# Boundedness of Convex Polytopes Networks via Local Fractional Metric Dimension

Authors: Muhammad Javaid; Hassan Zafar; Amer Aljaedi; Abdulaziz Mohammad Alanazi
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Metric dimension is one of the distance-based parameter which is frequently used to study the structural and chemical properties of the different networks in the various fields of computer science and chemistry such as image processing, pattern recognition, navigation, integer programming, optimal transportation models, and drugs discovery. In particular, it is used to find the locations of robots with respect to shortest distance among the destinations, minimum consumption of time, and lesser number of the utilized nodes and to characterize the chemical compounds having unique presentation in molecular networks. The fractional metric dimension being a latest developed weighted version of the metric dimension is used in the distance-related problems of the aforementioned fields to find their nonintegral optimal solutions. In this paper, we have formulated the local resolving neighborhoods with their cardinalities for all the edges of the convex polytopes networks to compute their local fractional metric dimensions in the form of exact values and sharp bounds. Moreover, the boundedness of all the obtained results is also proved.


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## Body

## 1. Introduction

There are various important tools in the subject of graph theory that are used to model real life problems such as school bus routing, networks of communication, allocation of the frequencies to the radio stations, and social networks consisting of establishing and working relationships between people. Slater formally defined the concept of resolving (locating) sets for a connected network and designed an algorithm to compute them [1]. Harary and Melter also studied the resolving sets of a connected network, and they called the minimum cardinality of a resolving set as metric dimension (MD) [2]. Gary and Johnson proved that the formulating of MD in its general form for all the connected networks is an NP complete problem that leads to characterize the certain families of networks with fixed MD [3]. Tomescu and Melter applied a metric basis in digital geometry and derived some fundamental results about the existence of the rectangles having minimal metric bases of any size greater or equal to three [4].Chartrand et al. [5] used MD to solve integer programming problem (IPP) with specific conditions. They defined independence resolvent set and established the metric independence number for different families of connected networks. In particular, they also established bounds of the MD of unicyclic networks and proved that MD of a connected network G is 1 iff G≅Pn, where Pn (path). Metric dimension generalized Peterson multinetworks P2n,n are studied in [6], and bounded metric dimension of generalized Peterson network is computed in [7]. After that, local MD [8], strong MD [9], k-metric dimension [10], and sharp bounds of partition dimension of convex polytopes are studied [11] in the form of different mathematical expressions and computing algorithms. Moreover, constant MD of some generalized convex polytopes is studied by Zuo et al. [12], and unbounded MD of splitting networks Sℙn,Sℂn of a path and cycle is computed by Pan et al. [13]. For further study of MD and its applications, we refer [14–16].The fractional version of MD known as fractional metric dimension (FMD) is introduced by Currie and Oellermann, and they found the nonintegral solutions of IPP by using it [17]. Fahar et al. proposed the optimal solution of linear relaxation of the IPP by using FMD [18]. Arguman and Mathew formally defined the concept of FMD and introduced many basic properties of FMD about the certain families of the connected networks [19]. The bounds of FMD of all the connected networks are established by Alkhalidi et al. [20]. In particular, the FMDs of generalized Jahangir, trees, and unicyclic networks are computed in [21, 22]. Recently, Javaid et al. [23] characterized all the networks with FMD exactly one. The bounds of FMD of metal organic networks and cartesian product of connected networks are computed in [24, 25].Aisyah et al. [26] introduced the idea of local fractional metric dimension (LFMD), and they also calculated the exact values of LFMD for corona product of connected networks. For all the connected networks, Javaid et al. [27] established upper bound of LFMD. Furthermore, Javaid et al. [28] improved the lower bound of LFMD from unity and illustrated this result by computing the LFMD for the prism-related networks.Now, we present applications of metric-based dimensions in different fields. In chemistry, the various structures of chemical compounds are considered as the sets of functional groups, where atoms and bonds are presented by vertices and edges, respectively. By permuting the positions of the sets of functional groups, different common substructures are characterized. The concept of resolving sets determines the particular position when two compounds share the same functional groups. This investigation plays a supreme role in drug discovery and pharmaceutical activities. On the other hand, MD is an essential tool to operate movement of navigating agents (reboots) in large sets of landmarks where the resolving sets uniquely determine the positions of navigating agents in the graph space. The problem of minimum machines (robots) to be placed at certain nodes to trace each and every node exactly once is worth investigating in sonar and loran stations. For more details, we refer to [3, 29].In this paper, we computed LFMD of some families of convex polyposes networks in the form of exact values and sharp bounds. It is also obtained that in all the cases, LFMD of convex polytopes remains bounded. The remaining manuscript includes Sections2–4 consisting of preliminaries, main results, and conclusion, respectively.


## 2. Preliminaries

LetG=VG,EG be a network with VG as vertex-set and EG⊆VG×VG as edge-set. A path is a simple network in which vertices can be ordered such as two vertices are adjacent if and only of they are consecutive along the list. A network G is connected if each pair of vertices of G leads a path. For any two vertices u,v⊆VG, the distance du,v is the number of edges in a shortest path between them. For further study of preliminary concepts of the subject graph theory, see [30].For a connected networkG, a vertex x∈VG resolves u,v⊆VG if dx,u≠dx,v. Let U=u1,u2,u3,…,um⊆VG and u∈VG, then m-tuple representation of u with respect to U donated by ru|U is defined as ru|U=ru,u1,ru,u2,ru,u3,…,ru,um. If the distinct vertices of G have different representations with respect to U, then U is called a resolving set of G. The resolving sets of minimum cardinalities are called metric bases in G, and the cardinality of a metric basis is known as metric dimension (MD) of G that is defined as follows:(1)dimG=minU:Uis are solving set ofG2.For each pair of two adjacent verticesu,v∈VG, the local resolving neighborhood (LRN) set is defined as(2)LRuv=a∈VG:da,u≠da,v.A local resolving function (LRF) is a real-valued functionϜ:VG⟶0,1 such that ϜLRuv≥1 for each LRuv of G, where ϜLRuv=∑x∈LRuvϜx. A LRF Ϝ of G is called minimal if there exists some other function Ϝ′:VG⟶0,1 such that Ϝ′≤℧ and ℧k≠Ϝk for at least one u∈VG that is not LRF of G. If Ϝ presents all the minimal resolving functions of G, then its LFMD is defined as dimlfG=minϜ, where Ϝ=∑u∈VGϜu [26].The networks of convex polytopes are fundamental geometric objects that have been investigated since antiquity. They are playing an important role in mathematical subjects like algebraic geometry, combinatorial optimization, and graph theory. For a connected networkG, Baca et al. [29] constructed the convex polytopes network Rm by the combination of prism and antiprism networks, where m≥5. It consists of inner ui:1≤i≤m, middle vi:1≤i≤m, and outer wi:1≤i≤m cycle vertices with VRm=3m and ERm=uiui+1,vivi+1,wiwi+1,uivi,ui+1vi,viwi:1≤i≤m−1,i+1≅1modm. For more details, see Figure 1.Figure 1
Convex polytopeRm.A network of convex polytopeAm is obtained from Rm by inserting new edges viwi+1, where 1≤i≤m−1, i+1≅1modm, and VAm=3m (see Figure 2). A network of convex polytope Sm is formed by connecting the alternating bands of triangles of two copies of prism networks. It consists of inner ui, middle vi&wi, and outer zi cycle vertices with 1≤i≤m and VSm=4m. The edge-set of Sm is defined as ESm=uiui+1,vivi+1,wiwi+1,zizi+1,uivi,viwi,vi+1wi,wizi:1≤i≤m−1,i+1≅1modm. For more details, see Figure 3. The network of convex polytope Qm is obtained from Sm by deleting edges wiwi+1 as shown in Figure 4.Figure 2
Convex polytopeAm.Figure 3
Convex polytopeSm.Figure 4
Convex polytopeQm.Now, we define some important results which will be frequently used in the main results as follows.Theorem 1 (see [27]).
For connected networkG=VG,EG and the local resolving neighborhood (LRN) set LRe of the edge e, if LRe∩A≥γ,∀e∈EG, then(3)1≤dimlfG≤Aγ,where 2≤γ≤VG, γ=minLRe:e∈EG, and A=∪LRe:LRe=γ.Theorem 2 (see [28]).
LetG=VG,EG be a connected network with VG be the vertex-set and LRe be the LRN set for the edge e of G. Then,(4)VGΓ≤dimlfG,where 2≤Γ≤VG and Γ=maxLRe:e∈EG.Corollary 1 (see [28]).
LetG=VG,EG be a connected network and LRe be LRN of e∈EG. If γ=δ and X=VG, then(5)dimlfG=VGδ,δ=maxLRe:e∈EG, γ=minLRe:e∈EG, and X=∪LRe:LReγ.


## 3. Main Results

In this section, we investigate boundedness of LFMD of different convex polytopes networks.


### 3.1. LRN Sets and LFMD of Convex PolytopeRm

This subsection covers the results related to the LRN sets and LFMD of the convex polynope network that is presented byRm for any integral value m≥5.Table 1
Cardinalities of the LRN sets ofRm for m≅1mod2.
Cardinalities of LRN setsComparisonLRwiwi+1=3m+33m>3m−3>3m+3/2LRvivi+1=3m+33m>3m−3>3m+3/2LRuiui+1=3m+33m>3m−3>3m+3/2Table 2
Cardinalities of LRN sets ofRm with m≅0mod2.
Cardinalities of LRN setsComparisonLRwiwi+1=3m/23m>3m−2>3m/2LRvivi+1=3m/23m>3m−2>3m/2LRuiui+1=3m/23m>3m−4>3m/2Lemma 1.
LetRm be a network of convex polytope, where m≥5 and m≅1mod2. Then, for 1≤i≤m,(a)
LRei=LRuivi=LRviui+1=3m+3/2, LRviwi=3m, and ∪i=1mLRei=3m(b)
LRei≤LRx≤LRviwi, ∀x∈ERm(c)
LRx∩∪i=1mLRei≥LRei,∀x∈ERmProof.
Assume thatui, vi, and wi are inner, middle, and outer vertices of Rm, respectively, where 1≤i≤m and m+1≅1modm.(a)
SinceLRuivi=ui,um+2i+1/2,um+2i+3/2,…,um,vi,vi+1,vi+2,…,vm+1/2,wi,wi+1,wi+2,…,wm+1/2 and LRuivi+1=ui+1,ui+2,ui+3,…,um+2i+1/2,vi,vm+2i+1/2,vm+2i+3/2,vm+2i+5/2,…,vm,wi,wm+2i+1/2,wm+2i+3/2,wm+2i+5/2,…,wm, therefore LRei=3m+3/2. Also, LRviwi=VRm, which implies that LRviwi=3m. Now, ∪i=1mLRei=VRm=3m.(b)
The LRN sets other thanLRei and LRviwi are LRwiwi+1=VRm−ui+1,vm+2i+1/2,wm+2i+1/2, LRvivi+1=VRm−ui+2,vm+2i+1/2,wm+2i+1/2, and LRuiui+1=VRm−um+2i+1/2,vi,wi.Now, we arrange the cardinalities of these LRN sets in Table1.From Table1, it is clear that LRei≤LRx≤LRviwi, ∀x∈EG with 1≤i≤m.(c)
Since∪i=1mLRei=VRm, therefore LRx∩LRei=LRx≥LRei.Lemma 2.
LetRm with m≥6 be a network of convex polytope, where m≅0mod2. Then,(a)
LRei=LRviui+1=LRuivi=3m/2 and LRviwi=3m with ∪i=1mLRei=VRm(b)
LRei≤LRx≤LRviwi, ∀x∈ERm(c)
LRx∩∪i=1mLRei≥LRei,∀x∈ERmProof.
Assume thatui, vi, and wi are inner, middle, and outer vertices, respectively, of Rm, where 1≤i≤m and m+1≅1modm.(a)
ConsiderLRuivi+1=ui+1,ui+2,ui+3,…,um+2i/2,vi,vm+2i+2/2,vm+2i+4/2,vm+2i+6/2,…,vm,wi,wm+2i/2,wm+2i+2/2,wm+2i+4/2,…,wm, LRuivi=ui,um+2i+2/2,um+2i+4/2,um+2i+6/2,…,um,vi,vi+1,vi+2,….,vm/2,vm,wi,wi+1,wi+2,….,wm/2 therefore LRei=3m/2, and ∪i=1mLRei=3m. Also, LRviwi=VRm which implies that LRviwi=VRm=3m. Furthermore, ∪i=1mLRei=VRm=3m.(b)
Other LRN sets areLRwiwi+1=VRm−ui+1,um+2i+2/2, LRvivi+1=VRm−ui+1,um+2i+2/2, and LRuiui+1=VRm−vm+2i/2,vi,wi,wm+2i/2.The cardinalities of these LRN sets are obtained as given in Table2.From Table2, it is clear that LRei≤LRx≤LRviwi, ∀x∈ERm.(c)
Since∪i=1mLRei=VRm, therefore LRx∩∪i=1mLRei=LRx≥LRei.Theorem 3.
LetRm with m≥5 be a network of convex polytope, where m≅1mod2. Then,(6)1<dimflRm≤2mm+1.Proof.
To prove the result, we have following cases:Case 1. For m=5, the LRN sets are as follows:(7)LR1=LRu1u2=VR5−u4,v1,w1,LR2=LRu2u3=VR5−u5,v2,w2,LR3=LRu3u4=VR5−u1,v3,w3,LR4=LRu4u5=VR5−u2,v4,w4,LR5=LRu5u1=VR5−u3,v5,w5,LR6=LRw1w2=VR5−u2,v2,w4,LR7=LRw2w3=VR5−u3,v3,w5,LR8=LRw3w4=VR5−u4,v4,w1,LR9=LRw4w4=VR5−u5,v5,w2,LR10=LRw4w5=VR5−u1,v1,w3,LR11=LRv1v2=VR5−v4,w4,LR12=LRv2v3=VR5−v5,w5,LR13=LRv3v4=VR5−v1,w1,LR14=LRv4v5=VR5−v2,w2,LR15=LRv5v1=VR5−v3,w1,LR16=LRv1w1=VR5,LR17=LRv2w2=VR5,LR18=LRv3w3=VR5,LR19=LRv4w4=VR5,LR20=LRv5w5=VR5,LR21=LRv1u2=VR5−v3,v2,u1,u5,w2,w3,LR22=LRv2u3=VR5−v3,v4,u1,u5,w2,w3,LR23=LRv3u4=VR5−v5,v4,u1,u5,w2,w3,LR24=LRv4u5=VR5−v1,v5,u1,u5,w2,w3,LR25=LRv5u1=VR5−v1,v2,u1,u5,w2,w3,LR26=LRu1v1=VR5−u2,u3,v2,v3,w4,w5,LR27=LRu2v2=VR5−u3,u4,v5,v1,w5,w1,LR28=LRu3v3=VR5−u4,u5,v1,v2,w3,w2,LR29=LRu4v4=VR5−u5,u1,v2,v3,w2,w1,LR30=LRu5v5=VR5−u1,u2,v3,v4,w3,w2.As, for1≤i≤5, LRei=LRuiui=LRviui+1=9, LRei≤LRx, ∀x∈ER5. Furthermore, ∪i=15LRviui+1=15 and LRx∩∪i=15LRei=9,∀x∈ER5. There exists an upper LRF ϕ:VR5⟶0,1 defined as ϕv=1/9,∀v∈VR5. Consequently, by Theorem 1, dimflR5≤5/3. Similarly, for 1≤i≤5, LRviwi=15 and LRviwi≥LRx, ∀x∈ER5. Thus, there exists a lower LRF such that ϕ′:VR5⟶0,1 defined as ϕ′v=1/15,∀v∈VR5. Consequently, by Theorem 2, dimflR5≥1. Since R5 is a nonbipartite network, therefore dimflR5 must be greater than 1. Hence, we have(8)1<dimflR5≤53.Case 2. For m>5, by Lemma 1, LRei=3m+3/2 and LRei≤LRx, ∀x∈ERm. Similarly, LRuiwi=3m and LRuiwi≥LRx, ∀x∈ERm. Therefore, ϕ:VRm⟶0,1 is defined by ϕv=2/3m+3,∀v∈VRm is an upper LRF and ϕ′:VRm⟶0,1 is defined by ϕ′v=1/3m,∀v∈VRm is a lower LRF. Moreover, Rm is a nonbipartite network. Therefore, by Theorems 1 and 2, we have(9)1<dimflRm≤2mm+1.Theorem 4.
LetRm with m≥6 be a network of convex polytope, where m≅0mod2. Then,(10)1<dimflRm≤2.Proof.
To prove the result, we have following two cases:Case 1. For m=6, the LRN sets are as follows:(11)LR1=LRu1u2=VR6−v1,v4,w1,w4,LR2=LRu2u3=VR6−v5,v2,w2,w5,LR3=LRu3u4=VR6−v3,v6,w3,w6,LR4=LRu4u5=VR6−v4,v1,w4,w1,LR5=LRu5u6=VR6−v5,v2,w5,w2,LR6=LRu6u1=VR6−v6,v3,w6,w3,LR7=LRw1w2=VR6−u2,u5,w2,w5,LR8=LRw2w3=VR6−u3,u6,w3,w6,LR9=LRw3w4=VR6−u4,u1,w4,w1,LR10=LRw4w5=VR6−u5,u2,w5,w2,LR11=LRw5w6=VR6−u6,u3,w6,w3,LR12=LRw6w1=VR6−u1,u4,w1,w4,LR13=LRv1v2=VR6−u2,u5,LR14=LRv2v3=VR6−u3,u6,LR15=LRv3v4=VR6−u4,u1,LR16=LRv4v5=VR6−u5,u2,LR17=LRv5v6=VR6−u6,u3,LR18=LRv6v1=VR6−u1,u4,LR19=LRv1w1=VR6,LR20=LRv2w2=VR6,LR21=LRv3w3=VR6,LR22=LRv4w4=VR6,LR23=LRv5w5=VR6,LR24=LRv6w6=VR6,LR25=LRu1v1=u1,u5,u6,v1,v2,v3,w1,w2,w3,LR26=LRu2v2=u2,u6,u1,v2,v3,v4,w2,w3,w4,LR27=LRu3v3=u3,u1,u2,v3,v4,v5,w3,w4,w5,LR28=LRu4v4=u4,u2,u3,v4,v5,v6,w4,w5,w6,LR29=LRu5v5=u5,u3,u4,v5,v6,v1,w5,w6,w1,LR30=LRu6v6=u6,u4,u5,v6,v1,v2,w6,w1,w2,LR31=LRv1u2=u2,u3,u4,v1,v5,v6,w1,w5,w6,LR32=LRv2u3=u3,u4,u5,v1,v2,v6,w1,w2,w6,LR33=LRv3u4=u4,u5,u6,v1,v2,v3,w1,w3,w5,LR34=LRv4u5=u5,u6,u1,v2,v3,v4,w4,w2,w3,LR35=LRv5u6=u6,u1,u2,v3,v4,v5,w3,w4,w5,LR36=LRv6u1=u1,u2,u3,v4,v5,v6,w4,w5,w6.For1≤i≤6, LRei=LRuiui=LRviui+1=9 and LRei≤LRx, ∀x∈ER6. Furthermore, ∪i=16LRei=18 and LRx∩∪i=16LRei=9, ∀x∈ER6. Therefore, there exists a lower LRF ϕ:VR6⟶0,1 defined as ϕv=1/9 for each v∈VRm. Consequently, by Theorem 1, dimflR6≤2.Similarly, for1≤i≤6, LRviwi=18 and LRviwi≥LRx, ∀x∈ER6. Thus, there exists an upper LRF ϕ′:VR6⟶0,1 defined as ϕv=1/18 for each v∈VRm. Consequently, by Theorem 2, dimflR6≥1. Furthermore, R6 is a nonbipartite network, so dimflR6 must be greater than 1. Hence, we have(12)1<dimflR6≤2.Case 2. For m>6, by Lemma 2, LRei=3m/2 for 1≤i≤m and LRei≤LRx, ∀x∈ERm. Therefore, ϕ:VRm⟶0,1 defined by ϕv=2/3m, ∀v∈VRm is an upper LRF. Thus, by Theorem 1, the dimflRm≤2. Similarly, as LRviwi=3m and LRviwi≥LRx, ∀x∈ERm, where 1≤i≤m. Therefore, ϕ′:VRm⟶0,1 defined by ϕ′v=1/3m,∀v∈VRm is a lower LRF. Thus, by Theorem 2, dimflRm≥1. Moreover, as Rm is a nonbipartite network, it implies that dimflRm must be greater than 1. Hence, we have(13)1<dimflRm≤2.


### 3.2. LRN Sets and LFMD of Convex PolytopeAm

This subsection deals with the results for the LRN sets and LFMD of the convex polytopes networkAm with m≥5.Table 3
Cardinalities of LRN sets ofAm for m≅1mod2.
Cardinalities of LRN setsComparisonLRwiwi+1=3m−53m>3m−5>3m+3/2LRuiui+1=3m−53m>3m−5>3m+3/2Table 4
Cardinalities of LRN sets ofAm for m≅0mod2.
Cardinalities of LRN setsComparisonLRwiwi+1=3m−43m>3m−4>3m+2/2LRuiui+1=3m−43m>3m−4>3m+2/2Lemma 3.
LetAm be a network of convex polytope, where m≥5 and m≅1mod2. Then, for 1≤i≤m,(a)
LRei=LRviui=LRviui+1=LRviwi+1==LRviwi=3m+3/2 and LRvivi+1=3m−3 with ∪i=1mLRei=VRm=3m(b)
LRei≤LRx≤LRvivi+1, ∀x∈EAm(c)
LRx∩∪i=1mLRei≥LRei,∀x∈EAmProof.
Assume thatui, vi, and wi are inner, middle, and outer vertices of Am, respectively, where 1≤i≤m and m+1≅1modm.(a)
SinceLRviui+1=ui+1,ui+2,ui+3,…,um+2i+1/2,vi,vm+2i+1/2,vm+2i+3/2,…,vm,wi,wi+1,wi+2,…,wm+2i+1/2,wm+2i+5/2,…,wm, LRviwi=ui,ui+1,ui+2,ui+3,…,um+2i−1/2,vi,vi+1,vi+2,…,vm+2i−1/2,wi,wm+2i+1/2,wm+2i+3/2,…,wm, LRuivi=ui,um+2i+1/2,um+2i+3/2,…,um,vi,vi+1,vi+2,….,vm+2i−1/2,wi,wi+1,wi+2,….,wm+2i−1/2, and LRviwi+1=ui,ui+1,um+6i−2/2,um+6i−2/2+1,…,um,vi,vm+6i−2/2,vm+6i−2/2+1,…,vmwi+1,wi+2,wi+3,….,wm+6i−4/2, therefore LRei=LRviui=LRviui+1=LRviwi+1=LRviwi=3m+3/2 with ∪i=1mLRei=VRm=3m. Also, =LRvivi+1=VAm−ui,vm+2i+1/2,wi; therefore, LRvivi+1=VRm−3=3m−3.(b)
The LRN sets of the otherLRei and LRvivi+1 are LRwiwi+1=VAm−ui,ui+1,um+2i−1/2,vi,wm+2i−1/2 and LRuiui+1=VAm−ui,ui+1,um+2i+1/2,vi,wm+2i+1/2.Now, we arrange the cardinalities of these LRN set in Table3.From Table3, it is clear that LRei≤LRx≤LRvivi+1, ∀x∈EAm.(c)
Since∪i=1mLRei=VAm, therefore LRx∩∪i=1mLRei=LRx≥LRei.Lemma 4.
LetAm be a network of convex polytope, where m≥6 and m≅0mod2. Then, for 1≤i≤m,(a)
LRei=LRviui=LRviui+1=LRviwi+1=LRviwi=3m+2/2 and LRvivi+1=3m−3 with ∪i=1mLRei=3m(b)
LRei≤LRx≤LRvivi+1,∀x∈EAm(c)
LRx∩∪i=1mLRei=LRx≥LRei,∀x∈EAmProof.
Assume thatui, vi, and wi are inner, middle, and outer vertices of Am, respectively, where 1≤i≤m and m+1≅1modm.(a)
ConsiderLRviui+1=ui+1,ui+2,ui+3,…,um+2i/2,vi,vm+6i−2/2,vm+6i−2/2+1,…,vm,wi,wi+1,wi+2,…,wm+6i−2/2,wm+6i−2/2+1,…,wm, LRviwi=ui,ui+1,ui+2,ui+3,…,um+4i−2/2,vi,vi+1,vi+2,…,vm+2i−2/2,wi,wm+6i−2/2,wm+6i/2,…,wm, LRuivi=ui,um+6i−2/2,um+6i−2/2+1,…,um,vi,vi+1,vi+2,….,vm+2i−2/2,wi,wi+1,wi+2,….,wm+2i−2/2, LRviwi+1=ui,ui+1,um+6i−2/2,um+6i/2,…,um,vi,vm+6i−2/2,vm+6i/2,…,vm,wi+1,wi+2,wi+3,….,wm+6i−4/2, and LRvivi+1=VAm−ui+1,wi+2,wm+6i−2/2. Furthermore, LRei=3m+2/2 and LRvivi+1=3m−3 with ∪i=1mLRei=VAm=3m.(b)
The LRN sets ofAm other than LRei and LRvivi+1 are LRwiwi+1=VAm−ui,ui+1,vi,vm+2i/2 and LRuiui+1=VAm−vm+4i−2/2,vi,wi,wi+1. Now, we arrange cardinalities of these LRN set in Table 4.From Table4, it is clear that LRei≤LRx≤LRvivi+1,∀x∈EAm.(c)
Since∪i=1mLRei=VAm, therefore LRx∩∪i=1mLRei=LRx≥LRei.Theorem 5.
LetAm with m≥5 be a network of convex polytope, where m≅1mod2. Then,(14)mm−1≤dimflAm≤2mm+1.Proof.
To prove the result, we have following cases:Case 1. For m=5, the LRN sets are as follows:(15)LR1=LRu1v1=VA5−u2,u3,v4,v5,w4,w5,LR2=LRu2v2=VA5−u3,u4,v1,v2,w1,w2,LR3=LRu3v3=VA5−u5,u1,v3,v4,w3,w4,LR4=LRu4v4=VA5−u2,u3,v5,v1,w5,w1,LR5=LRu5v5=VA5−u4,u5,v2,v3,w2,w3,LR6=LRv1u2=VA5−u1,u2,v2,v3,w3,w4,LR7=LRv2u3=VA5−u2,u3,v3,v4,w4,w5,LR8=LRv3u4=VA5−u3,u4,v4,v5,w5,w1,LR9=LRv4u5=VA5−u4,u5,v5,v1,w1,w2,LR10=LRv5u1=VA5−u5,u1,v1,v2,w2,w3,LR11=LRv1w1=VA5−u4,u5,v4,v5,w2,w3,LR12=LRv2w2=VA5−u5,u1,v5,v1,w3,w4,LR13=LRv3w3=VA5−u1,u2,v1,v2,w4,w5,LR14=LRv4w4=VA5−u2,u3,v2,v3,w5,w1,LR15=LRv5w5=VA5−u3,u4,v3,v4,w1,w5,LR16=LRv1w2=VA5−u3,u4,v2,v3,w1,w5,LR17=LRv2w3=VA5−u4,u5,v3,v4,w2,w1,LR18=LRv3w4=VA5−u5,u1,v4,v5,w3,w2,LR19=LRv4w5=VA5−u1,u2,v5,v1,w4,w3,LR20=LRv5w1=VA5−u2,u3,v1,v2,w5,w4,LR21=LRu1u2=VA5−u4,v1,w1,w2,w4,LR22=LRu2u3=VA5−u5,v2,w2,w3,w5,LR23=LRu3u4=VA5−u1,v3,w1,w3,w4,LR24=LRu4u5=VA5−u2,,v4,w1,w4,w2,LR25=LRu5u1=VA5−u3,v5,w5,w1,w3,LR26=LRw1w2=VA5−u1,u2,u4,v1,w4,LR27=LRw2w3=VA5−u2,u3,u5,v2,w5,LR28=LRw3w4=VA5−u3,u4,u1,v3,w1,LR29=LRw4w5=VA5−u4,u5,u2,v4,w2,LR30=LRw5w1=VA5−u5,u1,u3,v5,w3,LR31=LRv1v2=VA5−u2,w2,v4,LR32=LRv2v3=VA5−u3,w3,v5,LR33=LRv3v4=VA5−u4,w4,v1,LR34=LRv4v5=VA5−u5,w5,v2,LR35=LRv5v1=VA5−u6,w6,v3.For1≤i≤5, LRei=LRviwi=LRviwi+1=LRuivi=LRviui+1=9 and LRei≤LRx,∀x∈EA5. Furthermore, ∪i=15LRei=15 and LRx∩∪i=15LRei=9,∀x∈EA5 There exists an upper LRF ϕ:VA5⟶0,1 defined as ϕv=1/9, ∀v∈VA5. Consequently, by Theorem 1, dimflA5≤5/3. Similarly, for 1≤i≤5, LRvivi+1=12, LRvivi+1≥LRx,∀x∈EA5. Thus, there exist a lower LRF such that ϕ′:VA6⟶0,1 is defined as ϕ′v=1/12,∀v∈VA5. Consequently, by Theorem 2, dimflA5≤5/4. Hence, we have(16)54≤dimflA5≤53.Case 2. For m>5, by Lemma 3, LRei=LRviwi=LRviwi+1=LRuivi=LRviui+1=3m+3/2 and LRei≤LRx,∀x∈EA5, ∀x∈EAm. Similarly, LRvivi+1=3m−3 and LRvivi+1≥LRx, ∀x∈EAm. Therefore, ϕ:VAm⟶0,1 is defined by ϕv=2/3m+3, ∀v∈VAm is an upper LRF and ϕ′:VAm⟶0,1 is defined by ϕ′v=1/3m−3,∀v∈VAm is a lower LRF. Therefore, by Theorems 1 and 2, we have(17)mm−1≤dimflAm≤2mm+1.Theorem 6.
LetAm with m≥6 be a network of convex polytope, where m≅0mod2. Then,(18)mm−1≤dimflAm≤6m3m+2.Proof.
To prove the result, we have following cases:Case 1. For m=6, the LRN sets are as follows:(19)LR1=LRu1v1=VA6−u2,u3,u4,v4,v5,v6,w5,w6,LR2=LRu2v2=VA6−u2,u3,u4,v5,v6,v1,w6,w1,LR3=LRu3v3=VA6−u3,u4,u5,v6,v1,v2,w1,w2,LR4=LRu4v4=VA6−u4,u5,u6,v1,v2,v3,w2,w3,LR5=LRu5v5=VA6−u5,u6,u1,v2,v3,v4,w3,w4,LR6=LRu6v6=VA6−u6,u1,u2,v3,v4,v5,w4,w5,LR7=LRv1u2=VA6−u1,u5,u6,v2,v3,v4,w3,w4,LR8=LRv2u3=VA6−u2,u6,u1,v3,v4,v5,w4,w5,LR9=LRv3u4=VA6−u3,u1,u2,v4,v5,v1,w5,w6,LR10=LRv4u5=VA6−u4,u2,u3,v5,v6,v1,w6,w1,LR11=LRv5u6=VA6−u5,u3,u4,v6,v1,v2,w1,w2,LR12=LRv6u1=VA6−u6,u4,u5,v1,v5,w3,w2,w3,LR13=LRv1w1=VA6−u5,u6,v4,v5,v6,w2,w3,w4,LR14=LRv2w2=VA6−u6,u1,v5,v6,v1,w3,w4,w5,LR15=LRv3w3=VA6−u1,u2,v6,v1,v2,w4,w5,w6,LR16=LRv4w4=VA6−u2,u3,v1,v2,v3,w5,w6,w1,LR17=LRv5w5=VA6−u3,u4,v2,v3,v4,w6,w1,w2,LR18=LRv6w6=VA6−u4,u5,v3,v4,v5,w1,w2,w3,LR19=LRv1w2=VA6−u3,u4,v2,v3,v4,w1,w5,w6,LR20=LRv2w3=VA6−u4,u5,v3,v4,v5,w2,w6,w1,LR21=LRv3w4=VA6−u5,u6,v4,v5,v6,w3,w1,w2,LR22=LRv4w5=VA6−u6,u1,v5,v6,v1,w4,w2,w3,LR23=LRv5w6=VA6−u1,u2,v6,v1,v2,w5,w3,w4,LR24=LRv6w1=VA6−u2,u3,v1,v2,v3,w6,w4,w5,LR25=LRu1u2=VA6−v1,v4,w1,w2,LR26=LRu2u3=VA6−v2,v5,w2,w3,LR27=LRu3u4=VA6−v3,v6,w3,w4,LR28=LRu4u5=VA6−v4,v1,w4,w5,LR29=LRu5u6=VA6−v5,v2,w4,w1,LR30=LRu6u1=VA6−v6,v3,w4,w1,LR31=LRw1w2=VA6−u1,u2,v1,v4,LR32=LRw2w3=VA6−u2,u3,v2,v5,LR33=LRw3w4=VA6−u3,u4,v3,v6,LR34=LRw4w5=VA6−u4,u5,v4,v1,LR35=LRw5w6=VA6−u5,u6,v5,v2,LR36=LRw6w1=VA6−u6,u1,v6,v3,LR37=LRv1v2=VA6−u2,w2,w5,LR38=LRv2v3=VA6−u3,w3,w6,LR39=LRv3v4=VA6−u4,w4,w1,LR40=LRv4v5=VA6−u5,w5,w2,LR41=LRv5v6=VA6−u6,w6,w3,LR42=LRv6v1=VA6−u1,w1,w4.For1≤i≤6, LRei=LRviwi=LRviwi+1=LRuivi=LRviui+1=10 and LRei≤LRx,∀x∈EA5. Furthermore, ∪i=16LRei=18 and LRx∩∪i=16LRei=10,∀x∈EA6 There exists an upper LRF ϕ:VA6⟶0,1 defined as ϕv=1/10, ∀v∈VA6. Consequently, by Theorem 1, dimflA5≤9/5. Similarly, for 1≤i≤5, LRvivi+1=15 and LRvivi+1≥LRx,∀x∈EA6. Thus, there exists a lower LRF such that ϕ′:VA6⟶0,1 is defined as ϕ′v=1/15,∀v∈VA6. Consequently, by Theorem 2, dimflA6≥6/5. Hence, we have(20)65≤dimflA5≤95.Case 2. For m>6, by Lemma 4, LRei=LRuivi=LRviwi=LRviwi+1=LRviui+1=3m+2/2 and LRei≤LRx, ∀x∈EAm. Similarly, LRvivi+1=3m−3 and LRvivi+1≥LRx, ∀x∈EAm. Therefore, ϕ:VAm⟶0,1 is defined by ϕv=2/3m+2, ∀v∈VAm is an upper LRF andϕ′:VAm⟶0,1 is defined by ϕ′v=1/3m−3,∀v∈VAm is a lower LRF. Therefore, by Theorems 1 and 2, we have(21)mm−1≤dimflAm≤6m3m+2.


### 3.3. LRN Sets and LFMD of Convex PolytopeSm

This subsection deals with the results for the LRN sets and LFMD of the convex polytopes networkSm with m≥5.Table 5
Cardinalities of LRN sets ofSm for m≅1mod2.
Cardinalities of LRN setsComparisonLRvivi+1=4m−44m>4m−4>2m+1LRuiui+1=4m−44m>4m−4>2m+1LRwiwi+1=4m−44m>4m−4>2m+1LRzizi+1=4m−44m>4m−4>2m+1Table 6
Cardinalities of LRN sets ofSm for m≅0mod2.
Cardinalities of LRN setsComparisonLRvivi+1=4m−24m>4m−2>2mLRuiui+1=4m−44m>4m−4>2mLRwiwi+1=4m−44m>4m−4>2mLRzizi+1=4m−44m>4m−4>2mLemma 5.
LetSm be a network of convex polytope, where m≥5 and m≅1mod2. Then, for 1≤i≤m,(a)
LRei=LRviwi=LRwivi+1=2m+1 and LRei′=LRuivi=LRviwi=4m with ∪i=1mLRei=4m(b)
LRei≤LRx≤LRei′,∀x∈ESm(c)
LRx∩∪i=1mLRei≥LRei,∀x∈ESmProof.
Assume thatui, vi, wi, and zi are inner, middle, and outer vertices of Sm, respectively, where 1≤i≤m and m+1≅1modm.(a)
ConsiderLRviwi=ui,um+2i+1/2,um+2i+1/2+1,…,um,vi,vm+2i+1/2,vm+2i+1/2+1,…,vm,wi,wm+2i+1/2,wm+2i+1/2+1,…,wm,zi,zi+1,zi+2,…,zm+2i+1/2+1, LRwivi+1=ui+1,um+2i+1/2,um+2i+1/2+1,…,um,vi+1,vi+2,vi+3,…,vm+2i+1/2,wi,wm+2i+1/2,wm+2i+3/2,…,wm,zi,zm+2i+1/2,zm+2i+3/2,…,zm, and LRuivi=LRwizi=VSm. Therefore, LRei=2m+1 and LRei′=4m. Also, ∪i=1mLRei=VSm=4m.(b)
The LRN sets ofSm other than LRviwi, LRwivi+1, LRuivi, and LRwizi are LRvivi+1=VSm−vm+2i+1/2,um+2i+1/2,wi,zi, LRuiui+1=VSm−um+2i+1/2,vm+2i+1/2,wi,zi, LRwiwi+1=VSm−wm+2i+1/2,zm+2i+1/2,ui+1,vi+1, and LRzizi+1=VSm−ui+1,vi+1,wm+2i+1/2,zm+2i+1/2.The cardinalities of these LRN sets are obtained as given in Table5.From Table5, it is clear that LRei≤LRx≤LRei′,∀x∈ESm.(c)
Since∪i=1mLRei=VSm, therefore LRx∩∪i=1mLRei=LRx≥LRei.Lemma 6.
LetSm be a network of convex polytope, where m≥6 and m≅0mod2. Then, for 1≤i≤m,(a)
LRei=LRviwi=LRwivi+1=2m, LRei′=LRwizi=LRuivi=4m, and ∪i=1mLRei=4m(b)
LRviwi≤LRx≤LRei′, ∀x∈ESm(c)
LRx∩∪i=1mLRei≥LRei,∀x∈ESmProof.
Assume thatui, vi, wi, and zi are inner, middle and outer vertices of Sm, respectively, where 1≤i≤m and m+1≅1modm.(a)
ConsiderLRviwi=ui,um+2i+2/2,um+2i+2/2+1,…,um,vi,vm+2i+2/2,vm+2i+2/2+1,…,vm,wi,wi+1,wi+2,…,wm/2,zi,zi+1,zi+2,…,zm/2, LRwivi+1=ui,ui+1,ui+2,..,um+2i/2,vi+1,vi+2,vi+3,…,vm+2i/2,wi,wm+2i+2/2,wm+2i+3/2,…,wm,zi,zm+2i+2/2,zm+2i+3/2,…,zm, and LRwizi=LRuivi=VSm. Therefore, LRei=2m and LRei′=4m. Also, ∪i=1mLRei=4m.(b)
The LRN sets other thanLRviwi, LRwivi+1, LRuivi, and LRwizi are LRvivi+1=VSm−wi,wm+2i/2, LRwiwi+1=VSm−ui+1,um+2i+2/2,vi+1,vm+2i+2/2, LRuiui+1=VSm−wi,zi,wm+2i/2,zm+2i/2, and LRzizi+1=VSm−ui+1,um+2i+2/2,vi+1,vm+2i+2/2.The cardinalities of these LRN sets are obtained as given in Table6.From Table6, it is clear that LRei≤LRx≤LRei′,∀x∈ESm.(c)
Since∪i=1mLRei=VSm=4m, therefore LRx∩∪i=1mLRei=LRx≥LRei.Theorem 7.
LetSm with m≥5 be a network of convex polytope, where m≅1mod2. Then,(22)1<dimflSm≤2mm+1.Proof.
To prove the result, we have the following cases:Case 1. For m=5, the LRN sets are as follows:(23)LR1=LRv1w1=VS5−u2,u3,v2,v3,w5,w4,z4,z5,LR2=LRv2w2=VS5−u3,u4,v3,v4,w1,w5,z5,z1,LR3=LRv3w3=VS5−u4,u5,v4,v5,w2,w1,z1,z2,LR4=LRv4w4=VS5−u5,u1,v5,v1,w3,w2,z2,z3,LR5=LRv5w5=VS5−u1,u2,v1,v2,w4,w3,z3,z4,LR6=LRv2w1=VS5−u1,u5,v1,v5,w2,w3,z2,z3,LR7=LRv3w2=VS5−u2,u1,v2,v1,w3,w4,z3,z4,LR8=LRv4w3=VS5−u3,u2,v3,v2,w4,w5,z4,z5,LR9=LRv5w4=VS5−u4,u3,v4,v3,w5,w1,z5,z1,LR10=LRv1w5=VS5−u5,u4,v5,v4,w1,w2,z1,z2,LR11=LRz1z2=VS5−v2,u2,u5,w4,z4,LR12=LRz2z3=VS5−v3,u3,u2,w5,z5,LR13=LRz3z4=VS5−v4,u4,u3,w2,z2,LR14=LRz4z5=VS5−v5,u5,u4,w3,z3,LR15=LRz5z1=VS5−v1,u1,u5,w4,z4,LR16=LRw1z1=VS5−w2,w3,w5,LR17=LRw2z2=VS5−w3,w4,w1,LR18=LRw3z3=VS5−w4,w5,w2,LR19=LRw4z4=VS5−w5,w1,w3,LR20=LRw5z5=VS5−w1,w2,w4,LR21=LRv1v2=VS5−v4,w1,z1,LR22=LRv2v3=VS5−v5,w2,z2,LR23=LRv3v4=VS5−v1,w3,z3,LR24=LRv4v5=VS5−v2,w4,z4,LR25=LRv5v1=VS5−v3,w5,z5,LR26=LRu1u2=VS5−u4,w1,z1,LR27=LRu2u3=VS5−u5,w2,z2,LR28=LRu3u4=VS5−u1,w3,z3,LR29=LRu4u5=VS5−u2,w4,z4,LR30=LRu5u1=VS5−u3,w4,z5,LR31=LRu1v1=VS5,LR32=LRu2v2=VS5,LR33=LRu3v3=VS5,LR34=LRu4v4=VS5,LR35=LRu5v5=VS5.For1≤i≤5, LRei=LRviwi=LRviwi+1=12 and LRei≤LRx,∀x∈ES5. Furthermore, ∪i=15LRei=20 and LRx∩∪i=15LRei=9,∀x∈ES5. There exists an upper LRF ϕ:VS5⟶0,1 defined as ϕv=1/12, ∀v∈VS5. Consequently, by Theorem 1, dimflS5≤5/3. Similarly, for 1≤i≤5, LRuivi=24 and LRuivi≥LRx,∀x∈ES5. Thus, there exist a lower LRF such that ϕ′:VS5⟶0,1 is defined as ϕ′v=1/24,∀v∈VS5. Consequently, by Theorem 2, dimflS5≥1. Since S5 is nonbipartite network, therefore dimflS5 must be greater than 1. Hence, we have(24)1<dimflS5≤53.Case 2. For m>5, by Lemma 5, LRei=LRviwi=LRviwi+1=2m+1 and LRei≤LRx,∀x∈ESm. Similarly, LRuivi=4m and LRuivi≥LRx, ∀x∈ESm. Therefore, ϕ:VSm⟶0,1 is defined by ϕv=1/2m+1, ∀v∈VSm is an upper LRF and ϕ′:VSm⟶0,1 is defined by ϕ′v=1/4m,∀v∈VAm is a lower LRF. Therefore, by Theorems 1 and 2, we have(25)1<dimflSm≤2mm+1.Theorem 8.
LetSm with m≥6 be a network of convex polytope, where m≅0mod2. Then,(26)1<dimflSm≤2.Proof.
To prove the result, we have following cases:Case 1. For m=6, the LRN sets are as follows:(27)LR1=LRu1u2=VS6−w1,w4,z1,z4,LR2=LRu2u3=VS6−w2,w5,z2,z5,LR3=LRu3u4=VS6−w3,w6,z3,z6,LR4=LRu4u5=VS6−w4,w1,z4,z1,LR5=LRu5u6=VS6−w5,w2,z5,z2,LR6=LRu6u1=VS6−w6,w3,z6,z3,LR7=LRw1w2=VS6−v2,v5,u2,u5,LR8=LRw2w3=VS6−v3,v6,u3,u6,LR9=LRw3w4=VS6−v4,v1,u4,u1,LR10=LRw4w5=VS6−v5,v2,u5,u2LR11=LRw5w6=VS6−v6,v3,u6,u3,LR12=LRw6w1=VS6−v1,v4,u1,u4,LR13=LRz1z2=VS6−v2,v5,u2,u5,LR14=LRz2z3=VS6−v3,v6,u3,u6,LR15=LRv1w1=u1,u5,u6,v1,v5,v6,w1,w2,w6,z1,z2,z3,LR16=LRv2w2=u2,u6,u1,v2,v6,v1,w2,w3,w1,z2,z3,z4,LR17=LRv3w3=u3,u1,u2,v3,v1,v2,w3,w4,w2,z3,z4,z5,LR18=LRv4w4=u4,u2,u3,v4,v2,v3,w4,w5,w3,z4,z5,z6,LR19=LRv5w5=u5,u3,u4,v5,v3,v4,w5,w6,w4,z5,z6,z1,LR20=LRv6w6=u6,u4,u5,v6,v4,v5,w6,w1,w5,z6,z1,z2,LR21=LRv2w1=u2,u3,u4,v2,v3,v4,w1,w5,w6,z1,z5,z6,LR22=LRv3w2=u3,u4,u5,v3,v4,v5,w2,w6,w1,z2,z6,z1,LR23=LRv4w3=u4,u5,u6,v4,v5,v6,w3,w1,w2,z3,z1,z2,LR24=LRv5w4=u5,u6,u1,v5,v6,v1,w4,w2,w3,z4,z2,z3,LR25=LRv6w5=u6,u1,u2,v6,v1,v2,w5,w3,w4,z5,z3,z4,LR26=LRv6w1=u1,u2,u3,v1,v2,v3,w6,w4,w5,z6,z4,z5,LR27=LRz3z4=VS6−v4,v1,u4,u1,LR28=LRz4z5=VS6−v5,v2,u5,u2,LR29=LRz5z6=VS6−v6,v3,u6,u3,LR30=LRz6z1=VS6−v1,v4,u1,u4,LR31=LRv1v2=VS6−w1,w4,z1,LR32=LRv2v3=VS6−w2,w5,z2,LR33=LRv3v4=VS6−w3,w6,z3,LR34=LRv4v5=VS6−w4,w1,z4,LR35=LRv5v6=VS6−w5,w2,z5,LR36=LRv6v1=VS6−w6,w3,z6,LR37=LRu1v1=VS6,LR38=LRu2v2=VS6,LR39=LRu3v3=VS6,LR40=LRu4v4=VS6,LR41=LRu5v5=VS6,LR42=LRu6v6=VS6,LR43=LRw1z1=VS6,LR44=LRw2z2=VS6,LR45=LRw3z3=VS6,LR46=LRw4z4=VS6,LR47=LRw5z5=VS6,LR48=LRw6z6=VS6.For1≤i≤6, LRei=LRviwi=LRviwi+1=12 and LRei≤LRx,∀x∈ES6. Furthermore, ∪i=16LRei=VS6=20 and LRx∩∪i=16LRei=12,∀x∈ES6. There exists an upper LRF ϕ:VS6⟶0,1 defined as ϕv=1/12, ∀v∈VS6. Consequently, by Theorem 1, dimflS6≤2. Similarly, for 1≤i≤6, LRuivi=24 and LRuivi≥LRx,∀x∈ES6. Thus, there exists a lower LRF such that ϕ′:VS6⟶0,1 is defined as ϕ′v=1/24,∀v∈VS56. Consequently, by Theorem 2, dimflS6≥1. Since S6 is nonbipartite network, therefore dimflS6 must be greater than 1. Hence, we have(28)1<dimflS6≤2.Case 2. For m>6, by Lemma 6, LRei=LRviwi=LRviwi+1=2m and LRei≤LRx,∀x∈ESm. Similarly, LRuivi=4m and LRuivi≥LRx, ∀x∈ESm. Therefore, ϕ:VSm⟶0,1 is defined by ϕv=1/2m, ∀v∈VSm is an upper LRF and ϕ′:VSm⟶0,1 is defined by ϕ′v=1/4m,∀v∈VAm is a lower LRF. Moreover, Sm is a nonbipartite network so dimflSm must be greater than 1. Therefore, by Theorems 1 and 2, we have(29)1<dimflSm≤2.


### 3.4. LRN Sets and LFMD of Convex PolytopeQm

This subsection deals with the results for the LRN sets and LFMD of the convex polytopes networkQm with m≥5.Lemma 7.
LetQm be a network of convex polytope, where m≥5 and m≅1mod2. Then, for 1≤i≤m,(a)
LRei=LRuiui+1=LRzizi+1=LRuivi=LRviwi=LRwizi=4m−4 with ∪i=1mLRei=4m.Proof.
Assume thatui, vi, and wi are inner and middle vertices, respectively, and zi are outer vertices, where 1≤i≤m and m+1≅1modm:(a)
LRuiui+1=VQm−um+2i+1/2,vm+2i+1/2,wi,zi, LRuivi=VQm−vi+1,vm,wi+1,zi+1, LRviwi=VQm−ui+1,vi+2,zm−1,zm, LRzizi+1=VQm−ui+1,vi+1,wm+2i−1/2,zm+2i−1/2, LRwizi=VQm−vm,wi+1,wm,vm+2i−1/2, and LRei=4m−4, where 1≤i≤m.Lemma 8.
LetQm with m≥6 be a network of convex polytope, where m≅0mod2. Then, for 1≤i≤m,(a)
LRei=LRviwi=LRwivi+1=2m+4, LRei′=LRuiui+1=LRuivi=LRzizi+1=LRwizi=4m−4, and ∪i=1mLRei=4m(b)
LRei′∩∪i=1mLRei≥LRei′,∀x∈EQmProof.
Assuming thatui, vi, wi, and zi are inner, middle, and outer vertices, respectively, for 1≤i≤m and m+1≅1modm, we have(a)
LRviwi=VQm−ui+1,ui+2,ui+3,…,um+2i/2,vi+3,vi+4,…,vm+2i/2,wm+2i/2,wm+2i/2+1,wm+2i/2+2,…,wm−2,zm+2i/2,zm+2i+3/2,zm+2i+5/2,…,zm, and LRwivi+1=VQn−ui,um+2i+2/2,um+2i+3/2,…,um,vm+2i+2/2,vm+2i+2/2+1,vm+2i+2/2+2,…,vm,wm+2i+2/2,zi+1,zi+2,…,zm+2i+2/2; therefore, LRei=2m+4, and also ∪i=1mLRei=VQm=4m. Moreover, LRuiui+1=VQm−wi,wm+2i/2,zi,zm+2i/2, LRuivi=VQm−vi+1,vm,wi+1,wm−1, LRzizi+1=VQm−ui+1,vi+1,vm+2i+2/2,um+2i+2/2, and LRwizi=VQm−vi+2,vm,wi+1,wm; therefore, LRei′=4m−4.(b)
Since∪i=1mLRei=VQm, therefore LRei′∩∪i=1mLRei=LRei′≥LRei.Theorem 9.
LetQm with m≥5 be a network of convex polytope, where m≅1mod2. Then,(30)dimflQm=mm−1.Proof.
Form=5, the LRN sets are as follows:(31)LR1=LRu1u2=VQ5−u4,v4,w1,z1,LR2=LRu2u3=VQ5−u5,v5,w2,z2,LR3=LRu3u4=VQ5−u1,v1,w3,z3,LR4=LRu4u5=VQ5−u2,v2,w4,z4,LR5=LRu5u1=VQ5−u3,v3,w5,z5,LR6=LRu1v1=VQ5−v2,v5,w2,w4,LR7=LRu2v2=VQ5−v3,v1,w3,w5,LR8=LRu3v3=VQ5−v4,v2,w4,w1,LR9=LRu4v4=VQ5−v5,v3,w5,w2,LR10=LRu5v5=VQ5−v1,v4,w1,w3,LR11=LRv1w1=VQ5−u2,u3,z4,z5,LR12=LRv2w2=VQ5−u3,u4,z5,z1,LR13=LRv3w3=VQ5−u4,u5,z1,z2,LR14=LRv4w4=VQ5−u5,u1,z2,z3,LR15=LRv5w5=VQ5−u1,u2,z3,z4,LR16=LRw1v2=VQ5−u1,u5,z2,z3,LR17=LRw2v3=VQ5−u2,u1,z3,z4,LR18=LRw3v4=VQ5−u3,u2,z4,z5,LR19=LRw4v5=VQ5−u4,u3,z5,z1,LR20=LRw5v1=VQ5−u5,u4,z1,z2,LR21=LRw1z1=VQ5−v2,v5,w2,w5,LR22=LRw2z2=VQ5−v3,v1,w3,w1,LR23=LRw3z3=VQ5−v4,v2,w4,w2,LR24=LRw4z4=VQ5−v5,v3,w5,w3,LR25=LRw5z5=VQ5−v1,v4,w1,w4,LR26=LRz1z2=VQ5−u2,v2,w4,z4,LR27=LRz2z3=VQ5−v3,u3,w5,z5,LR28=LRz3z4=VQ5−v4,u4,w1,z1,LR29=LRz4z5=VQ5−v5,u5,w2,z2,LR30=LRz5z1=VQ5−v1,u1,w3,z3.
EachLRei=16, where 1≤i≤30. Furthermore, ∪i=130LRei=20. Therefore, by Corollary 1,(32)dimflQ5=54.
Form>5, by Lemma 7, LRei=LRuiui+1=LRzizi+1=LRuivi=LRviwi=LRwizi=4m−4 and ∪i=1mLRei=4m, where 1≤i≤m. Hence, from Corollary 1, we have(33)dimflQm=mm−1.Theorem 10.
LetQm with m≥6 be a network of convex polytope, where m≅0mod2.(34)mm−1≤dimflQm≤2mm+2.Proof.
To prove the result, we have following cases:Case 1. For m=6, the LRN sets are as follows:(35)LR1=LRu1u2=VQ6−w1,w4,z1,z4,LR2=LRu2u3=VQ6−w2,w5,z2,z5,LR3=LRu3u4=VQ6−w3,w6,z3,z6,LR4=LRu4u5=VQ6−w4,w1,z4,z1,LR5=LRu5u6=VQ6−w5,w2,z5,z2,LR6=LRu6u1=VQ6−w6,w3,z6,z3,LR7=LRu1v1=VQ6−v2,v6,w2,w5,LR8=LRu2v2=VQ6−v3,v1,w3,w6,LR9=LRu3v3=VQ6−v4,v2,w4,w1,LR10=LRu4v4=VQ6−v5,v3,w5,w2,LR11=LRu5v5=VQ6−v6,v4,w6,w3,LR12=LRu6v6=VQ6−v1,v5,w1,w4,LR13=LRw1z1=VQ6−v3,v6,w2,w6,LR14=LRw2z2=VQ6−v4,v1,w3,w1,LR15=LRw3z3=VQ6−v5,v2,w4,w2,LR16=LRw4z4=VQ6−v6,v3,w5,w3,LR17=LRw5z5=VQ6−v1,v4,w6,w4,LR18=LRw6z6=VQ6−v2,v3,w1,w5,LR19=LRz1z2=VQ6−u2,u5,v2,v5,LR20=LRz2z3=VQ6−u3,u6,v3,v6,LR21=LRz3z4=VQ6−u4,u1,v4,v1,LR22=LRz4z5=VQ6−u5,u2,v5,v2,LR23=LRz5z6=VQ6−u6,u3,v6,v3,LR24=LRz6z1=VQ6−u1,u4,v1,v4,LR25=LRv1w1=VQ6−u2,u3,u4,v2,v4,z4,z5,z6,LR26=LRv2w2=VQ6−u3,u4,u5,v3,v5,z5,z6,z1,LR27=LRv3w3=VQ6−u4,u5,u6,v4,v6,z6,z1,z2,LR28=LRv4w4=VQ6−u5,u6,u1,v5,v1,z1,z2,z3,LR29=LRv5w5=VQ6−u6,u1,u2,v6,v2,z2,z3,z4,LR30=LRv6w6=VQ6−u1,u2,u3,v1,v3,z3,z4,z5,LR31=LRw1v2=VQ6−u1,u5,u6,v5,w4,z2,z3,z4,LR32=LRw2v3=VQ6−u2,u6,u1,v6,w5,z3,z4,z5,LR33=LRw3v4=VQ6−u3,u1,u2,v1,w6,z4,z5,z6,LR34=LRw4v5=VQ6−u4,u2,u3,v2,w1,z5,z6,z1,LR35=LRw5v6=VQ6−u5,u3,u4,v3,w2,z6,z1,z2,LR36=LRw6v1=VQ6−u5,u4,u5,v4,w3,z1,z2,z3.For1≤i≤6, LRei=LRviwi=LRwivi+1=16 and LRei≤LRx,∀x∈EQ6. Furthermore, ∪i=16LRei=24. Thus, there exists an upper LRF ϕ:VQ6⟶0,1 defined as ϕv=1/16,∀v∈VQ6. Consequently, by Theorem 1, dimflQ6≤3/2. Similarly, for 1≤i≤6, LRei′=LRuivi=LRwizi=LRzizi+1=20 and LRei′≥LRx,∀x∈EQ6. Thus, there exists a LRF ϕ′:VQ6⟶0,1 defined as ϕ′v=1/20 which is a lower LRF ∀v∈VQ6. Consequently, by Theorem 2, dimflQ6≥6/5. Hence, we have(36)65≤dimflQ6≤32.Case 2. For m>6, by Lemma 8, LRei=LRviwi=LRwivi+1=2m+4 and LRei≤LRx,∀x∈EQm. Furthermore, LRei′=LRuiui+1,LRuivi,LRzizi+1 with LRwizi=4m−4 and LRei′≥LRx, ∀x∈EQm. Therefore, ϕ:VQm⟶0,1 defined by ϕv=1/2m+4, ∀v∈VRm is an upper LRF and ϕ′:VQm⟶0,1 defined by ϕ′v=1/4m−4, ∀v∈VQm is a lower LRF. Therefore, by Theorems 1 and 2, we have(37)mm−1≤dimflQm≤2mm+2.


## 3.1. LRN Sets and LFMD of Convex PolytopeRm

This subsection covers the results related to the LRN sets and LFMD of the convex polynope network that is presented byRm for any integral value m≥5.Table 1
Cardinalities of the LRN sets ofRm for m≅1mod2.
Cardinalities of LRN setsComparisonLRwiwi+1=3m+33m>3m−3>3m+3/2LRvivi+1=3m+33m>3m−3>3m+3/2LRuiui+1=3m+33m>3m−3>3m+3/2Table 2
Cardinalities of LRN sets ofRm with m≅0mod2.
Cardinalities of LRN setsComparisonLRwiwi+1=3m/23m>3m−2>3m/2LRvivi+1=3m/23m>3m−2>3m/2LRuiui+1=3m/23m>3m−4>3m/2Lemma 1.
LetRm be a network of convex polytope, where m≥5 and m≅1mod2. Then, for 1≤i≤m,(a)
LRei=LRuivi=LRviui+1=3m+3/2, LRviwi=3m, and ∪i=1mLRei=3m(b)
LRei≤LRx≤LRviwi, ∀x∈ERm(c)
LRx∩∪i=1mLRei≥LRei,∀x∈ERmProof.
Assume thatui, vi, and wi are inner, middle, and outer vertices of Rm, respectively, where 1≤i≤m and m+1≅1modm.(a)
SinceLRuivi=ui,um+2i+1/2,um+2i+3/2,…,um,vi,vi+1,vi+2,…,vm+1/2,wi,wi+1,wi+2,…,wm+1/2 and LRuivi+1=ui+1,ui+2,ui+3,…,um+2i+1/2,vi,vm+2i+1/2,vm+2i+3/2,vm+2i+5/2,…,vm,wi,wm+2i+1/2,wm+2i+3/2,wm+2i+5/2,…,wm, therefore LRei=3m+3/2. Also, LRviwi=VRm, which implies that LRviwi=3m. Now, ∪i=1mLRei=VRm=3m.(b)
The LRN sets other thanLRei and LRviwi are LRwiwi+1=VRm−ui+1,vm+2i+1/2,wm+2i+1/2, LRvivi+1=VRm−ui+2,vm+2i+1/2,wm+2i+1/2, and LRuiui+1=VRm−um+2i+1/2,vi,wi.Now, we arrange the cardinalities of these LRN sets in Table1.From Table1, it is clear that LRei≤LRx≤LRviwi, ∀x∈EG with 1≤i≤m.(c)
Since∪i=1mLRei=VRm, therefore LRx∩LRei=LRx≥LRei.Lemma 2.
LetRm with m≥6 be a network of convex polytope, where m≅0mod2. Then,(a)
LRei=LRviui+1=LRuivi=3m/2 and LRviwi=3m with ∪i=1mLRei=VRm(b)
LRei≤LRx≤LRviwi, ∀x∈ERm(c)
LRx∩∪i=1mLRei≥LRei,∀x∈ERmProof.
Assume thatui, vi, and wi are inner, middle, and outer vertices, respectively, of Rm, where 1≤i≤m and m+1≅1modm.(a)
ConsiderLRuivi+1=ui+1,ui+2,ui+3,…,um+2i/2,vi,vm+2i+2/2,vm+2i+4/2,vm+2i+6/2,…,vm,wi,wm+2i/2,wm+2i+2/2,wm+2i+4/2,…,wm, LRuivi=ui,um+2i+2/2,um+2i+4/2,um+2i+6/2,…,um,vi,vi+1,vi+2,….,vm/2,vm,wi,wi+1,wi+2,….,wm/2 therefore LRei=3m/2, and ∪i=1mLRei=3m. Also, LRviwi=VRm which implies that LRviwi=VRm=3m. Furthermore, ∪i=1mLRei=VRm=3m.(b)
Other LRN sets areLRwiwi+1=VRm−ui+1,um+2i+2/2, LRvivi+1=VRm−ui+1,um+2i+2/2, and LRuiui+1=VRm−vm+2i/2,vi,wi,wm+2i/2.The cardinalities of these LRN sets are obtained as given in Table2.From Table2, it is clear that LRei≤LRx≤LRviwi, ∀x∈ERm.(c)
Since∪i=1mLRei=VRm, therefore LRx∩∪i=1mLRei=LRx≥LRei.Theorem 3.
LetRm with m≥5 be a network of convex polytope, where m≅1mod2. Then,(6)1<dimflRm≤2mm+1.Proof.
To prove the result, we have following cases:Case 1. For m=5, the LRN sets are as follows:(7)LR1=LRu1u2=VR5−u4,v1,w1,LR2=LRu2u3=VR5−u5,v2,w2,LR3=LRu3u4=VR5−u1,v3,w3,LR4=LRu4u5=VR5−u2,v4,w4,LR5=LRu5u1=VR5−u3,v5,w5,LR6=LRw1w2=VR5−u2,v2,w4,LR7=LRw2w3=VR5−u3,v3,w5,LR8=LRw3w4=VR5−u4,v4,w1,LR9=LRw4w4=VR5−u5,v5,w2,LR10=LRw4w5=VR5−u1,v1,w3,LR11=LRv1v2=VR5−v4,w4,LR12=LRv2v3=VR5−v5,w5,LR13=LRv3v4=VR5−v1,w1,LR14=LRv4v5=VR5−v2,w2,LR15=LRv5v1=VR5−v3,w1,LR16=LRv1w1=VR5,LR17=LRv2w2=VR5,LR18=LRv3w3=VR5,LR19=LRv4w4=VR5,LR20=LRv5w5=VR5,LR21=LRv1u2=VR5−v3,v2,u1,u5,w2,w3,LR22=LRv2u3=VR5−v3,v4,u1,u5,w2,w3,LR23=LRv3u4=VR5−v5,v4,u1,u5,w2,w3,LR24=LRv4u5=VR5−v1,v5,u1,u5,w2,w3,LR25=LRv5u1=VR5−v1,v2,u1,u5,w2,w3,LR26=LRu1v1=VR5−u2,u3,v2,v3,w4,w5,LR27=LRu2v2=VR5−u3,u4,v5,v1,w5,w1,LR28=LRu3v3=VR5−u4,u5,v1,v2,w3,w2,LR29=LRu4v4=VR5−u5,u1,v2,v3,w2,w1,LR30=LRu5v5=VR5−u1,u2,v3,v4,w3,w2.As, for1≤i≤5, LRei=LRuiui=LRviui+1=9, LRei≤LRx, ∀x∈ER5. Furthermore, ∪i=15LRviui+1=15 and LRx∩∪i=15LRei=9,∀x∈ER5. There exists an upper LRF ϕ:VR5⟶0,1 defined as ϕv=1/9,∀v∈VR5. Consequently, by Theorem 1, dimflR5≤5/3. Similarly, for 1≤i≤5, LRviwi=15 and LRviwi≥LRx, ∀x∈ER5. Thus, there exists a lower LRF such that ϕ′:VR5⟶0,1 defined as ϕ′v=1/15,∀v∈VR5. Consequently, by Theorem 2, dimflR5≥1. Since R5 is a nonbipartite network, therefore dimflR5 must be greater than 1. Hence, we have(8)1<dimflR5≤53.Case 2. For m>5, by Lemma 1, LRei=3m+3/2 and LRei≤LRx, ∀x∈ERm. Similarly, LRuiwi=3m and LRuiwi≥LRx, ∀x∈ERm. Therefore, ϕ:VRm⟶0,1 is defined by ϕv=2/3m+3,∀v∈VRm is an upper LRF and ϕ′:VRm⟶0,1 is defined by ϕ′v=1/3m,∀v∈VRm is a lower LRF. Moreover, Rm is a nonbipartite network. Therefore, by Theorems 1 and 2, we have(9)1<dimflRm≤2mm+1.Theorem 4.
LetRm with m≥6 be a network of convex polytope, where m≅0mod2. Then,(10)1<dimflRm≤2.Proof.
To prove the result, we have following two cases:Case 1. For m=6, the LRN sets are as follows:(11)LR1=LRu1u2=VR6−v1,v4,w1,w4,LR2=LRu2u3=VR6−v5,v2,w2,w5,LR3=LRu3u4=VR6−v3,v6,w3,w6,LR4=LRu4u5=VR6−v4,v1,w4,w1,LR5=LRu5u6=VR6−v5,v2,w5,w2,LR6=LRu6u1=VR6−v6,v3,w6,w3,LR7=LRw1w2=VR6−u2,u5,w2,w5,LR8=LRw2w3=VR6−u3,u6,w3,w6,LR9=LRw3w4=VR6−u4,u1,w4,w1,LR10=LRw4w5=VR6−u5,u2,w5,w2,LR11=LRw5w6=VR6−u6,u3,w6,w3,LR12=LRw6w1=VR6−u1,u4,w1,w4,LR13=LRv1v2=VR6−u2,u5,LR14=LRv2v3=VR6−u3,u6,LR15=LRv3v4=VR6−u4,u1,LR16=LRv4v5=VR6−u5,u2,LR17=LRv5v6=VR6−u6,u3,LR18=LRv6v1=VR6−u1,u4,LR19=LRv1w1=VR6,LR20=LRv2w2=VR6,LR21=LRv3w3=VR6,LR22=LRv4w4=VR6,LR23=LRv5w5=VR6,LR24=LRv6w6=VR6,LR25=LRu1v1=u1,u5,u6,v1,v2,v3,w1,w2,w3,LR26=LRu2v2=u2,u6,u1,v2,v3,v4,w2,w3,w4,LR27=LRu3v3=u3,u1,u2,v3,v4,v5,w3,w4,w5,LR28=LRu4v4=u4,u2,u3,v4,v5,v6,w4,w5,w6,LR29=LRu5v5=u5,u3,u4,v5,v6,v1,w5,w6,w1,LR30=LRu6v6=u6,u4,u5,v6,v1,v2,w6,w1,w2,LR31=LRv1u2=u2,u3,u4,v1,v5,v6,w1,w5,w6,LR32=LRv2u3=u3,u4,u5,v1,v2,v6,w1,w2,w6,LR33=LRv3u4=u4,u5,u6,v1,v2,v3,w1,w3,w5,LR34=LRv4u5=u5,u6,u1,v2,v3,v4,w4,w2,w3,LR35=LRv5u6=u6,u1,u2,v3,v4,v5,w3,w4,w5,LR36=LRv6u1=u1,u2,u3,v4,v5,v6,w4,w5,w6.For1≤i≤6, LRei=LRuiui=LRviui+1=9 and LRei≤LRx, ∀x∈ER6. Furthermore, ∪i=16LRei=18 and LRx∩∪i=16LRei=9, ∀x∈ER6. Therefore, there exists a lower LRF ϕ:VR6⟶0,1 defined as ϕv=1/9 for each v∈VRm. Consequently, by Theorem 1, dimflR6≤2.Similarly, for1≤i≤6, LRviwi=18 and LRviwi≥LRx, ∀x∈ER6. Thus, there exists an upper LRF ϕ′:VR6⟶0,1 defined as ϕv=1/18 for each v∈VRm. Consequently, by Theorem 2, dimflR6≥1. Furthermore, R6 is a nonbipartite network, so dimflR6 must be greater than 1. Hence, we have(12)1<dimflR6≤2.Case 2. For m>6, by Lemma 2, LRei=3m/2 for 1≤i≤m and LRei≤LRx, ∀x∈ERm. Therefore, ϕ:VRm⟶0,1 defined by ϕv=2/3m, ∀v∈VRm is an upper LRF. Thus, by Theorem 1, the dimflRm≤2. Similarly, as LRviwi=3m and LRviwi≥LRx, ∀x∈ERm, where 1≤i≤m. Therefore, ϕ′:VRm⟶0,1 defined by ϕ′v=1/3m,∀v∈VRm is a lower LRF. Thus, by Theorem 2, dimflRm≥1. Moreover, as Rm is a nonbipartite network, it implies that dimflRm must be greater than 1. Hence, we have(13)1<dimflRm≤2.


## 3.2. LRN Sets and LFMD of Convex PolytopeAm

This subsection deals with the results for the LRN sets and LFMD of the convex polytopes networkAm with m≥5.Table 3
Cardinalities of LRN sets ofAm for m≅1mod2.
Cardinalities of LRN setsComparisonLRwiwi+1=3m−53m>3m−5>3m+3/2LRuiui+1=3m−53m>3m−5>3m+3/2Table 4
Cardinalities of LRN sets ofAm for m≅0mod2.
Cardinalities of LRN setsComparisonLRwiwi+1=3m−43m>3m−4>3m+2/2LRuiui+1=3m−43m>3m−4>3m+2/2Lemma 3.
LetAm be a network of convex polytope, where m≥5 and m≅1mod2. Then, for 1≤i≤m,(a)
LRei=LRviui=LRviui+1=LRviwi+1==LRviwi=3m+3/2 and LRvivi+1=3m−3 with ∪i=1mLRei=VRm=3m(b)
LRei≤LRx≤LRvivi+1, ∀x∈EAm(c)
LRx∩∪i=1mLRei≥LRei,∀x∈EAmProof.
Assume thatui, vi, and wi are inner, middle, and outer vertices of Am, respectively, where 1≤i≤m and m+1≅1modm.(a)
SinceLRviui+1=ui+1,ui+2,ui+3,…,um+2i+1/2,vi,vm+2i+1/2,vm+2i+3/2,…,vm,wi,wi+1,wi+2,…,wm+2i+1/2,wm+2i+5/2,…,wm, LRviwi=ui,ui+1,ui+2,ui+3,…,um+2i−1/2,vi,vi+1,vi+2,…,vm+2i−1/2,wi,wm+2i+1/2,wm+2i+3/2,…,wm, LRuivi=ui,um+2i+1/2,um+2i+3/2,…,um,vi,vi+1,vi+2,….,vm+2i−1/2,wi,wi+1,wi+2,….,wm+2i−1/2, and LRviwi+1=ui,ui+1,um+6i−2/2,um+6i−2/2+1,…,um,vi,vm+6i−2/2,vm+6i−2/2+1,…,vmwi+1,wi+2,wi+3,….,wm+6i−4/2, therefore LRei=LRviui=LRviui+1=LRviwi+1=LRviwi=3m+3/2 with ∪i=1mLRei=VRm=3m. Also, =LRvivi+1=VAm−ui,vm+2i+1/2,wi; therefore, LRvivi+1=VRm−3=3m−3.(b)
The LRN sets of the otherLRei and LRvivi+1 are LRwiwi+1=VAm−ui,ui+1,um+2i−1/2,vi,wm+2i−1/2 and LRuiui+1=VAm−ui,ui+1,um+2i+1/2,vi,wm+2i+1/2.Now, we arrange the cardinalities of these LRN set in Table3.From Table3, it is clear that LRei≤LRx≤LRvivi+1, ∀x∈EAm.(c)
Since∪i=1mLRei=VAm, therefore LRx∩∪i=1mLRei=LRx≥LRei.Lemma 4.
LetAm be a network of convex polytope, where m≥6 and m≅0mod2. Then, for 1≤i≤m,(a)
LRei=LRviui=LRviui+1=LRviwi+1=LRviwi=3m+2/2 and LRvivi+1=3m−3 with ∪i=1mLRei=3m(b)
LRei≤LRx≤LRvivi+1,∀x∈EAm(c)
LRx∩∪i=1mLRei=LRx≥LRei,∀x∈EAmProof.
Assume thatui, vi, and wi are inner, middle, and outer vertices of Am, respectively, where 1≤i≤m and m+1≅1modm.(a)
ConsiderLRviui+1=ui+1,ui+2,ui+3,…,um+2i/2,vi,vm+6i−2/2,vm+6i−2/2+1,…,vm,wi,wi+1,wi+2,…,wm+6i−2/2,wm+6i−2/2+1,…,wm, LRviwi=ui,ui+1,ui+2,ui+3,…,um+4i−2/2,vi,vi+1,vi+2,…,vm+2i−2/2,wi,wm+6i−2/2,wm+6i/2,…,wm, LRuivi=ui,um+6i−2/2,um+6i−2/2+1,…,um,vi,vi+1,vi+2,….,vm+2i−2/2,wi,wi+1,wi+2,….,wm+2i−2/2, LRviwi+1=ui,ui+1,um+6i−2/2,um+6i/2,…,um,vi,vm+6i−2/2,vm+6i/2,…,vm,wi+1,wi+2,wi+3,….,wm+6i−4/2, and LRvivi+1=VAm−ui+1,wi+2,wm+6i−2/2. Furthermore, LRei=3m+2/2 and LRvivi+1=3m−3 with ∪i=1mLRei=VAm=3m.(b)
The LRN sets ofAm other than LRei and LRvivi+1 are LRwiwi+1=VAm−ui,ui+1,vi,vm+2i/2 and LRuiui+1=VAm−vm+4i−2/2,vi,wi,wi+1. Now, we arrange cardinalities of these LRN set in Table 4.From Table4, it is clear that LRei≤LRx≤LRvivi+1,∀x∈EAm.(c)
Since∪i=1mLRei=VAm, therefore LRx∩∪i=1mLRei=LRx≥LRei.Theorem 5.
LetAm with m≥5 be a network of convex polytope, where m≅1mod2. Then,(14)mm−1≤dimflAm≤2mm+1.Proof.
To prove the result, we have following cases:Case 1. For m=5, the LRN sets are as follows:(15)LR1=LRu1v1=VA5−u2,u3,v4,v5,w4,w5,LR2=LRu2v2=VA5−u3,u4,v1,v2,w1,w2,LR3=LRu3v3=VA5−u5,u1,v3,v4,w3,w4,LR4=LRu4v4=VA5−u2,u3,v5,v1,w5,w1,LR5=LRu5v5=VA5−u4,u5,v2,v3,w2,w3,LR6=LRv1u2=VA5−u1,u2,v2,v3,w3,w4,LR7=LRv2u3=VA5−u2,u3,v3,v4,w4,w5,LR8=LRv3u4=VA5−u3,u4,v4,v5,w5,w1,LR9=LRv4u5=VA5−u4,u5,v5,v1,w1,w2,LR10=LRv5u1=VA5−u5,u1,v1,v2,w2,w3,LR11=LRv1w1=VA5−u4,u5,v4,v5,w2,w3,LR12=LRv2w2=VA5−u5,u1,v5,v1,w3,w4,LR13=LRv3w3=VA5−u1,u2,v1,v2,w4,w5,LR14=LRv4w4=VA5−u2,u3,v2,v3,w5,w1,LR15=LRv5w5=VA5−u3,u4,v3,v4,w1,w5,LR16=LRv1w2=VA5−u3,u4,v2,v3,w1,w5,LR17=LRv2w3=VA5−u4,u5,v3,v4,w2,w1,LR18=LRv3w4=VA5−u5,u1,v4,v5,w3,w2,LR19=LRv4w5=VA5−u1,u2,v5,v1,w4,w3,LR20=LRv5w1=VA5−u2,u3,v1,v2,w5,w4,LR21=LRu1u2=VA5−u4,v1,w1,w2,w4,LR22=LRu2u3=VA5−u5,v2,w2,w3,w5,LR23=LRu3u4=VA5−u1,v3,w1,w3,w4,LR24=LRu4u5=VA5−u2,,v4,w1,w4,w2,LR25=LRu5u1=VA5−u3,v5,w5,w1,w3,LR26=LRw1w2=VA5−u1,u2,u4,v1,w4,LR27=LRw2w3=VA5−u2,u3,u5,v2,w5,LR28=LRw3w4=VA5−u3,u4,u1,v3,w1,LR29=LRw4w5=VA5−u4,u5,u2,v4,w2,LR30=LRw5w1=VA5−u5,u1,u3,v5,w3,LR31=LRv1v2=VA5−u2,w2,v4,LR32=LRv2v3=VA5−u3,w3,v5,LR33=LRv3v4=VA5−u4,w4,v1,LR34=LRv4v5=VA5−u5,w5,v2,LR35=LRv5v1=VA5−u6,w6,v3.For1≤i≤5, LRei=LRviwi=LRviwi+1=LRuivi=LRviui+1=9 and LRei≤LRx,∀x∈EA5. Furthermore, ∪i=15LRei=15 and LRx∩∪i=15LRei=9,∀x∈EA5 There exists an upper LRF ϕ:VA5⟶0,1 defined as ϕv=1/9, ∀v∈VA5. Consequently, by Theorem 1, dimflA5≤5/3. Similarly, for 1≤i≤5, LRvivi+1=12, LRvivi+1≥LRx,∀x∈EA5. Thus, there exist a lower LRF such that ϕ′:VA6⟶0,1 is defined as ϕ′v=1/12,∀v∈VA5. Consequently, by Theorem 2, dimflA5≤5/4. Hence, we have(16)54≤dimflA5≤53.Case 2. For m>5, by Lemma 3, LRei=LRviwi=LRviwi+1=LRuivi=LRviui+1=3m+3/2 and LRei≤LRx,∀x∈EA5, ∀x∈EAm. Similarly, LRvivi+1=3m−3 and LRvivi+1≥LRx, ∀x∈EAm. Therefore, ϕ:VAm⟶0,1 is defined by ϕv=2/3m+3, ∀v∈VAm is an upper LRF and ϕ′:VAm⟶0,1 is defined by ϕ′v=1/3m−3,∀v∈VAm is a lower LRF. Therefore, by Theorems 1 and 2, we have(17)mm−1≤dimflAm≤2mm+1.Theorem 6.
LetAm with m≥6 be a network of convex polytope, where m≅0mod2. Then,(18)mm−1≤dimflAm≤6m3m+2.Proof.
To prove the result, we have following cases:Case 1. For m=6, the LRN sets are as follows:(19)LR1=LRu1v1=VA6−u2,u3,u4,v4,v5,v6,w5,w6,LR2=LRu2v2=VA6−u2,u3,u4,v5,v6,v1,w6,w1,LR3=LRu3v3=VA6−u3,u4,u5,v6,v1,v2,w1,w2,LR4=LRu4v4=VA6−u4,u5,u6,v1,v2,v3,w2,w3,LR5=LRu5v5=VA6−u5,u6,u1,v2,v3,v4,w3,w4,LR6=LRu6v6=VA6−u6,u1,u2,v3,v4,v5,w4,w5,LR7=LRv1u2=VA6−u1,u5,u6,v2,v3,v4,w3,w4,LR8=LRv2u3=VA6−u2,u6,u1,v3,v4,v5,w4,w5,LR9=LRv3u4=VA6−u3,u1,u2,v4,v5,v1,w5,w6,LR10=LRv4u5=VA6−u4,u2,u3,v5,v6,v1,w6,w1,LR11=LRv5u6=VA6−u5,u3,u4,v6,v1,v2,w1,w2,LR12=LRv6u1=VA6−u6,u4,u5,v1,v5,w3,w2,w3,LR13=LRv1w1=VA6−u5,u6,v4,v5,v6,w2,w3,w4,LR14=LRv2w2=VA6−u6,u1,v5,v6,v1,w3,w4,w5,LR15=LRv3w3=VA6−u1,u2,v6,v1,v2,w4,w5,w6,LR16=LRv4w4=VA6−u2,u3,v1,v2,v3,w5,w6,w1,LR17=LRv5w5=VA6−u3,u4,v2,v3,v4,w6,w1,w2,LR18=LRv6w6=VA6−u4,u5,v3,v4,v5,w1,w2,w3,LR19=LRv1w2=VA6−u3,u4,v2,v3,v4,w1,w5,w6,LR20=LRv2w3=VA6−u4,u5,v3,v4,v5,w2,w6,w1,LR21=LRv3w4=VA6−u5,u6,v4,v5,v6,w3,w1,w2,LR22=LRv4w5=VA6−u6,u1,v5,v6,v1,w4,w2,w3,LR23=LRv5w6=VA6−u1,u2,v6,v1,v2,w5,w3,w4,LR24=LRv6w1=VA6−u2,u3,v1,v2,v3,w6,w4,w5,LR25=LRu1u2=VA6−v1,v4,w1,w2,LR26=LRu2u3=VA6−v2,v5,w2,w3,LR27=LRu3u4=VA6−v3,v6,w3,w4,LR28=LRu4u5=VA6−v4,v1,w4,w5,LR29=LRu5u6=VA6−v5,v2,w4,w1,LR30=LRu6u1=VA6−v6,v3,w4,w1,LR31=LRw1w2=VA6−u1,u2,v1,v4,LR32=LRw2w3=VA6−u2,u3,v2,v5,LR33=LRw3w4=VA6−u3,u4,v3,v6,LR34=LRw4w5=VA6−u4,u5,v4,v1,LR35=LRw5w6=VA6−u5,u6,v5,v2,LR36=LRw6w1=VA6−u6,u1,v6,v3,LR37=LRv1v2=VA6−u2,w2,w5,LR38=LRv2v3=VA6−u3,w3,w6,LR39=LRv3v4=VA6−u4,w4,w1,LR40=LRv4v5=VA6−u5,w5,w2,LR41=LRv5v6=VA6−u6,w6,w3,LR42=LRv6v1=VA6−u1,w1,w4.For1≤i≤6, LRei=LRviwi=LRviwi+1=LRuivi=LRviui+1=10 and LRei≤LRx,∀x∈EA5. Furthermore, ∪i=16LRei=18 and LRx∩∪i=16LRei=10,∀x∈EA6 There exists an upper LRF ϕ:VA6⟶0,1 defined as ϕv=1/10, ∀v∈VA6. Consequently, by Theorem 1, dimflA5≤9/5. Similarly, for 1≤i≤5, LRvivi+1=15 and LRvivi+1≥LRx,∀x∈EA6. Thus, there exists a lower LRF such that ϕ′:VA6⟶0,1 is defined as ϕ′v=1/15,∀v∈VA6. Consequently, by Theorem 2, dimflA6≥6/5. Hence, we have(20)65≤dimflA5≤95.Case 2. For m>6, by Lemma 4, LRei=LRuivi=LRviwi=LRviwi+1=LRviui+1=3m+2/2 and LRei≤LRx, ∀x∈EAm. Similarly, LRvivi+1=3m−3 and LRvivi+1≥LRx, ∀x∈EAm. Therefore, ϕ:VAm⟶0,1 is defined by ϕv=2/3m+2, ∀v∈VAm is an upper LRF andϕ′:VAm⟶0,1 is defined by ϕ′v=1/3m−3,∀v∈VAm is a lower LRF. Therefore, by Theorems 1 and 2, we have(21)mm−1≤dimflAm≤6m3m+2.


## 3.3. LRN Sets and LFMD of Convex PolytopeSm

This subsection deals with the results for the LRN sets and LFMD of the convex polytopes networkSm with m≥5.Table 5
Cardinalities of LRN sets ofSm for m≅1mod2.
Cardinalities of LRN setsComparisonLRvivi+1=4m−44m>4m−4>2m+1LRuiui+1=4m−44m>4m−4>2m+1LRwiwi+1=4m−44m>4m−4>2m+1LRzizi+1=4m−44m>4m−4>2m+1Table 6
Cardinalities of LRN sets ofSm for m≅0mod2.
Cardinalities of LRN setsComparisonLRvivi+1=4m−24m>4m−2>2mLRuiui+1=4m−44m>4m−4>2mLRwiwi+1=4m−44m>4m−4>2mLRzizi+1=4m−44m>4m−4>2mLemma 5.
LetSm be a network of convex polytope, where m≥5 and m≅1mod2. Then, for 1≤i≤m,(a)
LRei=LRviwi=LRwivi+1=2m+1 and LRei′=LRuivi=LRviwi=4m with ∪i=1mLRei=4m(b)
LRei≤LRx≤LRei′,∀x∈ESm(c)
LRx∩∪i=1mLRei≥LRei,∀x∈ESmProof.
Assume thatui, vi, wi, and zi are inner, middle, and outer vertices of Sm, respectively, where 1≤i≤m and m+1≅1modm.(a)
ConsiderLRviwi=ui,um+2i+1/2,um+2i+1/2+1,…,um,vi,vm+2i+1/2,vm+2i+1/2+1,…,vm,wi,wm+2i+1/2,wm+2i+1/2+1,…,wm,zi,zi+1,zi+2,…,zm+2i+1/2+1, LRwivi+1=ui+1,um+2i+1/2,um+2i+1/2+1,…,um,vi+1,vi+2,vi+3,…,vm+2i+1/2,wi,wm+2i+1/2,wm+2i+3/2,…,wm,zi,zm+2i+1/2,zm+2i+3/2,…,zm, and LRuivi=LRwizi=VSm. Therefore, LRei=2m+1 and LRei′=4m. Also, ∪i=1mLRei=VSm=4m.(b)
The LRN sets ofSm other than LRviwi, LRwivi+1, LRuivi, and LRwizi are LRvivi+1=VSm−vm+2i+1/2,um+2i+1/2,wi,zi, LRuiui+1=VSm−um+2i+1/2,vm+2i+1/2,wi,zi, LRwiwi+1=VSm−wm+2i+1/2,zm+2i+1/2,ui+1,vi+1, and LRzizi+1=VSm−ui+1,vi+1,wm+2i+1/2,zm+2i+1/2.The cardinalities of these LRN sets are obtained as given in Table5.From Table5, it is clear that LRei≤LRx≤LRei′,∀x∈ESm.(c)
Since∪i=1mLRei=VSm, therefore LRx∩∪i=1mLRei=LRx≥LRei.Lemma 6.
LetSm be a network of convex polytope, where m≥6 and m≅0mod2. Then, for 1≤i≤m,(a)
LRei=LRviwi=LRwivi+1=2m, LRei′=LRwizi=LRuivi=4m, and ∪i=1mLRei=4m(b)
LRviwi≤LRx≤LRei′, ∀x∈ESm(c)
LRx∩∪i=1mLRei≥LRei,∀x∈ESmProof.
Assume thatui, vi, wi, and zi are inner, middle and outer vertices of Sm, respectively, where 1≤i≤m and m+1≅1modm.(a)
ConsiderLRviwi=ui,um+2i+2/2,um+2i+2/2+1,…,um,vi,vm+2i+2/2,vm+2i+2/2+1,…,vm,wi,wi+1,wi+2,…,wm/2,zi,zi+1,zi+2,…,zm/2, LRwivi+1=ui,ui+1,ui+2,..,um+2i/2,vi+1,vi+2,vi+3,…,vm+2i/2,wi,wm+2i+2/2,wm+2i+3/2,…,wm,zi,zm+2i+2/2,zm+2i+3/2,…,zm, and LRwizi=LRuivi=VSm. Therefore, LRei=2m and LRei′=4m. Also, ∪i=1mLRei=4m.(b)
The LRN sets other thanLRviwi, LRwivi+1, LRuivi, and LRwizi are LRvivi+1=VSm−wi,wm+2i/2, LRwiwi+1=VSm−ui+1,um+2i+2/2,vi+1,vm+2i+2/2, LRuiui+1=VSm−wi,zi,wm+2i/2,zm+2i/2, and LRzizi+1=VSm−ui+1,um+2i+2/2,vi+1,vm+2i+2/2.The cardinalities of these LRN sets are obtained as given in Table6.From Table6, it is clear that LRei≤LRx≤LRei′,∀x∈ESm.(c)
Since∪i=1mLRei=VSm=4m, therefore LRx∩∪i=1mLRei=LRx≥LRei.Theorem 7.
LetSm with m≥5 be a network of convex polytope, where m≅1mod2. Then,(22)1<dimflSm≤2mm+1.Proof.
To prove the result, we have the following cases:Case 1. For m=5, the LRN sets are as follows:(23)LR1=LRv1w1=VS5−u2,u3,v2,v3,w5,w4,z4,z5,LR2=LRv2w2=VS5−u3,u4,v3,v4,w1,w5,z5,z1,LR3=LRv3w3=VS5−u4,u5,v4,v5,w2,w1,z1,z2,LR4=LRv4w4=VS5−u5,u1,v5,v1,w3,w2,z2,z3,LR5=LRv5w5=VS5−u1,u2,v1,v2,w4,w3,z3,z4,LR6=LRv2w1=VS5−u1,u5,v1,v5,w2,w3,z2,z3,LR7=LRv3w2=VS5−u2,u1,v2,v1,w3,w4,z3,z4,LR8=LRv4w3=VS5−u3,u2,v3,v2,w4,w5,z4,z5,LR9=LRv5w4=VS5−u4,u3,v4,v3,w5,w1,z5,z1,LR10=LRv1w5=VS5−u5,u4,v5,v4,w1,w2,z1,z2,LR11=LRz1z2=VS5−v2,u2,u5,w4,z4,LR12=LRz2z3=VS5−v3,u3,u2,w5,z5,LR13=LRz3z4=VS5−v4,u4,u3,w2,z2,LR14=LRz4z5=VS5−v5,u5,u4,w3,z3,LR15=LRz5z1=VS5−v1,u1,u5,w4,z4,LR16=LRw1z1=VS5−w2,w3,w5,LR17=LRw2z2=VS5−w3,w4,w1,LR18=LRw3z3=VS5−w4,w5,w2,LR19=LRw4z4=VS5−w5,w1,w3,LR20=LRw5z5=VS5−w1,w2,w4,LR21=LRv1v2=VS5−v4,w1,z1,LR22=LRv2v3=VS5−v5,w2,z2,LR23=LRv3v4=VS5−v1,w3,z3,LR24=LRv4v5=VS5−v2,w4,z4,LR25=LRv5v1=VS5−v3,w5,z5,LR26=LRu1u2=VS5−u4,w1,z1,LR27=LRu2u3=VS5−u5,w2,z2,LR28=LRu3u4=VS5−u1,w3,z3,LR29=LRu4u5=VS5−u2,w4,z4,LR30=LRu5u1=VS5−u3,w4,z5,LR31=LRu1v1=VS5,LR32=LRu2v2=VS5,LR33=LRu3v3=VS5,LR34=LRu4v4=VS5,LR35=LRu5v5=VS5.For1≤i≤5, LRei=LRviwi=LRviwi+1=12 and LRei≤LRx,∀x∈ES5. Furthermore, ∪i=15LRei=20 and LRx∩∪i=15LRei=9,∀x∈ES5. There exists an upper LRF ϕ:VS5⟶0,1 defined as ϕv=1/12, ∀v∈VS5. Consequently, by Theorem 1, dimflS5≤5/3. Similarly, for 1≤i≤5, LRuivi=24 and LRuivi≥LRx,∀x∈ES5. Thus, there exist a lower LRF such that ϕ′:VS5⟶0,1 is defined as ϕ′v=1/24,∀v∈VS5. Consequently, by Theorem 2, dimflS5≥1. Since S5 is nonbipartite network, therefore dimflS5 must be greater than 1. Hence, we have(24)1<dimflS5≤53.Case 2. For m>5, by Lemma 5, LRei=LRviwi=LRviwi+1=2m+1 and LRei≤LRx,∀x∈ESm. Similarly, LRuivi=4m and LRuivi≥LRx, ∀x∈ESm. Therefore, ϕ:VSm⟶0,1 is defined by ϕv=1/2m+1, ∀v∈VSm is an upper LRF and ϕ′:VSm⟶0,1 is defined by ϕ′v=1/4m,∀v∈VAm is a lower LRF. Therefore, by Theorems 1 and 2, we have(25)1<dimflSm≤2mm+1.Theorem 8.
LetSm with m≥6 be a network of convex polytope, where m≅0mod2. Then,(26)1<dimflSm≤2.Proof.
To prove the result, we have following cases:Case 1. For m=6, the LRN sets are as follows:(27)LR1=LRu1u2=VS6−w1,w4,z1,z4,LR2=LRu2u3=VS6−w2,w5,z2,z5,LR3=LRu3u4=VS6−w3,w6,z3,z6,LR4=LRu4u5=VS6−w4,w1,z4,z1,LR5=LRu5u6=VS6−w5,w2,z5,z2,LR6=LRu6u1=VS6−w6,w3,z6,z3,LR7=LRw1w2=VS6−v2,v5,u2,u5,LR8=LRw2w3=VS6−v3,v6,u3,u6,LR9=LRw3w4=VS6−v4,v1,u4,u1,LR10=LRw4w5=VS6−v5,v2,u5,u2LR11=LRw5w6=VS6−v6,v3,u6,u3,LR12=LRw6w1=VS6−v1,v4,u1,u4,LR13=LRz1z2=VS6−v2,v5,u2,u5,LR14=LRz2z3=VS6−v3,v6,u3,u6,LR15=LRv1w1=u1,u5,u6,v1,v5,v6,w1,w2,w6,z1,z2,z3,LR16=LRv2w2=u2,u6,u1,v2,v6,v1,w2,w3,w1,z2,z3,z4,LR17=LRv3w3=u3,u1,u2,v3,v1,v2,w3,w4,w2,z3,z4,z5,LR18=LRv4w4=u4,u2,u3,v4,v2,v3,w4,w5,w3,z4,z5,z6,LR19=LRv5w5=u5,u3,u4,v5,v3,v4,w5,w6,w4,z5,z6,z1,LR20=LRv6w6=u6,u4,u5,v6,v4,v5,w6,w1,w5,z6,z1,z2,LR21=LRv2w1=u2,u3,u4,v2,v3,v4,w1,w5,w6,z1,z5,z6,LR22=LRv3w2=u3,u4,u5,v3,v4,v5,w2,w6,w1,z2,z6,z1,LR23=LRv4w3=u4,u5,u6,v4,v5,v6,w3,w1,w2,z3,z1,z2,LR24=LRv5w4=u5,u6,u1,v5,v6,v1,w4,w2,w3,z4,z2,z3,LR25=LRv6w5=u6,u1,u2,v6,v1,v2,w5,w3,w4,z5,z3,z4,LR26=LRv6w1=u1,u2,u3,v1,v2,v3,w6,w4,w5,z6,z4,z5,LR27=LRz3z4=VS6−v4,v1,u4,u1,LR28=LRz4z5=VS6−v5,v2,u5,u2,LR29=LRz5z6=VS6−v6,v3,u6,u3,LR30=LRz6z1=VS6−v1,v4,u1,u4,LR31=LRv1v2=VS6−w1,w4,z1,LR32=LRv2v3=VS6−w2,w5,z2,LR33=LRv3v4=VS6−w3,w6,z3,LR34=LRv4v5=VS6−w4,w1,z4,LR35=LRv5v6=VS6−w5,w2,z5,LR36=LRv6v1=VS6−w6,w3,z6,LR37=LRu1v1=VS6,LR38=LRu2v2=VS6,LR39=LRu3v3=VS6,LR40=LRu4v4=VS6,LR41=LRu5v5=VS6,LR42=LRu6v6=VS6,LR43=LRw1z1=VS6,LR44=LRw2z2=VS6,LR45=LRw3z3=VS6,LR46=LRw4z4=VS6,LR47=LRw5z5=VS6,LR48=LRw6z6=VS6.For1≤i≤6, LRei=LRviwi=LRviwi+1=12 and LRei≤LRx,∀x∈ES6. Furthermore, ∪i=16LRei=VS6=20 and LRx∩∪i=16LRei=12,∀x∈ES6. There exists an upper LRF ϕ:VS6⟶0,1 defined as ϕv=1/12, ∀v∈VS6. Consequently, by Theorem 1, dimflS6≤2. Similarly, for 1≤i≤6, LRuivi=24 and LRuivi≥LRx,∀x∈ES6. Thus, there exists a lower LRF such that ϕ′:VS6⟶0,1 is defined as ϕ′v=1/24,∀v∈VS56. Consequently, by Theorem 2, dimflS6≥1. Since S6 is nonbipartite network, therefore dimflS6 must be greater than 1. Hence, we have(28)1<dimflS6≤2.Case 2. For m>6, by Lemma 6, LRei=LRviwi=LRviwi+1=2m and LRei≤LRx,∀x∈ESm. Similarly, LRuivi=4m and LRuivi≥LRx, ∀x∈ESm. Therefore, ϕ:VSm⟶0,1 is defined by ϕv=1/2m, ∀v∈VSm is an upper LRF and ϕ′:VSm⟶0,1 is defined by ϕ′v=1/4m,∀v∈VAm is a lower LRF. Moreover, Sm is a nonbipartite network so dimflSm must be greater than 1. Therefore, by Theorems 1 and 2, we have(29)1<dimflSm≤2.


## 3.4. LRN Sets and LFMD of Convex PolytopeQm

This subsection deals with the results for the LRN sets and LFMD of the convex polytopes networkQm with m≥5.Lemma 7.
LetQm be a network of convex polytope, where m≥5 and m≅1mod2. Then, for 1≤i≤m,(a)
LRei=LRuiui+1=LRzizi+1=LRuivi=LRviwi=LRwizi=4m−4 with ∪i=1mLRei=4m.Proof.
Assume thatui, vi, and wi are inner and middle vertices, respectively, and zi are outer vertices, where 1≤i≤m and m+1≅1modm:(a)
LRuiui+1=VQm−um+2i+1/2,vm+2i+1/2,wi,zi, LRuivi=VQm−vi+1,vm,wi+1,zi+1, LRviwi=VQm−ui+1,vi+2,zm−1,zm, LRzizi+1=VQm−ui+1,vi+1,wm+2i−1/2,zm+2i−1/2, LRwizi=VQm−vm,wi+1,wm,vm+2i−1/2, and LRei=4m−4, where 1≤i≤m.Lemma 8.
LetQm with m≥6 be a network of convex polytope, where m≅0mod2. Then, for 1≤i≤m,(a)
LRei=LRviwi=LRwivi+1=2m+4, LRei′=LRuiui+1=LRuivi=LRzizi+1=LRwizi=4m−4, and ∪i=1mLRei=4m(b)
LRei′∩∪i=1mLRei≥LRei′,∀x∈EQmProof.
Assuming thatui, vi, wi, and zi are inner, middle, and outer vertices, respectively, for 1≤i≤m and m+1≅1modm, we have(a)
LRviwi=VQm−ui+1,ui+2,ui+3,…,um+2i/2,vi+3,vi+4,…,vm+2i/2,wm+2i/2,wm+2i/2+1,wm+2i/2+2,…,wm−2,zm+2i/2,zm+2i+3/2,zm+2i+5/2,…,zm, and LRwivi+1=VQn−ui,um+2i+2/2,um+2i+3/2,…,um,vm+2i+2/2,vm+2i+2/2+1,vm+2i+2/2+2,…,vm,wm+2i+2/2,zi+1,zi+2,…,zm+2i+2/2; therefore, LRei=2m+4, and also ∪i=1mLRei=VQm=4m. Moreover, LRuiui+1=VQm−wi,wm+2i/2,zi,zm+2i/2, LRuivi=VQm−vi+1,vm,wi+1,wm−1, LRzizi+1=VQm−ui+1,vi+1,vm+2i+2/2,um+2i+2/2, and LRwizi=VQm−vi+2,vm,wi+1,wm; therefore, LRei′=4m−4.(b)
Since∪i=1mLRei=VQm, therefore LRei′∩∪i=1mLRei=LRei′≥LRei.Theorem 9.
LetQm with m≥5 be a network of convex polytope, where m≅1mod2. Then,(30)dimflQm=mm−1.Proof.
Form=5, the LRN sets are as follows:(31)LR1=LRu1u2=VQ5−u4,v4,w1,z1,LR2=LRu2u3=VQ5−u5,v5,w2,z2,LR3=LRu3u4=VQ5−u1,v1,w3,z3,LR4=LRu4u5=VQ5−u2,v2,w4,z4,LR5=LRu5u1=VQ5−u3,v3,w5,z5,LR6=LRu1v1=VQ5−v2,v5,w2,w4,LR7=LRu2v2=VQ5−v3,v1,w3,w5,LR8=LRu3v3=VQ5−v4,v2,w4,w1,LR9=LRu4v4=VQ5−v5,v3,w5,w2,LR10=LRu5v5=VQ5−v1,v4,w1,w3,LR11=LRv1w1=VQ5−u2,u3,z4,z5,LR12=LRv2w2=VQ5−u3,u4,z5,z1,LR13=LRv3w3=VQ5−u4,u5,z1,z2,LR14=LRv4w4=VQ5−u5,u1,z2,z3,LR15=LRv5w5=VQ5−u1,u2,z3,z4,LR16=LRw1v2=VQ5−u1,u5,z2,z3,LR17=LRw2v3=VQ5−u2,u1,z3,z4,LR18=LRw3v4=VQ5−u3,u2,z4,z5,LR19=LRw4v5=VQ5−u4,u3,z5,z1,LR20=LRw5v1=VQ5−u5,u4,z1,z2,LR21=LRw1z1=VQ5−v2,v5,w2,w5,LR22=LRw2z2=VQ5−v3,v1,w3,w1,LR23=LRw3z3=VQ5−v4,v2,w4,w2,LR24=LRw4z4=VQ5−v5,v3,w5,w3,LR25=LRw5z5=VQ5−v1,v4,w1,w4,LR26=LRz1z2=VQ5−u2,v2,w4,z4,LR27=LRz2z3=VQ5−v3,u3,w5,z5,LR28=LRz3z4=VQ5−v4,u4,w1,z1,LR29=LRz4z5=VQ5−v5,u5,w2,z2,LR30=LRz5z1=VQ5−v1,u1,w3,z3.
EachLRei=16, where 1≤i≤30. Furthermore, ∪i=130LRei=20. Therefore, by Corollary 1,(32)dimflQ5=54.
Form>5, by Lemma 7, LRei=LRuiui+1=LRzizi+1=LRuivi=LRviwi=LRwizi=4m−4 and ∪i=1mLRei=4m, where 1≤i≤m. Hence, from Corollary 1, we have(33)dimflQm=mm−1.Theorem 10.
LetQm with m≥6 be a network of convex polytope, where m≅0mod2.(34)mm−1≤dimflQm≤2mm+2.Proof.
To prove the result, we have following cases:Case 1. For m=6, the LRN sets are as follows:(35)LR1=LRu1u2=VQ6−w1,w4,z1,z4,LR2=LRu2u3=VQ6−w2,w5,z2,z5,LR3=LRu3u4=VQ6−w3,w6,z3,z6,LR4=LRu4u5=VQ6−w4,w1,z4,z1,LR5=LRu5u6=VQ6−w5,w2,z5,z2,LR6=LRu6u1=VQ6−w6,w3,z6,z3,LR7=LRu1v1=VQ6−v2,v6,w2,w5,LR8=LRu2v2=VQ6−v3,v1,w3,w6,LR9=LRu3v3=VQ6−v4,v2,w4,w1,LR10=LRu4v4=VQ6−v5,v3,w5,w2,LR11=LRu5v5=VQ6−v6,v4,w6,w3,LR12=LRu6v6=VQ6−v1,v5,w1,w4,LR13=LRw1z1=VQ6−v3,v6,w2,w6,LR14=LRw2z2=VQ6−v4,v1,w3,w1,LR15=LRw3z3=VQ6−v5,v2,w4,w2,LR16=LRw4z4=VQ6−v6,v3,w5,w3,LR17=LRw5z5=VQ6−v1,v4,w6,w4,LR18=LRw6z6=VQ6−v2,v3,w1,w5,LR19=LRz1z2=VQ6−u2,u5,v2,v5,LR20=LRz2z3=VQ6−u3,u6,v3,v6,LR21=LRz3z4=VQ6−u4,u1,v4,v1,LR22=LRz4z5=VQ6−u5,u2,v5,v2,LR23=LRz5z6=VQ6−u6,u3,v6,v3,LR24=LRz6z1=VQ6−u1,u4,v1,v4,LR25=LRv1w1=VQ6−u2,u3,u4,v2,v4,z4,z5,z6,LR26=LRv2w2=VQ6−u3,u4,u5,v3,v5,z5,z6,z1,LR27=LRv3w3=VQ6−u4,u5,u6,v4,v6,z6,z1,z2,LR28=LRv4w4=VQ6−u5,u6,u1,v5,v1,z1,z2,z3,LR29=LRv5w5=VQ6−u6,u1,u2,v6,v2,z2,z3,z4,LR30=LRv6w6=VQ6−u1,u2,u3,v1,v3,z3,z4,z5,LR31=LRw1v2=VQ6−u1,u5,u6,v5,w4,z2,z3,z4,LR32=LRw2v3=VQ6−u2,u6,u1,v6,w5,z3,z4,z5,LR33=LRw3v4=VQ6−u3,u1,u2,v1,w6,z4,z5,z6,LR34=LRw4v5=VQ6−u4,u2,u3,v2,w1,z5,z6,z1,LR35=LRw5v6=VQ6−u5,u3,u4,v3,w2,z6,z1,z2,LR36=LRw6v1=VQ6−u5,u4,u5,v4,w3,z1,z2,z3.For1≤i≤6, LRei=LRviwi=LRwivi+1=16 and LRei≤LRx,∀x∈EQ6. Furthermore, ∪i=16LRei=24. Thus, there exists an upper LRF ϕ:VQ6⟶0,1 defined as ϕv=1/16,∀v∈VQ6. Consequently, by Theorem 1, dimflQ6≤3/2. Similarly, for 1≤i≤6, LRei′=LRuivi=LRwizi=LRzizi+1=20 and LRei′≥LRx,∀x∈EQ6. Thus, there exists a LRF ϕ′:VQ6⟶0,1 defined as ϕ′v=1/20 which is a lower LRF ∀v∈VQ6. Consequently, by Theorem 2, dimflQ6≥6/5. Hence, we have(36)65≤dimflQ6≤32.Case 2. For m>6, by Lemma 8, LRei=LRviwi=LRwivi+1=2m+4 and LRei≤LRx,∀x∈EQm. Furthermore, LRei′=LRuiui+1,LRuivi,LRzizi+1 with LRwizi=4m−4 and LRei′≥LRx, ∀x∈EQm. Therefore, ϕ:VQm⟶0,1 defined by ϕv=1/2m+4, ∀v∈VRm is an upper LRF and ϕ′:VQm⟶0,1 defined by ϕ′v=1/4m−4, ∀v∈VQm is a lower LRF. Therefore, by Theorems 1 and 2, we have(37)mm−1≤dimflQm≤2mm+2.


## 4. Conclusion

In this article, we studied the boundedness of convex polytopes networks with the help of distance-based parameter that is called by local fractional metric dimension (LFMD). The lower bounds of convex polytopes networks are also improved from unity. The LFMD of the convex polytopeQm, where m≅1mod2 is m/m−1. Furthermore, other families of convex polytopes such as Rm, Am, Sm, and Qm attain both upper and lower bounds between 1 and 2, respectively, as their order approaches to ∞.Now, we close our discussion with an open problem and characterize all classes of connected networks those remain bounded via LFMD whenm⟶∞.Limiting values of convex polytopes networks are illustrated in Tables7 and 8.Table 7
Limiting values of LFMDs of convex polytopes networks.
m≅LFMDsLimiting LFMDs form⟶∞Comment1mod21<dimflRm≤2m/m+11<dimflRm≤2Bounded1mod21<dimflAm≤2m/m+11<dimflSm≤2Bounded0mod2m/m−1≤dimflAm≤2m/m+21<dimflSm≤2Bounded1mod21≤dimflSm≤2m/m+11<dimflSm≤2Bounded0mod2m/m−1≤dimflQm≤2m/m+21<dimflSm≤2BoundedTable 8
Constant and bounded values of LFMD of convex polytopes networks.
m≅LFMDsComment0mod21<dimflRm≤2Bounded0mod21<dimflSm≤2Bounded



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