# On the Fractional Metric Dimension of Convex Polytopes

Authors: M. K. Aslam; Muhammad Javaid; Q. Zhu; Abdul Raheem
Journal: Mathematical Problems in Engineering
 (2021)

---

## Abstract

In order to identify the basic structural properties of a network such as connectedness, centrality, modularity, accessibility, clustering, vulnerability, and robustness, we need distance-based parameters. A number of tools like these help computer and chemical scientists to resolve the issues of informational and chemical structures. In this way, the related branches of aforementioned sciences are also benefited with these tools as well. In this paper, we are going to study a symmetric class of networks called convex polytopes for the upper and lower bounds of fractional metric dimension (FMD), where FMD is a latest developed mathematical technique depending on the graph-theoretic parameter of distance. Apart from that, we also have improved the lower bound of FMD from unity for all the arbitrary connected networks in its general form.


---

## Body

## 1. Introduction

A networkℵ=Vℵ,Eℵ is a mathematical structure consisting of two set of vertices Vℵ and set of edges Eℵ⊆Vℵ×Vℵ, where Vℵ=v and Eℵ=e are called order and size of ℵ, respectively. In ℵ, the length of the shortest path between any two vertices a,b∈Vℵ is called distance which is usually denoted by da,b. For more information regarding graph-theoretic terminologies, we refer to [1–3].With every passing day, technological boom is reshaping our lives in such a way that the replacement of manpower is being done by robots, devices, and machineries. At the same instance, we cannot compromise on the constraints of employing minimum number of these and their operational cost. In order to overcome these constraints with ease, we have to get the aid of distance-based parameters such as metric dimension (MD). ConsiderM=m1,m2,…,mk⊆Vℵ; then, M becomes an ordered set of vertices bearing some ordering imposed by us. For any b∈Vℵ, the distance of b from all the elements of M in k-tuple metric form is given by rb|M=db,m1,db,m2,db,m3,…,db,mk. The set M becomes a resolving set if, for any pair of distinct vertices a,b∈Vℵ−M, we have ra|M≠rb|ℵ. The resolving set bearing minimum number of vertices in ℵ forms the metric basis of ℵ, and its cardinality represents the MD of ℵ denoted by λℵ [4, 5].Slater gave the terminology of resolving sets by proclaiming them as the locating set for any connected network [6, 7]. After studying these terminologies by themselves, Harary and Melter gave them the name of MD of a network. Afterwards, a large number of researchers computed the MD of different families of networks. The families of networks having constant and bounded MD have been the topic of [8]. Chartrand et al., on the contrary, proved the MD of path and cycle [4]. In the same manner, ℙn,2, Am, and ℂm2 have been proved to be the networks with constant MD in [8]. Moreover, in [9], the MD of generalized Petersen network is proved to be bounded.Chartrand et al. utilized MD to present the solution of integer programming problem (IPP) [4]. Later on, for the sake of acquiring a higher accuracy solution of an IPP, Currie and Oellermann [10] founded the concept of fractional metric dimension (FMD). Fehr et al. employed FMD to get the optimal solution of a certain linear programming relaxation problem [11]. Arumugam and Mathew brought some undiscovered features of FMD to light [12]. Afterwards, a large number of results appeared related to the FMD of several networks that are formed as an aftermath of graph products, namely, Cartesian, hierarchial, corona, lexicographic, and comb products, see [12], [13–15], and [16]. Similarly, Liu et al. computed the FMD of generalized Jahangir network [17]. In a recent time, Raza et al. evaluated the FMD of metal organic networks [18].Aisyah et al. pioneered the terminology of local fractional metric dimension (LFMD) and calculated for the corona product of two networks [19]. In the same manner, Liu et al. calculated the LFMD of rotationally symmetric and planar networks [20]. More recently, Javaid et al. evaluated the sharp extremal values of LFMD of connected networks [21]. In this paper, firstly, we improved the lower bound of FMD from unity, and secondly, the FMD of a family of network bearing rotational symmetry called convex polytopes is computed in the form of bounds. The flux of this paper is in the following sequence. Section 1 is introduction. Section 2 is related to the MD’s role in Robotics and Chemistry. Section 3 is of preliminaries. In Section 4, the theoretical development of improved lower bound of FMD of a connected network is done. Section 5 contains the main results regarding the FMD of convex polytopes. Section 6 winds up the paper with a hand full of concluding remarks.


## 2. Applications

The increasing demand of networking is nurturing the development in distance-based dimensions. All such tools help in the allocation of an interpolar to a suitable region for employing it effectively [6, 22]. In the same way, allocating the robots in some production units and in public health facilities have been discussed in [23]. Moreover, for various techniques for the rectification of example and picture handling and information handling, we refer to [24]. Similarly, a chemical compound in the graph-theoretic form is regarded as a molecular graph having nodes as atoms and links between them as bonds [4]. With the aid of picturesque form of a compound and distance-based parameters, chemists are now able to not only remove discrepancies in some chemical structure but also are able to find the sites showing similar properties in them. All these techniques are the topic of [5, 6, 24, 25]. More recently, the same tools are found helpful in solving resolvability problems in nanotechnology and polymer-based industries [26, 27].


## 3. Preliminaries

In a networkℵ=Vℵ,Eℵ, for a,b⊆Vℵ, a vertex c resolves a,b in ℵ if da,c≠db,c. For any pair of vertices a,b∈Vℵ, the resolving neighbourhood set (RNs) of a,b is given by Ra,b=c∈Vℵ|da,c≠db,c.Suppose that, in a connected network,ℵ=Vℵ,Eℵ, bearing v as its order. A function ψ:Vℵ⟶0,1 is called the upper resolving function (URF) of ℵ if ψRa,b≥1,∀Ra,b in ℵ, where ψRa,b=∑c∈Ra,bψc.A URFψ is called the minimal upper resolving function (MURF) if ∃ is another URF η such that η≤ψ and ηc≠ψc for at least one c∈Vℵ. Similarly, a function defined as κ:Vℵ⟶0,1 is called lower resolving function (LRF) if κRa,b≤1,∀Ra,b in ℵ. An LRF κ is called the maximal lower resolving function (MLRF) if ∃ is another LRF μ such that μ≥κ and μc≠κc for at least one c∈Vℵ. The FMD of a connected network is defined as dimfℵ=χ, where χ=minψ:ψis the upper minimal resolving function or χ=maxκ:κis the lower maximal resolving function.


### 3.1. Convex Polytopes

The networkℵ≅Bm of convex polytope type I bearing 3m 3-sided faces, m 4-sided faces, n 5-sided faces, and a pair of n-sided faces is obtained by the combination of the network of convex polytope ℚn and prism network Dn [28]. The sets VBm and EBm are VBm=vr,wr,xr,yr,zr|1≤r≤m and EBm=vrvr+1,wrwr+1,xrxr+1,yryr+1,zrzr+1|1≤r≤m∪vrwr,wrxr,wr+1xr,xryr,yrzr|1≤r≤m, respectively. In the same manner, the cycles induced by vr|1≤r≤m, wr|1≤r≤m, xr|1≤r≤m, yr|1≤r≤m, and zr|1≤r≤m are called by inner, interior, exterior, and outer cycle, respectively. Figure 1 illustrates Bm.Figure 1
Type I convex polytopeBm.The network of convex polytope type IIℵ≅ℂm is created out of the network of Bm by adding new edges yr+1zr. It comprises of 3m 3-sided faces, m 4-sided faces, m 5-sided faces, and a pair of m-sided faces. Similarly, the cycles induced by vr|1≤r≤m, wr|1≤r≤m, xr|1≤r≤m, yr|1≤r≤m, and zr|1≤r≤m are called by inner, interior, exterior, and outer cycle, respectively. Figure 2 illustrates ℂm.Figure 2
Type II convex polytopeℂm.The setsVℂm and Eℂm are given by Vℂm=vr,wr,xr,yr,zr|1≤r≤m and Eℂm=vrvr+1,wrwr+1,xrxr+1,yryr+1,zrzr+1|1≤r≤m∪vrwr,wrxr,wr+1xr,xryr,yr+1zr|1≤r≤m, respectively.


## 3.1. Convex Polytopes

The networkℵ≅Bm of convex polytope type I bearing 3m 3-sided faces, m 4-sided faces, n 5-sided faces, and a pair of n-sided faces is obtained by the combination of the network of convex polytope ℚn and prism network Dn [28]. The sets VBm and EBm are VBm=vr,wr,xr,yr,zr|1≤r≤m and EBm=vrvr+1,wrwr+1,xrxr+1,yryr+1,zrzr+1|1≤r≤m∪vrwr,wrxr,wr+1xr,xryr,yrzr|1≤r≤m, respectively. In the same manner, the cycles induced by vr|1≤r≤m, wr|1≤r≤m, xr|1≤r≤m, yr|1≤r≤m, and zr|1≤r≤m are called by inner, interior, exterior, and outer cycle, respectively. Figure 1 illustrates Bm.Figure 1
Type I convex polytopeBm.The network of convex polytope type IIℵ≅ℂm is created out of the network of Bm by adding new edges yr+1zr. It comprises of 3m 3-sided faces, m 4-sided faces, m 5-sided faces, and a pair of m-sided faces. Similarly, the cycles induced by vr|1≤r≤m, wr|1≤r≤m, xr|1≤r≤m, yr|1≤r≤m, and zr|1≤r≤m are called by inner, interior, exterior, and outer cycle, respectively. Figure 2 illustrates ℂm.Figure 2
Type II convex polytopeℂm.The setsVℂm and Eℂm are given by Vℂm=vr,wr,xr,yr,zr|1≤r≤m and Eℂm=vrvr+1,wrwr+1,xrxr+1,yryr+1,zrzr+1|1≤r≤m∪vrwr,wrxr,wr+1xr,xryr,yr+1zr|1≤r≤m, respectively.


## 4. Lower Bound of FMD of Connected Network

In this section, we develop criteria for the improved lower bound of FMD of connected network. Before going further, we give the following proposition.Proposition 1.
Suppose thatℵ is a connected network and R=Ra,b is the resolving neighbourhood set for any a,b⊂Vℵ. For κ=maxR, if Y=∪R:R=κ⊆Vℵ, then R∩Y≤κ for each resolving neighbourhood set of ℵ.Lemma 1.
Letℵ be a connected network and R be the resolving neighbourhood set. Then,(1)Vℵκ≤dimfℵ,where κ=maxR and 2≤κ≤Vℵ.Proof.
Here, we define a mappingη:Vℵ⟶0,1 such that ηb=1/κ for b∈Vℵ. By Proposition 1, for a,b⊂Vℵ, we have(2)ηR=∑x∈Rηx=∑x∈R∩Vℵ1κ=R∩Vℵ1κ≤1.
This shows the fact thatη is a lower resolving function (LRF). In order to show that η is maximal LRF, suppose that there exists another LRF ϕ such that ϕx≥ηx, where ϕx≠ηx for at least one x∈Vℵ. For all x∈R such that R=κ, we have(3)ϕR=∑x∈Rϕx>∑x∈Rηx=1.
Hence,ϕR>1. This shows that ϕ is not LRF, and consequently, η is maximal LRF. Let η¯ be another maximal LRF of ℵ. Then,(4)η¯=∑x∈Vℵηx.
Now, we assume the following three cases: (a)η¯x>1/κ,∀x∈Vℵ, (b) η¯x≤1/κ,∀x∈Vℵ, and (c) η¯x>1/κ for some x∈Vℵ.Case (a): ifη¯x>1/κ,∀x∈Vℵ. For R⊆Y such that R=κ, we have η¯R>1. This shows that η¯ is not LRF. Hence, this case does not hold.Case (b): suppose thatη¯x≤1/κ,∀x∈Vℵ. Then,(5)η¯=∑x∈Vℵη¯x≤Vℵκ=η.Consequently,(6)dimfℵ=Vℵκ.Case (c): ifη¯x>1/κ for some x∈Vℵ, suppose that S=t∈Vℵ|η¯t>1/κ and Y=∪R:R=κ. We observe that S∩Y=Φ; otherwise, for κ=R, ηx¯>1 which implies that η¯ is not a LRF. Consider(7)η¯=∑x∈Vℵη¯x=∑x∈Yη¯x+∑x∈Vℵ−Yη¯x.As∑x∈Vℵ−Yη¯x≥∑x∈Vℵ−Yηx, hence,(8)η¯=∑x∈Vℵη¯x=∑x∈Yη¯x+∑x∈Vℵ−Yη¯x≥∑x∈Yηx+∑x∈Vℵ−Yηx=Vℵκ=η.Consequently,(9)dimfℵ=η¯≥η=Vℵκ.Therefore, from all the above case, we have(10)Vℵκ≤dimfℵ.This completes the proof.


## 5. Main Results

In this part of paper, we discuss the main results of our findings regarding the networks under consideration. Lemmas2 and 3 concern with RNs of Bm and ℂm, respectively. Similarly, Theorems 1 and 2 give the upper and lower bounds of FMD of the aforementioned networks.Lemma 2.
Letℵ≅Bm be a type I convex polytope, with m≥6 and m≡0mod2. For 1≤l,r≤m, p≥3, and s≥2, s≡0mod2 and p≡1mod2; then,(a)
Rl=Rwl,xl=Rr=Rwr,xr−1=7m/2+1, ∪l=1mRl=5m, ∪r=1mRr=5m, and ∪r=1mRr∪∪r=1mRr=5m(b)
Rr<R′1=Rvr,xr=R′2=Rvr,xr−1 and R′u∩∪r=1mRr≥Rr(c)
Rt<R′3=Rvr,vr+1=R′4=Rvr,vr+p=R′5=Rwr,wr+1=R′6=Rwr,wr+p=R′7=Rxr,xr+s=R′8=Ryr,yr+s=R′9=Rzr,zr+s=5m−6 and R′u∩∪r=1mRr≥Rr with vm+1=v1(d)
Rr<R′10=Rvr,vr+s=|R′11|=|Rwr,wr+s|=R′12=Rxr,xr+1=R′13=Rxr,xr+p=R′14=Ryr,yr+1=R′15=Ryr,yr+p=R′16=Rzr,zr+1=R′17=Rzr,zr+p=R′18=Rvr,wr+p=R′19=Ryr,zr+p=5m−4, where p≥3 and R′u∩∪r=1mRr≥Rr(e)
Rt<R20=Rwr,xr+p=R21=Rwr,xr+s=R′22=Rvr,zr+p=R′23=Rvr,yr+s=5m−3, where R′u∩∪r=1mRr≥Rr(f)
Rt<R′24=Rvr,zr+s=R′25=Rwr,zr+p=R′26=Rwr,zr+s=R′27=Rvr,yr+p=R′28=Rxr,yr=R′29=Ryr,zr5m−2 and R′u∩∪r=1mRr≥Rr(g)
Rt<R′30=Rxr,zr and R′u∩∪r=1mRr≥Rr(h)
Rt<R′24=Rvr,zr+s=R′25=Rwr,zr+p=R′26=Rwr,zr+s=R′27=Rvr,yr+p=R′28=Rxr,yr=R′29=Ryr,zr5m−2 and R′u∩∪r=1mRr≥Rr(i)
Rt<R′39=Rvr,wr=5m and R′u∩∪r=1mRr≥RrProof.
(a)
The RNs ofwl,xl and wr,xr−1 are Rwl,xl=VBm−vh|h≡l+1,l+2,…,l+m/2modm∪wh|h≡l+1,l+2,…,l+m/2modm∪xh|h≡l+1,l+2,…,l+m/2−1modmand Rwr,xr−1=VBm−vh|h≡r−1,r−2,…,r−m/2modm∪wh|h≡r−1,r−2,…,r−m/2modm∪xh|h≡r−2,r−3,…,r−m/2+1modm. We note that ∪l=1mRl=VBm, ∪r=1mRr=VBm and ∪l=1mRl=∪r=1mRr=5m, and ∪l=1mRr∪∪r=1mRr=5m,(b)
The RNs ofvr,xr and vr,xr−1 are Ŕ1=Rvr,xr=VG−wh|h≡r,r−1,…,r−m/2+1modm∪xh|h≡r,r−1,…,r−m/2modm and Ŕ2=Rvr,xr−1=VG−wh|h≡r+1,r+2,…,r+m/2modm∪xh|h≡r+1,r+2,…,r+m/2modm, respectively. It is clear from the above that Rr<Ŕu and Ŕu∩∪r=1mRr≥Rr.(c)
The RNs ofvr,vr+1,vr,vr+p,wr,wr+1,wr,wr+p,xr,xr+s,xr,xr+s,yr,yr+s, and zr,zr+s are R′3=Rvr,vr+1=VBm−xh|h≡r,r+m/2modm∪yh|h≡r,r+m/2modm∪zh|h≡r,r+m/2modm=R′5=Rwr,wr+p, R′4=Rvr,vr+p=VBm−xh|h≡r+p−1/2,r+p+m−1/2modm∪yh|h≡r+p−1/2,r+p+m−1/2modm∪zh|h≡r+p−1/2,r+p+m−1/2modm=R′6=Rwr,wr+p, R′7=Rxr,xr+s=VBm−xh|h≡r+s/2,r+m+s/2modm∪yh|h≡r+s/2,r+m+s/2modm∪zh|h≡r+s/2,r+m+s/2modm=R′8=Ryr,yr+s=R′9=Rzr,zr+s, respectively. Clearly, R′u=5m−6. Since Rr=7m/2+1<R′u, then R′u∩∪r=1mRr=5m−6≥Rr,(d)
The RN’s ofvr,vr+s,wr,wr+s,xr,xr+1,xr,xr+p,yr,yr+1,yr,yr+p,zr,zr+1,zr,zr+p,vr,wr+p, and yr,zr+p are given by Ŕ10=Rvr,vr+s=VBm−vh|h≡r+s/2,r+s+m/2modm∪wh|h≡r+s/2,r+s+m/2modm=Ŕ11=Rwr,wr+s=Ŕ13=Rxr,xr+p=Ŕ15=Ryr,yr+p=Ŕ17=Rzr,zr+p, Ŕ18=Rvr,wr+p=VBm−vh|h≡r+p+1/2,r+p+m+1/2modm∪wh|h≡r+p−1/2,r−p−1/2modm, Ŕ19=Ryr,zr+p=VBm−xh|h≡r+p+1/2,r+p+m+1/2modm∪yh|h≡r+p+1/2,r+p+m+1/2modm, where Ŕ12=Rxr,xr+1=Ŕ14=Ryr,yr+1=Ŕ16=Rzr,zr+1=Rvr,vr+s. We can see that Ŕu=5m−4>Rt and Ŕu∩∪t=12Rt≥Rt.(e)
The RNs ofwr,xr+p, wr,xr+s, vr,zr+p, and vr,yr+s are Ŕ20=Rwr,xr+p=VG−vh|h≡r+p+1/2modm∪wh|h≡r+p+1/2modm∪xh|h≡r+p+m+1/2modm, Ŕ21=Rwr,xr+s=VG−vh|h≡r+m+s/2modm∪wh|h≡r+m+s/2modm∪xh|h≡r+s/2modm, Ŕ22=Rvr,yr+s=VBm−wh|h≡r+s−2/2modm∪yh|h≡r+s−2/2modm∪zhh≡r+s+2/2modm, and Ŕ23=Rvr,zr+p=VBm−wh|h≡r+p+1/2modm, and Ŕ28=Rxr,yr=VBm−vh|h≡r−1,r+1modm=Ŕ29=Ryr,zr, respectively. We can see that Ŕu=5m−3>Rt and Ŕu∩∪t=12Rt≥Rt.(g)
The resolving neighbourhood ofxr,zr is Ŕ30=Rxr,zr=VBm−yh|1≤h≤6. We can see that Ŕu=4m>Rt and Ŕu∩∪t=1mRt>Rt.(h)
The resolving neighbourhoods ofvr,wr+1 and vr,zr+1 are Ŕ31=Rvr,wr+1=VBm−vh|h≡r+1,r+2,…,r+m/2modn, Ŕ32=Rvr,zr+1=VBm−zh|h≡r+1,r+2,…,r+m/2modn, Ŕ33=Rvr,xr+1=VBm−zh|h≡r+2,r+3,…,r+m/2modm∪xh|h≡rmodm, Ŕ34=Rwr,zr+1=VBm−wh|h≡r+2,r+3,…,r+m/2modm∪yh|h≡rmodm, Ŕ35=Rvr,wr−1=VBm−vh|h≡r−1,r−2,…,r−m/2modn, Ŕ36=Rvr,zr−1=VBm−zh|h≡r−1,r−2,…,r−m/2modn, Ŕ37=Rvr,xr−1=VBm−zh|h≡r−2,r−3,…,r−m/2modm∪xh|h≡r−1modm, and Ŕ38=Rwr,zr−1=VBm−wh|h≡r−2,r−3,…,r−m/2modm∪yh|h≡r−1modm, respectively. We can see that Ŕu=9m/2>Rt and Ŕu∩∪t=1mRt>Rt.(i)
The RN ofvr,wr is Ŕ39=Rvr,wr=VBm. Clearly, Rwrxr|=3m/2+1<Rvrwr and Rvrwr∩∪r=1mRr=3n≥Rr.Theorem 1.
Ifℵ≅Bm with m≥6 and m≡0mod2, then dimfBm<10m/7m+2.Proof.
Case I
m=6.The RNs are given as follows.In the same way, from Lemma2, we can see that Rxr,xr+s=Ryr,yr+s=Rzr,zr+s=Rvr,vr+p−1.Similarly, from Lemma2, we can see that Rvr,vr+s=Ryr,yr+s=Rzr,zr+p.Tables1 and 2 represent the RNs having cardinality of 24, whereas Tables 3–6 show the RNs with cardinalities of 26, 27, 28, and 30, respectively. On the contrary, Table 7 bears RNs with minimum cardinality of 22. Also, it is observed that ∪r=112Rr=VB6; this implies ∪r=112Rr=30 and Rr¯∩∪r=16Rr≥Rr.Now, we define a functionμ:VB6⟶0,1 such that μvr=μwr=μxr=μyr=μzr=1/22. As Rr for 1≤t≤12 of B6 are pairwise overlapping, hence, ∃ is another minimal resolving function κ¯ of B6 such that μ¯<μ. As a result, dimfB6<∑r=1181/10<30/22.Similarly, Table7 shows the RNs with maximum cardinality of 30=κ; hence, by Lemma 1, VB6/κ=30/30=1<dimfB6.Therefore,(11)1<dimfB6<3022.Case II
m≥8.We have seen from Lemma2 that the RNs with minimum cardinality of 7m/2+1 are Rwl,xl and Rwr,xr−1 and ∪t=1mRt=VBm. Let λ=7m/2+1 and δ=∪t=1mRt=5m. Now, we define a mapping μ:VBm⟶0,1 such that(12)μa=1λ,fora∈∪t=1mRt,0,fora∈VB−∪t=1mRt.We can see thatμ is a RF for Bm with m≥3 because μRu,v≥1,∀u,v∈VBm. On the contrary, assume that there is another resolving function ρ, such that ρu≤μu, for at least one u∈VBm,ρu≠μu. As a consequence, ρRu,v<1, where Ru,v is a RN of Bm with minimum cardinality λ. This implies that ρ is not a resolving function which is contradiction. Therefore, μ is a minimal resolving function that attains minimum μ for Bm. Since all Rr are having pairwise nonempty intersection, so there is another minimal resolving function of μ¯ of Bm such that μ¯≤μ. Hence, assigning 1/λ to the vertices of Bm in ∪t=12mRt and calculating the summation of all the weights, we obtain(13)dimfBm=∑t=1δ1λ≤5m7m/2+1=10m7m+2.Also, the RN with maximum cardinality of5m is Rvr,wr. Let VBm=ω and Rvr,wr=κ; thus, from Lemma 2, we have VBm/κ=ω/κ=5m/5m=1<dimfB.Therefore, we conclude the following:(14)1<dimfBm<10m7m+2.Table 1
The representation ofR′u for 1≤u≤41.
RNsElementsEqualityRv1,v2VB6−x1,x4∪y1,y4∪z1,z4Rv4,v5,Rw1,w2,Rw3,w6,Rv3,v6,Rw4,w7,Rx4,x7,Rx3,x5Rv2,v3VB6−x2,x5∪y2,y5∪z2,z5Rv5,v6,Rw2,w3,Rw5,w6,Rv1,v4,Rw1,w4,Rv5,v8,Rw5,w8,Rx1,x3,Rw5,w8,Rx1,x3,Rx4,x6Rv3,v4VB6−x3,x6∪y3,y6∪z3,z6Rv1,v6,Rw1,w6,Rw1,w8,Rv2,v5,Rw2,w5,Rv1,v6,Rw1,w6,Rx2,x4,Rx1,x5Rv1,x1VB6−w1,w5,w6∪x4,x5,x6Rv2,x2VB6−w1,w2,w6∪x1,x5,x6Rv3,x3VB6−w1,w2,w3∪x1,x2,x6Rv4,x4VB6−w2,w3,w4∪x1,x2,x3Rv5,x5VB6−w3,w4,w5∪x2,x3,x4Rv6,x6VB6−w4,w5,w6∪x3,x4,x5Rv2,x1VB6−w2,w3,w4∪x2,x3,x4Rv3,x2VB6−w3,w4,w5∪x3,x4,x5Rv4,x3VB6−w4,w5,w6∪x4,x5,x6Rv5,x4VB6−w1,w5,w6∪x1,x5,x6Rv6,x5VB6−w1,w2,w6∪x1,x2,x6Table 2
The representation ofR′u for 42≤u≤47.
RNsElementsRx1,z1Rx2,z2VB6−yh|1≤h≤6Rx3,z3Rx4,z4Rx5,z5Rx5,z5Table 3
The representation ofR′u for 48≤u≤81.
RNsElementsEqualityRv1,v3VB6−v2,v5∪w2,w5Rv5,v7,Rw1,w3,Rw5,w7,Rv4,v8,Rw4,w8,Rx1,x2,Rx4,x5,Rx5,x8Rv2,v4VB6−v3,v6∪w3,w6Rv6,v8,Rw2,w4,Rw6,w8,Rv1,v5,Rw1,w5,Rx2,x3,Rx5,x6,Rx1,x4,Rx1,x6Rv3,v5VB6−v4,v7∪w4,w7Rv1,v7,Rw3,w5,Rw1,w7,Rw2,w6,Rx3,x4,Rx6,x7,Rx2,x5,Rx2,x7,Rv2,v6Ry1,z4VB6−x3,x6∪y3,y6Ry2,z5VB6−x1,x4∪y1,y4Ry3,z6VB6−x2,x5∪y2,y5Rv1,w4VB6−v3,v6∪w2,w6Rv2,w5VB6−v1,v4∪w1,y2Rv3,w6VB6−v2,v5∪w2,w4Table 4
The representation ofR′u for 82≤u≤136.
RNsElementsEqualityRv1,y3VB6−w1∪y1∪z3Rv4,y6Rv2,y4VB6−w2∪y2∪z4Rv1,y5Rv3,y5VB6−w3∪y3∪z5Rv2,y6Rv1,z4VB6−w5∪x5∪y6Rv2,z5VB6−w6∪y6∪y1Rv3,z6VB6−w1∪y1∪y2Rv1,w2VB6−v2,v3,v4Rv5,w4Rv2,w3VB6−v3,v4,v5Rv6,w5Rv3,w4VB6−v4,v5,v6Rv4,w5VB6−v1,v5,v6Rv2,w1Rv5,w6VB6−v1,v2,v6Rv3,w2Rv1,w6VB6−z1,v2,v3Rv4,w3Rv1,z2VB6−z2,z3,z4Rv5,z4Rv2,z3VB6−z3,z4,z5Rv6,z5Rv3,z4VB6−z4,z5,z6Rv4,z5VB6−z1,z5,z6Rv2,z1Rv5,z6VB6−z1,z2,z6Rv3,z2Rv1,z6VB6−z1,z2,z3Rv4,z3Rv1,x2VB6−z3,z4∪x1Rv2,x3VB6−z4,z5∪x2Rv3,x4VB6−z5,z6∪x3Rv4,x5VB6−z1,z6∪x4Rv5,x6VB6−z1,z2∪x5Rv1,x6VB6−z2,z3∪x6Rv2,x1VB6−z5,z6∪x1Rv3,x2VB6−z1,z6∪x2Rv4,x3VB6−z1,z2∪x3Rv5,x4VB6−z2,z3∪x4Rv6,x5VB6−z3,z4∪x5Rw1,z2VB6−z3,z4∪y1Rw2,z3VB6−w4,z5∪y2Rw3,z4VB6−w5,w6∪y3Rw4,z5VB6−w1,w6∪y4Rw5,z6VB6−w1,w2∪y5Rw1,z6VB6−w2,w3∪y6Rw2,z1VB6−w5,w6∪y1Rw3,z2VB6−w1,w6∪y2Rw4,z3VB6−w1,w2∪y3Rw5,z4VB6−w2,w3∪y4Rw6,z5VB6−w3,w4∪y5Rw1,z6VB6−w4,w5∪y6Table 5
The representation ofR′u for 137≤u≤174.
RNsElementsEqualityRv1,z3VB6−w3∪y1Rv4,y6Rv2,z4VB6−w4∪y2Rv1,y5Rv3,z5VB6−w5∪y3Rv4,z6VB6−w6∪y4Rv1,z5VB6−w2∪y5Rv2,z6VB6−w1∪y6Rw1,z4VB6−x3∪y2Rw2,z5VB6−x4∪y2Rw3,z6VB6−x5∪y3Rw1,z3VB6−x4∪y5Rw2,z4VB6−x4∪y6Rw3,z5VB6−x5∪y1Rw4,z6VB6−x6∪y2Rw1,z5VB6−x1∪y3Rw2,z6VB6−x2∪y4Rv1,y4VB6−w3∪y1Rv2,y5VB6−w4∪y2Rv3,y6Rv1,y4Rv1,y3VB6−v4∪w5Rv2,y4VB6−v5∪w6Rv3,y5VB6−v6∪w1Rv4,y6VB6−v1∪w2Rv1,y5VB6−v2∪w3Rv2,y6VB6−v3∪w4Rx1,y1VB6−v2,v6Ry1,z1Rx2,y2VB6−v1,v3Ry2,z2Rx3,y3VB6−v2,v4Ry3,z3Rx4,y4VB6−v3,v5Ry4,z4Rx5,y5VB6−v4,v6Ry5,z5Rx6,y6VB6−v1,v5Ry6,z6Table 6
The representation ofR′u for 175≤u≤180.
RNsElementsRv1,w1Rv2,w2Rv3,w3Rv4,w4VB6Rv5,w5Rv5,w5Table 7
The representation ofRr for 1≤r≤12.
RNsElementsR1=Rw1,x1VB6−v2,v3,v4∪w2,w3,w4∪x2,x3R2=Rw2,x2VB6−v3,v4,v5∪w3,w4,w5∪x3,x4R3=Rw3,x3VB6−v4,v5,v6∪w4,w5,w6∪x4,x5R4=Rw4,x4VB6−v1,v5,v6∪w1,w5,w6∪x5,x6R5=Rw5,x5VB6−v1,v2,v6∪w1,w2,w6∪x1,x6R6=Rw6,x6VB6−v1,v2,v3∪w1,w2,w3∪x1,x2R7=Rw2,x1VB6−v1,v5,v6∪w1,w5,w6∪x5,x6R8=Rw3,x2VB6−v1,v2,v6∪w1,w2,w6∪x1,x6R9=Rw4,x3VB6−v1,v2,v3∪w1,w2,w3∪x1,x2R10=Rw5,x4VB6−v2,v3,v4∪w2,w3,w4∪x2,x3R11=Rw6,x5VB6−v3,v4,v5∪w3,w4,w5∪x3,x4R12=Rw1,x6VB6−v4,v5,v6∪w4,w5,w6∪x4,x5Lemma 3.
Letℵ≅ℂm be a type II convex polytope, where m≥6 and m≡0mod2. For 1≤l,r≤m, p≥3, s≥2, s≡0mod2, and p≡1mod2, then(a)
Rl=Ryl,zl=Rr=Ryr,zr−1=3m+1, ∪l=1mRl=5m, ∪r=1mRr=5m, and ∪r=1mRr∪∪r=1mRr=5m(b)
Rr<Ŕ1=Rwr,xr=Ŕ2=Rwr,xr−1=7m/2+1 and ∪r=1mRr=5m(c)
Rr<R′3=Rvr,vr+1=R′4=Rvr,vr+p=R′5=Rwr,wr+1=R′6=Rwr,wr+p=R′7=Rxr,xr+s=R′8=Rxr,xr+1=R′9=Rxr,xr+p=R′10=Ryr,yr+1=R′11=Ryr,yr+p=R′12=Rzr,zr+s=R′13=Rvr,wr+p=5m−6 and R′u∩∪r=1mRr≥Rr,(d)
Rr<R′14=Rzr,zr+p=R′15=Rvr,zr+s=R′16=Rwr,xr+p=R′17=Rwr,yr+s=R′18=Rxr,yr+s=R′19=Rxr,zr+s=5m−4 and R′u∩∪r=1mRr≥Rr, where p≥3(e)
Rr<R′20=Rvr,xr+p=R′21=Rvr,xr+s=R′22=Rvr,yr+s=R′23=Rwr,yr+p=R′24=Rwr,zr+s=R′25=Ryr,zr+s=R′26=Rxr,zr+p=5m−3 and R′u∩∪r=1mRr≥Rr with vm+1=v1, where p≥3(f)
Rr<Ŕ27=Rwr,yr=Ŕ28=Rwr,zr=Ŕ29=Rvr,yr+p=Ŕ30=Rxr,yr+p=5m−2 and Ŕu∩∪r=1mRr≥Rr, where p≥3(g)
Rr<Ŕ31=Ryr,zr+p=Ŕ32=Ryr,zr+p=5n−1 and Ŕu∩∪r=1mRr≥Rr, where p≥3(h)
Rr<Ŕ33=Rvr,xr=Ŕ34=Rvr,xr−1=4m+1 and Ŕu∩∪r=1mRr≥Rr, where p≥3(i)
Rr<Ŕ35=Rxr,yr+1=4m+2 and Ŕu∩∪r=1mRr≥Rr(j)
Rr<Ŕ36=Rxr,zr+1=4m and Ŕu∩∪r=1mRr≥Rr(k)
Rr<Ŕ37=Rvr,wr+1=Ŕ38=Rvr,yr=9m/2 and Ŕu∩∪r=1mRr≥Rr(l)
Rr<Ŕ39=Rvr,xr+1=9m/2−1 and Ŕu∩∪r=1mRr≥Rr(m)
Rr<Ŕ40=Rvr,zr=Ŕ41=Rxr,zr=9m/2+1 and Ŕu∩∪r=1mRr≥Rr(n)
Rr<Ŕ42=Rvr,wr=Ŕ43=Rxr,yr=5m and Ŕu∩∪r=1mRr≥RrProof.
(a)
The RNs ofyl,zl and yr,zr−1 are Ryl,zl=Vℂm−vh|h≡l+2,l+3,…,l+m/,2modm∪wh|h≡l+2,l+3,…,l+m/2modm∪yh|h≡l−1,l−2,…,l−m/2+1modm∪zh|h≡l−1,l−2,…,l−m/2modm and Ryr,zr−1=Vℂm−vh|h≡r−2,r−3,…,r−m/2modm∪wh|h≡r−2,r−3,…,r−m/2modm∪yh|h≡r+1,r+2,…,r+m/2−1modm∪zh|h≡r+1,r+2,…,r+m/2modm, respectively. We note that Rr=Rl=3m+1,∪l=1mRl=Vℂm,∪r=1mRr=Vℂm and ∪l=1mRl∪∪r=1mRr=5m. Also, ∪l=1mRl=∪r=1mRr=5m.(b)
The RNs ofwr,xr and wr,xr−1 are Ŕ1=Rwr,xr=Vℂm−vh|h≡r+1,r+2,…,r+m/2modm∪wh|h≡r+1,r+2,…,r+m/2modm∪xh|h≡r+1,r+2,…,r+m/2−1modm, Ŕ2=Rwr,xr−1=Vℂm−vh|h≡r−1,r−2,…,r−m/2modm∪wh|h≡r−1,r−2,…,r−m/2modm∪xh|h≡r−2,r−3,…,r−m/2+1modm, and Ryr,zr=Vℂm−xh|h≡r−1,r−2,…,r−m/2+1modm, respectively. Clearly, Ŕu=7m/2+1. Since Rr=7m/2+1<Ŕu, then Ŕu∩∪r=1mRr≥Rr, Rr<Ŕ3=Rvr,vr+1=Ŕ4=Rvr,vr+p=Ŕ5=Rwr,wr+1=Ŕ6=Rwr,wr+p=Ŕ7=Rxr,xr+s=Ŕ8=Rxr,xr+1=Ŕ9=Rxr,xr+p=Ŕ10=Ryr,yr+1=Ŕ11=Ryr,yr+p=Ŕ12=Rzr,zr+s=Ŕ13=Rvr,wr+p=Ŕ14=Rvr,wr+s=Ŕ15=Rwr,zr+p=5m−6.(c)
The RNs ofvr,vr+1,vr,vr+p,wr,wr+1,wr,wr+p,xr,xr+s,xr,xr+1,xr,xr+p,yr,yr+1,yr,yr+p,zr,zr+s and vr,wr+p are Ŕ3=Rvr,vr+1=Vℂm−xh|h≡r+p−1/2,r+p+m−1/2modn∪yh|h≡r+p−1/2,r+p+m−1/2modn∪zh|h≡r+p−1/2,r−p−1/2modn=Ŕ5=Rwr,wr+1Ŕ4=Rvr,vr+p=Vℂm−xh|h≡r+p−1/2,r+p+m−1/2modn∪yh|h≡r+p−1/2,r+p+m−1/2modn∪zh|h≡r+p−1/2,r−p−1/2modn=Ŕ6=Rwr,wr+p, Ŕ7=Rvr,vr+s=Vℂm−vh|r+s/2,r+s+m/2modm∪wh|r+s/2,r+s+m/2modm∪zh|r+s−2/2,r+s+m−2/2modm=Ŕ8=Rwr,wr+s=Ŕ9=Rxr,xr+p=Ŕ11=Ryr,yr+p,Ŕ12=Rzr,zr+s=Vℂm−vh|h≡r+s+2/2,r+s+m+2/2modm∪wh|h≡r+s+2/2,r+s+m+2/2modm∪zh|r+s/2,r+s+m/2modm, Ŕ13=Rvr,wr+p=Vℂm−vh|h≡r+p+1/2,r+p+m−1/2modm∪wh|h≡r+p−1/2,r−p−1/2modm∪zh|h≡r+p−3/2,r+p+m−1/2modm, and Ŕ8=Rxr,xr+1=Rvr,vr+2=Ŕ10=Ryr,yr+1. Clearly, Ŕu=5m−6. Since Rr=7m/2+1<Ŕu, then Ŕu∩∪r=1mRr=5m−6≥Rr.(d)
The RN’s ofzr,zr+p,vr,zr+s,wr,xr+p,wr,yr+s,xr,yr+s and xr,zr+s are Ŕ14=Rzr,zr+p=Vℂm−xh|h≡r+p+1/2,r+p+1+m/2modn∪yh||h≡r+p+1/2,r+p+1+m/2modn, Ŕ15=Rvr,zr+s=Vℂm−xh|h≡r+s/2,r+s+m/2modn∪wh|h≡r+s+2/2,r+s+m/2modn, Ŕ16=Rwr,xr+p=Vℂm−wh|h≡r+p+1/2modm∪vh|h≡r+p+1/2modm∪yh|h≡r+p−1/2modm∪xh||h≡r+p+m−1/2modm, Ŕ17=Rwr,yr+s=Vℂm−xh|h≡r+s/2modm∪vh|h≡r+s+2/2modm∪zh|h≡r−s/2,r−s+2/2modm, Ŕ18=Rxr,yr+s=Vℂm−vh|h≡r+s+2/2,r+m+s/2modn∪wh|h≡r+s+2/2,r+m+s/2modn and Ŕ19=Rxr,zr+s=Vℂm−vh|h≡r+s+4/2,r+m+s/2modn∪wh|h≡r+s+4/2,r+m+s/2modn, respectively. We can see that, for 9≤u≤14, Ŕu=5m−4>Rt and Ŕu∩∪t=1mRt≥Rt,(e)
The RNs ofvr,xr+p,vr,xr+s,vr,yr+s,wr,yr+p,wr,zr+s,yr,zr+s, and xr,zr+p are Ŕ20=Rvr,xr+p=Vℂm−vh|h≡r+m+p−1/2modn∪wh|h≡r+m+p+1/2modn∪xh|h≡r+p−1/2modn, Ŕ21=Rvr,xr+s=Vℂm−vh|h≡r+s+2/2modn∪wh|h≡r+s/2modn∪xh|h≡r+m+s/2modn, Ŕ22=Rvr,yr+s=Vℂm−vh|h≡r+s+m−2/2modn∪xh|h≡r+s+m/2modn∪yh|h≡r+s−2/2modn, Ŕ23=Rwr,yr+p=Vℂm−wh|h≡r+p+1/2modn∪yh|h≡r−p−1/2modn∪zh|h≡r+p−3/2modn, Ŕ24=Rwr,zr+s=Vℂm−vh|h≡r+s+1modn∪wh|h≡r+s+1modn∪zn, Ŕ25=Ryr,zr+s=Vℂm−vh|h≡r+s+2/2modn∪wh|h≡r+s+2/2modn∪xh|h≡r+s/2modn, and Ŕ26=Rxr,zr+p=Vℂm−vh|h≡r+pmodn∪wh|h≡r+pmodn∪xh|h≡r+p−1modn, respectively. We can see that, for 17≤u≤23, Ŕu=5m−3>Rt and Ŕu∩∪t=1mRt≥Rt.(f)
The RNs ofwr,yr,vr,yr+p,wr,zr and xr,yr+p are Ŕ27=Rwr,yr=Vℂ−xh|h≡rmodm∪yh|h≡r−1modm, Ŕ28=Rwr,zr=Vℂ−xh|h≡r+1,r+2modm, Ŕ29=Rvr,yr+p=Vℂ−wh|h≡r+p+1/2modm∪xh|h≡r+p−1/2modm and Ŕ30=Rxr,yr+p=Vℂ−xh|h≡r+p+1/2,r−p+1/2modm, respectively. Clearly, Ŕu=5m−2 for 27≤u≤30 and Ŕu∩∪r=1mRr≥Rr.(g)
The RNs ofwr,zr+p and yr,zr+p is Ŕ31=Vℂm−vh|r≡r+p+3/2,r+p+m+1/2modm∪wh|r≡r+p+3/2,r+p+m+1/2modm∪zh|r≡r+p−1/2modm and Ŕ32=Rwr,zr+p=Vℂm−xh|h≡r+p+1/2,r+p+m+1/2modm∪yh|h≡r+p−1/2,r−p−1/2modm∪wh|h≡r+p+3/2modm, respectively. Clearly, Ŕ29>Rt and Ŕu∩∪r=1mRr≥Rr.(h)
The RNs ofvr,xr and vr,xr−1 are Ŕ33=Rvr,xr=Vℂm−wh|h≡r,r−1,r−2,…,r−n/2+1modm∪xh|h≡r−1,r−2…,r−n/2modm and Ŕ34=Rvr,xr−1=Vℂm−wh|h≡r,r+1,r+2,…,r+n/2−1modm∪xh|h≡r+1,r+2…,r+n/2modm, respectively. Clearly, Ŕu=4m>Rt and Ŕu∩∪r=1mRr≥Rr.(i)
The RN ofxr,yr+1 is Ŕ35=Rxr,yr+1=Vℂm−yh|h≡r−1,r−2,…,r−m/2+1modm∪zh|h≡r−1,r−2,…,r−m/2+1modm. Clearly, Ŕ35=4m+2>Rt and Ŕu∩∪r=1mRr≥Rr.(j)
The RN ofxr,zr+1 is Ŕ36=Rxr,zr+1=Vℂm−vh|h≡r+3,r+4,…,r+m/2modm∪wh|h≡r+3,r+4,…,r+m/2modm as we can see that Ŕ36>Rt and Ŕu∩∪r=1mRr≥Rr.(k)
The RNs ofvr,wr+1 and vr,yr are Ŕ37=Rvr,wr+1=Vℂm−vh|h≡r+1,r+2,…,r+m/2modm and Ŕ38=Rvr,yr=Vℂm−wh|h≡r+1,r+2,…,r+m/2modm, as we can see that Ru=9m/2>Rt and Ŕu∩∪r=1mRr≥Rr.(l)
The RN ofvr,xr+1 is Ŕ39=Rvr,xr+1=Vℂm−vh|h≡r+1,r+2,…,r+n/2modm∪xh|h≡rmodm∪wh|h≡r+m+2/2modm. Clearly, Ŕ39=9m/2−1>Rt and Ŕu∩∪r=1mRr≥Rr.(m)
The RNs ofvr,zr and xr,zr are Ŕ40=Rvr,zr=Vℂm−xh|h≡r,r+n/2modm∪wh|r+2,r+3,…,r+n/2modm and Ŕ41=Rxr,zr=Vℂm−xh|h≡r+2,r+3,…,r+n/2modmyr, respectively. Clearly, Ru=9m/2+1>Rt and Ŕu∩∪r=1mRr≥Rr.(n)
The RNs ofvr,wr and xr,yr are Ŕ42=Rvr,wr=Ŕ43=Rxr,yr=Vℂ, as both are equal to Vℂm; therefore, Ŕu∩∪r=1mRr≥Rr.Theorem 2.
Ifℵ≅ℂm with m≥6 and m≡0mod2, then dimfℂm<5m/3m+1.Table 8
The representation ofR′u for 1≤u≤12.
RNsElementsRw1,x1Vℂ6−v2,v3,v4∪w2,w3,w4∪x2,x3Rw2,x2Vℂ6−v3,v4,v5∪w3,w4,w5∪x3,x4Rw3,x3Vℂ6−v4,v5,v6∪w4,w5,w6∪x4,x5Rw4,x4Vℂ6−v1,v5,v6∪w1,w5,w6∪x5,x6Rw5,x5Vℂ6−v1,v2,v6∪w1,w2,w6∪x1,x6Rw6,x6Vℂ6−v1,v2,v3∪w1,w2,w3∪x1,x2Rw2,x1Vℂ6−v1,v5,v6∪w1,w5,w6∪x5,x6Rw3,x2Vℂ6−v1,v2,v6∪w1,w2,w6∪x1,x6Rw4,x3Vℂ6−v1,v2,v3∪w1,w2,w3∪x1,x2Rw5,x4Vℂ6−v2,v3,v4∪w2,w3,w4∪x2,x3Rw6,x5Vℂ6−v3,v4,v5∪w3,w4,w5∪x3,x4Rw1,x6Vℂ6−v4,v5,v6∪w4,w5,w6∪x4,x5Table 9
The representation ofR′u for 13≤u≤67.
RNsElementsEqualityRv1,v2Vℂ6−x1,x4∪y1,y4∪z1,z6Rv1,v2, Rw1,w2,Rw1,w2,Rv2,v3Vℂ6−x2,x5∪y2,y5∪z1,z2v1,v4, Rw2,w3,Rw1,w4,Rv3,v4Vℂ6−x3,x6∪y3,y6∪z2,z3Rv2,v5, Rw3,w4,Rw2,w5,Rv4,v7,Rw4,w7Rv4,v5Vℂ6−x1,x4∪y1,y4∪z3,z4Rv3,v6, Rw4,w5,Rw3,w6,Rv5,v6Vℂ6−x2,x5∪y2,y5∪z4,z5Rw5,w6,Rv1,v6Vℂ6−x3,x6∪y3,y6∪z5,z6Rw1,w6, Rv1,v4,Rv1,v3Vℂ6−v2,v5∪w2,w5∪z1,z4Rw1,w6, Rv4,v6, Ry1,y2Ry4,y5,Ry3,y6,Rz2,z6,Rz4,z6Rv2,v4Vℂ6−v3,v6∪w3,w6∪z2,z5Rw1,w6, Rv1,v5, Ry2,y3Ry5,y6,Ry1,y4,Rz1,z3Rv3,v5Vℂ6−v1,v4∪w1,w4∪z3,z6Rw1,w6, Rv2,v6, Ry3,y4Ry1,y6,Ry2,y5,Rz2,z4Rv1,w4Vℂ6−v1,v4∪w1,w4∪z3,z6Rw1,w6, Rv2,v6, Ry3,y4Ry1,y6,Ry2,y5,Rz2,z4Rv1,w4Vℂ6−v3,v5∪w2,w6∪z1,z5Rv2,w5Vℂ6−v2,v6∪w1,w3∪z2,z6Rv3,w6Vℂ6−v1,v5∪w2,w4∪z1,z3Rv1,x1Vℂ6−w1,w5,w6∪x4,x5,x6Rv2,x2Vℂ6−w1,w2,w6∪x1,x5,x6Rv3,x3Vℂ6−w1,w2,w3∪x1,x2,x6Rv4,x4Vℂ6−w2,w3,w4∪x1,x2,x3Rv5,x5Vℂ6−w3,w4,w5∪x2,x3,x4Rv6,x6Vℂ6−w4,w5,w6∪x3,x4,x5Rv2,x1Vℂ6−w2,w3,w4∪x3,x4,x5Rv3,x2Vℂ6−w3,w4,w4∪x4,x5,x6Rv4,x3Vℂ6−w4,w5,w6∪x1,x5,x6Rv5,x4Vℂ6−w1,w5,w6∪x1,x2,x6Rv6,x5Vℂ6−w1,w2,w6∪x1,x2,x3Rv1,x6Vℂ6−w1,w2,w3∪x2,x3,x4Table 10
The representation ofR′u for 68≤u≤79.
RNsElementsRy1,z4Vℂ6−v3,v6∪w3,w6∪z2Ry2,z5Vℂ6−v1,v4∪w1,w4∪z3Ry3,z6Vℂ6−v2,v5∪w2,w5∪z4Rw1,z4Vℂ6−x3,x6∪y2,y6∪w4Rw2,z5Vℂ6−x1,x4∪y1,y3∪w5Rw3,z6Vℂ6−x2,x5∪y1,y2∪w6Rv1,x2Vℂ6−v2,v3,v4∪x1∪w5Rv2,x3Vℂ6−v3,v4,v5∪x2∪w6Rv3,x4Vℂ6−v4,v5,v6∪x3∪w1Rv4,x5Vℂ6−v1,v5,v6∪x4∪w2Rv5,x6Vℂ6−v1,v2,v3∪x5∪w3Rv1,x6Vℂ6−v2,v3,v4∪x6∪w4Table 11
The representation ofR′u for 80≤u≤127.
RNsElementsEqualityRz1,z2Vℂ6−x2,x5∪y2,y5Rz4,z5,Rz3,z6Rz2,z3Vℂ6−x3,x6∪y3,y6Rz5,z6,Rz1,z4Rz3,z4Vℂ6−x1,x4∪y3,y6Rz1,z6,Rz2,z5Rv1,z3Vℂ6−x2,x5∪w3,w5Rw4,z6Rv2,z4Vℂ6−x3,x6∪w4,w6Rw1,z5Rv3,z5Vℂ6−x1,x4∪w1,w5Rw2,z6Rw1,x4Vℂ6−w3∪v3∪y2∪x5Rw2,x5Vℂ6−w4∪v4∪y3∪x6Rw3,x6Vℂ6−w5∪v5∪y4∪x1Rw1,x3Vℂ6−x2∪v3∪z5,z6Rw2,x4Vℂ6−x3∪v4∪z1,z6Rw3,x5Vℂ6−x4∪v5∪z1,z2Rw4,x6Vℂ6−x5∪v6∪z2,z3Rw1,x5Vℂ6−x6∪v1∪z3,z4Rw2,x6Vℂ6−x1∪v2∪z4,z5Rx1,y3Vℂ6−v3,v5∪w3,w5Rx2,y4Vℂ6−v4,v6∪w4,w6Rx3,y5Vℂ6−v1,v5∪w1,w5Rx4,y6Vℂ6−v2,v6∪w2,w6Rx1,y5Vℂ6−v1,v3∪w1,w3Rx2,y6Vℂ6−v2,v4∪w2,w4Rx1,z3Vℂ6−v4,v5∪w4,w5Rx2,z4Vℂ6−v5,v6∪w6,w6Rx3,z5Vℂ6−v1,v6∪w1,w6Rx4,z6Vℂ6−v1,v2∪w1,w2Rx1,z5Vℂ6−v2,v3∪w2,w3Rx2,z6Vℂ6−v3,v4∪w3,w4Rx1,y2Vℂ6−y5,y6∪z5,z6Rx2,y3Vℂ6−y1,y6∪z1,z6Rx3,y4Vℂ6−y1,y2∪z1,z2Rx4,y5Vℂ6−y2,y3∪z2,z3Rx5,y6Vℂ6−y3,y4∪z3,z4Rx1,y6Vℂ6−y4,y5∪z4,z5Rv1,z1Vℂ6−w3,w4∪x1,x4Rv2,z2Vℂ6−w4,w5∪x2,x5Rv3,z3Vℂ6−w5,w6∪x3,x6Rv4,z4Vℂ6−w1,w6∪x1,x4Rv5,z5Vℂ6−w1,w2∪x2,x5Rv6,z6Vℂ6−w2,w3∪x3,x6Table 12
The representation ofR′u for 128≤u≤163.
RNsElementsRv1,x4Vℂ6−v5∪x2∪w6Rv2,x5Vℂ6−v6∪x3∪w1Rv1,w3Vℂ6−v3∪x5∪w2Rv2,w4Vℂ6−v4∪x6∪w3Rv3,w5Vℂ6−v5∪x1∪w4Rv4,w6Vℂ6−v6∪x2∪w5Rv1,w5Vℂ6−v1∪x3∪w6Rv2,w6Vℂ6−v2∪x4∪w1Rv1,y3Vℂ6−v4∪w5∪y1Rv2,y4Vℂ6−v5∪w6∪y2Rv3,y5Vℂ6−v6∪w1∪y3Rv4,y6ℂ6−v1∪w2∪y4Rv1,y5Vℂ6−v2∪w3∪y5Rv2,y6Vℂ6−v3∪w4∪y6Rx1,z4Vℂ6−v4∪w4∪x3Rx2,z5Vℂ6−v5∪w5∪x4Rx3,z6Vℂ6−v6∪w6∪x5Rx1,z4Vℂ6−v1∪w1∪x6Rv1,w2Vℂ6−v2,v3,v4Rv2,w3Vℂ6−v3,v4,v5Rv3,w4Vℂ6−v4,v5,v6Rv4,w5Vℂ6−v1,v5,v6Rv5,w6Vℂ6−v1,v2,v3Rv1,w6Vℂ6−v2,v3,v4Rv1,y1Vℂ6−w2,w3,w4Rv2,y2Vℂ6−w3,w4,w5Rv3,y3Vℂ6−w4,w5,w6Rv4,y4Vℂ6−w1,w5,w6Rv5,y5Vℂ6−w1,w2,w3Rv6,y6Vℂ6−w2,w3,w4Rx1,z1Vℂ6−x3,x4∪y1Rx2,z2Vℂ6−x4,x5∪y2Rx3,z3Vℂ6−x5,x6∪y3Rx4,z4Vℂ6−x1,x6∪y4Rx5,z5Vℂ6−x1,x2∪y5Rx6,z6Vℂ6−x2,x3∪y6Table 13
The representation ofR′u for 164≤u≤186.
RNsElementsRw1,y1Vℂm−x1∪y6Rw2,y2Vℂm−x2∪y1Rw3,y3Vℂm−x3∪y2Rw4,y4Vℂm−x4∪y3Rw5,y5Vℂm−x5∪y4Rw6,y6Vℂm−x6∪y5Rv1,y4Vℂm−x2∪w3Rv2,y5Vℂm−x3∪w4Rv3,y6Vℂm−x4∪w5Rw1,z1Vℂm−x2,x3Rw2,z2Vℂm−x3,x4Rw3,z3Vℂm−x4,x5Rw4,z4Vℂm−x5,x6Rw5,z5Vℂm−x1,x6Rw6,z6Vℂm−x1,x2Rx1,y4Vℂm−x3,x5Rx2,y5Vℂm−x4,x6Rx3,y6Vℂm−x1,x5Rx1,z2Vℂ6−v4∪w4Rx2,z3Vℂ5−v5∪w5Rx3,z4Vℂ6−v6∪w6Rx4,z5Vℂ6−v1∪w1Rx5,z6Vℂ6−v2∪w2Rx1,z6Vℂ6−v3∪w3Table 14
The representation ofR′u for 187≤u≤198.
RNsElementsRv1,w1Rv2,w2Rv3,w3Rv4,w4Rv5,w5Rv5,w5Rx1,y1Rx2,y2Vℂ6Rx3,y3Rx4,y4Rx5,y5Rx5,y5Table 15
The representation ofRr for 1≤r≤12.
RNsElementsR1=Ry1,z1Vℂ6−v3,v4∪w3,w4∪y5,y6∪z4,z5,z6R2=Ry2,z2Vℂ6−v4,v5∪w4,w5∪y1,y6∪z1,z5,z6R3=Ry3,z3Vℂ6−v5,v6∪w5,w6∪y1,y2∪z1,z2,z6R4=Ry4,z4Vℂ6−v1,v6∪w1,w6∪y2,y3∪z1,z2,z3R5=Ry5,z5Vℂ6−v1,v2∪w1,w2∪y3,y4∪z2,z3,z4R6=Ry6,z6Vℂ6−v2,v3∪w2,w3∪y4,y5∪z3,z4,z5R7=Ry2,z1Vℂ6−v5,v6∪w5,w6∪y3,y4∪z3,z4,z5R8=Ry3,z2Vℂ6−v1,v6∪w1,w6∪y4,y5∪z4,z5,z6R9=Ry4,z3Vℂ6−v1,v2∪w1,w2∪y5,y6∪z1,z5,z6R11=Ry6,z5Vℂ6−v3,v4∪w3,w4∪y1,y2∪z1,z2,z3R12=Ry1,z6Vℂ6−v4,v5∪w4,w5∪y2,y3∪z2,z3,z4Proof.
Case I
m=6.The RNs are given as follows.From Lemma3, we see that Ryr,yr+p=Rxr,xr+p.In Tables8–14, the RNs have cardinalities of 22, 24, 25, 26, 27, 28, and 30, respectively. On the contrary, Table 15 represents RNs with minimum cardinality of 21. We can see that ∪r=112Rr=Vℂ6 this implies ∪r=112Rr=30 and Rr¯∩∪r=112Rr≥Rr, where 1≤u≤198.Now, we define a functionμ:VB6⟶0,1 such that μvr=μwr=μxr=μyr=μzr=1/22, as Rr for 1≤r≤12 of ℂ6 are pairwise overlapping; hence, ∃ is another minimal resolving function κ¯ of ℂ6 such that μ¯<μ. As a result, dimfℂ6<∑r=1121/10<30/21.On the contrary, Table4 shows the RNs with maximum cardinality of 30=κ; hence, by Lemma 1, Vℂ6/κ=30/30=1<dimfℂ6. Therefore,(15)1<dimfℂ6<3021.Case II
m≥8.We have seen from Lemma3 that the RNs with minimum cardinality of 3m+1 are Ryl,zl and Ryr,zr−1 and ∪r=1mRt=Vℂm. Let λ=3m+1 and δ=∪t=1mRt=5m. Now, we define a mapping μ:Vℂm⟶0,1 such that(16)μa=1λ,fora∈∪t=1mRt,0,fora∈Vℂ−∪t=1mRt.We can see thatμ is a RF for ℂm with m≥3 because μRu,v≥1,∀u,v∈Vℂm. On the contrary, assume that there is another resolving function ρ, such that ρu≤μu, for at least one u∈Vℂm,ρu≠μu. As a consequence, ρRu,v<1, where Ru,v is a RN of ℂm with minimum cardinality λ. This implies that ρ is not a resolving function which is contradiction. Therefore, μ is a minimal resolving function that attains minimum μ for ℂm. Since all Rr are having pairwise nonempty intersection, so there is another minimal resolving function of μ¯ of ℂm such that μ¯≤μ. Hence, assigning 1/λ to the vertices of ℂm in ∪r=1mRr and calculating the summation of all the weights, we obtain(17)dimfℂm=∑r=1δ1λ≤5m7m/2+1=10m7m+2.Also, the RNs with maximum cardinality of5m are Rvr,wr and Rxr,yr. Let Vℂm=ω and Rvr,wr=Rxr,yr=κ; thus, from Lemma 1, we have Vℂm/κ=ω/κ=5m/5m=1<dimfℂ.Therefore, we conclude the following:(18)1<dimfℂm<5m3m+1.


## 6. Conclusion

In this paper,(i)
We have found the improved lower bound of FMD of connected networks(ii)
Apart from that, we have calculated the lower and upper bounds of FMD of symmetric networks called by covex polytopes’ Type I and Type IIBm and ℂm(iii)
Table16 shows the summary of main results and Table 17 gives the values of FMDs as they tend to ∞Table 16
FMD of convex polytopes form≥6.
NetworkLower bound ofdimfUpper bound ofdimfCommentBm110m/7m+2Boundedℂm15m/3m+1BoundedTable 17
Upper bounds of FMD as they tend to∞, where m≥6∧m≡0mod2.
NetworkValues of FMD as they tend to∞Bmlimm⟶∞10m/7m+210/7ℂmlimm⟶∞5m/3m+15/3


### 6.1. Open Problem

To characterize the networks with lower bound of FMD greater than 1 is still an open problem.


## 6.1. Open Problem

To characterize the networks with lower bound of FMD greater than 1 is still an open problem.



---

*Source: 3925925-2021-09-23.xml*