n≥5Hn82,n=3;n/2,n≥5Hn9,Hn15,Hn16nHn123/2,n=3;2n/n+1,n≥5Hn132,n=3;2n/n+1,n≥5The boundedness comparison of sequences of local farctional metric dimension of rotationally symmetric hexagonal planar networks are shown in Figures6–8.Figure 6
Graph of sequences of rotationally symmetric hexagonal planar networksHni,i=1,2,5,6,11,14, and their comparison.
(a)(b)(c)Figure 7
Graph of sequences of rotationally symmetric hexagonal planar networksHni,i=8,9,15,16, and their comparison.
(a)(b)(c)Figure 8
Graph of sequences of rotationally symmetric hexagonal planar networksHni,i=3,4,7,10,12,13,17, and their comparison.
(a)(b)(c)


## 5. Conclusion

In this paper, we discuss the local fractional metric dimension of rotationally symmetric hexagonal planar networks. The sequences of their upper bounds are investigated. The sequences of upper bounds of each rotationally symmetric hexagonal planar network are plotted for compression of their bounds. We can see easily that there are two types of sequences: bounded as well as unbounded. In future work, researchers can find domination and spectral number of rotationally symmetric hexagonal planar networks.



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