# Local Fractional Locating Number of Convex Polytope Networks

Authors: Hassan Zafar; Muhammad Javaid; Ebenezer Bonyah
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The concept of locating number for a connected network contributes an important role in computer networking, loran and sonar models, integer programming and formation of chemical structures. In particular it is used in robot navigation to control the orientation and position of robot in a network, where the places of navigating agents can be replaced with the vertices of a network. In this note, we have studied the latest invariant of locating number known as local fractional locating number of an antiprism based convex polytope networks. Furthermore, it is also proved that these convex polytope networks posses boundedness under local fractional locating number.


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## Body

## 1. Introduction

Slater [1] introduced the methodology to compute the locating set of a connected network. He defined the minimum cardinality of a locating set as a locating number of a connected network. Melter and Harary independently studied the concept of location number but they used the different term called as metric dimension. They also briefly studied the locating number of serval type of networks such as cycles, complete and complete bipartite networks [2]. Applications of locating number can be found for navigation of reboots [3], chemical structures [4], combinatorial optimization [5] image processing & pattern recognition [6].Chartrand et al. [4] played a vital role in the study of locating number (LN), they characterized all those connected networks of order p having locating number 1, p−2, and p−1. Furthermore, they also presented a new technique to compute bounds of locating number of unicyclic networks. Since then researchers have computed locating number of many connected networks such as generalized Peterson network [7], Cartesian products [8], constant locating number of some convex polytopes and generalized convex polytopes [9, 10], Mobius ladders [11], Toeplitz networks [12], k—dimensional networks, and fan networks[13, 14]. Moreover, LN of corona product and partition dimension of different products of networks can be seen in [15, 16] and fault tolrent LN of some families of convex polytopes studied in [17, 18]. For the study of edge LN of wheel and k level wheel networks, we refer [19, 20]. There are various new invariants of LN which have been introduced in recent times such as partition dimension [21], Strong—LN [5], fault-tolerant LN [22], edge LN [23], mixed—LN [24], independent resolving sets [25], and K—LN [26].Chartared et al. use the concept of LN to solve an integer programming problem (IPP) with specific conditions [4] and Currie and Olllermann used the idea of fractional locating number (FLN) to find solution of specific IPP as well [27]. The FLN formally introduced in networking theory by Arguman and Mathew and they computed exact values of FLN of a path, cycles, wheels, complete and friendship networks. Furthermore, they also developed some new techniques to compute exact values of FLN of connected networks with specific conditions [28]. Later on Arguman et al. characterized all those networks have FLN exactly VG/2 and they also presented many results on FLN of Cartesian product of two networks [29]. Feng et al. computed FLN of distance regular and vertex transitive networks [30]. For the study of FLN of corona, lexicographic, and hierarchical products of connected networks see [31, 32]. Recently, Alkhalidi et al. established sharp bound of FLN of all the connected networks [33].The latest version of FLN called by local fractional locating number (LFLN) is defined by Aisyah et al. and they computed LFLN of different connected networks [34]. Javaid et al. developed sharp bounds of LFLN of all the connected networks and they computed upped bounds of local FLN of wheel-related networks. Furthermore, they also improved the lower bound of LFLN different from unity and also developed a technique to compute exact value of LFLN under specific conditions [35, 36]. For the study of LFLN of generalized gear, sunlet and convex polytope networks see [37–39].In this manuscript, our main objective is to compute LFLN of cretin family of convex polytopes in the form of sharp upper and lower bounds. It has been proved that in every case the convex polytopes remain bounded. The manuscript is organised as Section2 contains preliminaries and Sections 3 and 4 have main results and conclusion respectively.


## 2. Preliminaries

A networkG is an order pair VG,EG, where VG is the vertex set and EG is the edge set. A walk is a finite sequence of edges and vertices between two vertices. A trail is a walk in which all edges are distinct and a path is a trail in which all vertices are distinct. A network G is connected if there is a path between each pair vertices. The distance between two vertices a and b is donated by da,b is defined as the length of the shortest path between a and b. For further preliminary results of networking theory see [40]. A vertex c∈VG is said to resolve a pair a,b∈VG, if da,c≠db,c. Suppose that W=w1,w2,w3,…,wp⊆VG and x∈VG, then p tuple representation of x with respect to W is defined as dx|W=dx,w1,dx,w2,dx,w3,…,dx,wp. If distinct vertices of G have unique representation with respect to W then W is known as locating/resolving set. The minimum cardinality of W is called locating number (LN) of G that is defined as(1)LNG=minW:Wis the resolving set ofG.For an edgeab∈EG the local resolving neighbourhood set (LRN) is the collection of all vertices of G which resolve an edge ab and it is donated by ℝlab=c:dc,a≠dc,b, where c∈VG. A real valued function h:VG⟶0,1 becomes local resolving function of G if hℝlab≥1 for each ℝlab in G, where hℝlab=∑x∈ℝlabhx. A local resolving function (LRF) is called minimal LRF if there exist another function h′:VG⟶0,1 such that h′≤h and ha≠h′a for at least one a∈VG, that is not LRF of G. If h=∑v∈VGhv, then local fractional locating number (LFLN) of G is defined as(2)LFLNG=minh:his minimal local resolving function ofG.


## 3. Main Results

This section is devoted to the main results in which, we have examined the LFLN of cretin family convex polytope networksDp and Ep and it has been proved that these polytope networks remain bounded under LFLN when their order approaches to infinity.


### 3.1. LFLN of Convex Polytpoe NetworkDp

In this particular subsection, we have computed resolving local neighbourhood sets and LFLN ofDp in the form of exact values and sharp lower and upper bounds.The convex polytope network is introduced by Bača [41] and LN of Dp is 3 is proved in [10]. The vertex set VDp consists of inner ai:1≤i≤p, middle ai1:1≤i≤p, bi:1≤i≤p and outer vertices bi1:1≤i≤p. The edge set Dp is defined as EDp=aiai+1:1≤i≤p∪ai1ai:1≤i≤p∪ai1bi:1≤i≤p∪bi1bi:1≤i≤p∪bi1bi+11:1≤i≤p∪ai+11bi:1≤i≤p. Furthermore, the order and size of Dp are 4p and 6p respectively and for complete details see Figure 1.Figure 1
Convex polytope networkDp.Lemma 1.
LetDp be a convex polytope network, with p≥5 and p≅1mod2. Then.(i)
ℝlai1bi=ℝlbiai+11=5p+7/2 and ∪i=1pℝlai1bi=VDp.(ii)
ℝlai1bi≤ℝle and ℝle∩∪i=1pℝlai1bi≥ℝlai1bi∀e∈EDp.Proof.
Considerai inner, ai1,bi middle and bi1 are outer vertices of Dp, where 1≤i≤p and p+1≅1modp.(i)
ℝlai1bi=VDp−ai+1,ai+2,ai+3,….,ap+2i−1/2,ap+2i−1/21,bp+2i+1/21,bp+2i+3/21,bp+2i+1/51,…,bp+i−11 and ℝlbiai+11=VDp−ai,ap+2i+3/2,ap+2i+5/2,ap+2i+7/2,….,ap+i−1,bi+11,bi+21,…,bp+2i+1/51. Note that ∪i=1pℝlai1bi=VDp and ℝlai1bi=5p+7/2.(ii)
ℝlaiai+1=VDp−ap+2i+1/2,ap+2i+1/21,bi1,bi, ℝlbi1bi+11=VDp−ai+1,ai+11,bp+2i+1/21,bp+2i+1/2, ℝlaiai1=VDp−ai+21,ap1,bi+2,bp and ℝlbibi1=VDp−ai+31,ap1,bi+2,bp.
The cardinalities of all the LRN sets ofDp are illustrated in Table 1.
From Table1, we note that ℝlai1bi≤ℝle. Since ∪i=1pℝlai1bi=4p therefore ℝle∩∪i=1pℝlai1bi≥ℝlai1bi∀e∈EDp.Table 1
The cardinalities of all the LRN sets ofDp.
RLN setComparisonℝlaiai+14p−4>ℝlai1biℝlbi1bi+114p−4>ℝlai1biℝlaiai14p−4>ℝlai1biℝlbibi14p−4>ℝlai1biTheorem 1.
LetD3 be a convex polytope network. Then(3)65≤LFLND3≤32.Proof.
The LRN sets of convex polytope networkD3 are: ℝl1=ℝla1a2=VD3−a3,a31,b1,b11,(4)ℝl2=ℝla2a3=VD3−a1,a21,b2,b21ℝl3=ℝla3a1=VD3−a2,a11,b3,b31ℝl4=ℝla11b1=VD3−a2,a31,b2,b31ℝl5=ℝla21b2=VD3−a3,a11,b3,b11ℝl6=ℝla31b3=VD3−a1,a21,b1,b21ℝl7=ℝlb11b21=VD3−b31,b3,a21,a2ℝl8=ℝlb21b31=VD3−b11,b1,a31,a3ℝl9=ℝlb31b11=VD3−b21,b2,a11,a1ℝl10=ℝlb11b1=VD3−b2,b3,a31ℝl11=ℝlb21b2=VD3−b3,b1,a11ℝl12=ℝlb31b3=VD3−b1,b2,a21ℝl13=ℝla11a1=VD3−a21,a31ℝl14=ℝla21a2=VD3−a31,a11ℝl15=ℝla31a3=VD3−a11,a21ℝl16=ℝla21b1=VD3−a1,a21ℝl17=ℝla31b2=VD3−a2,a31ℝl18=ℝla11b3=VD3−a3,a11.
Since,ℝlaiai+1=ℝlbi1bi+11=ℝlbi1bi=ℝlai1ai=8 which is less then the other LRN sets, where 1≤i≤3. Furthermore, ∪i=13ℝlaiai+1=VD3 and ℝle∩∪i=13ℝlaiai+1≥ℝlaiai+1∀e∈ED3.
Therefore, we define an upper LRFh:D3⟶0,1 as hv=1/8∀v∈VD3. In order to show that h is minimal LRF consider another upper LRF h′:VD3⟶0,1 as hv<1/8∀v∈VD3 therefore hℝle<1 and h′<h which shows that h′ is not LRF of D3 hence LFLND3≤∑i=1121/8=3/2. Likewise for 1≤i≤3 cardinality of LRN set ℝlai1bi is 10 which is greater then the cardinalities of all other LRN sets of D3. Therefore, we define a maximal lower LRF g:Dp⟶0,1 as gv=1/10∀v∈VD3. Hence LFLND3≥∑i=1121/10=6/5. Consequently,(5)65≤LFLND3≤35.Theorem 2.
LetD5 be a convex polytope network. Then(6)LFLND5=54.Proof.
The LRN sets are given by:ℝl1=ℝla1a2=VD5−a4,a41,b1,b11,(7)ℝl2=ℝla2a3=VD5−a5,a51,b2,b21ℝl3=ℝla3a4=VD5−a1,a11,b3,b31ℝl4=ℝla4a5=VD5−a2,a21,b4,b41ℝl5=ℝla5a1=VD5−a3,a31,b5,b51ℝl6=ℝlb1b11=VD5−a31,a51,b2,b5ℝl7=ℝlb2b21=VD5−a41,a11,b3,b1ℝl8=ℝlb3b31=VD5−a51,a21,b4,b2ℝl9=ℝlb4b41=VD5−a11,a31,b5,b1ℝl10=ℝlb5b51=VD5−a21,a41,b1,b2ℝl11=ℝlb11b21=VD5−a21,a2,b4,b41ℝl12=ℝlb21b31=VD5−a31,a3,b5,b51ℝl13=ℝlb31b41=VD5−a41,a4,b1,b11ℝl14=ℝlb41b51=VD5−a51,a5,b2,b21ℝl15=ℝlb51b11=VD5−a11,a1,b3,b31ℝl16=ℝla1a11=VD5−a21,a51,b2,b4ℝl17=ℝla2a21=VD5−a31,a11,b3,b5ℝl18=ℝla3a31=VD5−a41,a21,b4,b1ℝl19=ℝla4a41=VD5−a51,a31,b5,b2ℝl20=ℝla5a51=VD5−a11,a41,b1,b3ℝl21=ℝla11b1=VD5−a2,a3,b41,b51ℝl22=ℝla21b2=VD5−a3,a4,b51,b11ℝl23=ℝla31b3=VD5−a4,a5,b11,b21ℝl24=ℝla41b4=VD5−a5,a1,b21,b31ℝl25=ℝla51b5=VD5−a1,a2,b31,b41ℝl26=ℝla21b1=VD5−a1,a5,b21,b31ℝl27=ℝla31b2=VD5−a2,a1,b31,b41ℝl28=ℝla41b3=VD5−a3,a2,b41,b51ℝl29=ℝla51b4=VD5−a4,a3,b51,b11ℝl30=ℝla11b5=VD5−a5,a4,b11,b21.
It is clear that the cardinality of each RLN set ofD5 is 16. Therefore, we define a constant function h:VD5⟶0,1 as hv=1/16∀v∈VD5. Hence LFLND5=∑i=1201/16=5/4.Theorem 3.
LetDp be a convex polytope network, with p≥7 and p≅1mod2. Then(8)pp−1≤LFLNDp≤8p5p+7.Proof.
To prove the result, we split it into two casesCase 1.
Forp=7 , we have following LRN sets;ℝl1=ℝla11b1=VD7−a2,a3,a4,a41,b51,b61,b71,(9)ℝl2=ℝla21b2=VD7−a3,a4,a5,a51,b61,b71,b11ℝl3=ℝla31b3=VD7−a4,a5,a6,a61,b71,b11,b21ℝl4=ℝla41b4=VD7−a5,a6,a7,a71,b11,b21,b31ℝl5=ℝla51b5=VD7−a6,a7,a1,a11,b21,b31,b41ℝl6=ℝla61b6=VD7−a7,a1,a2,a21,b31,b41,b51ℝl7=ℝla71b7=VD7−a1,a2,a3,a31,b41,b51,b61ℝl8=ℝla21b1=VD7−a1,a6,a7,b21,b31,b41,b51ℝl9=ℝla31b2=VD7−a2,a7,a1,b31,b41,b51,b61ℝl10=ℝla41b3=VD7−a3,a1,a2,b41,b51,b61,b71ℝl11=ℝla51b4=VD7−a4,a2,a3,b51,b61,b71,b11ℝl12=ℝla61b5=VD7−a5,a3,a4,b61,b71,b11,b21ℝl13=ℝla71b6=VD7−a6,a4,a5,b71,b11,b21,b31ℝl14=ℝla11b7=VD7−a7,a5,a6,b11,b21,b31,b41,ℝl15=ℝla1a2=VD7−a5,a51,b1,b11ℝl16=ℝla2a3=VD7−a6,a61,b2,b21,ℝl17=ℝla3a4=VD7−a7,a71,b3,b31,ℝl18=ℝla4a5=VD7−a1,a11,b4,b41,ℝl19=ℝla5a6=VD7−a2,a21,b5,b51,ℝl20=ℝla6a7=VD7−a3,a31,b6,b61,ℝl21=ℝla7a1=VD7−a4,a41,b7,b71,ℝl22=ℝlb11b21=VD7−a2,a21,b5,b51ℝl23=ℝlb21b31=VD7−a3,a31,b6,b61,ℝl24=ℝlb31b41=VD7−a4,a41,b7,b71,ℝl25=ℝlb41b51=VD7−a5,a51,b1,b11,ℝl26=ℝlb51b61=VD7−a6,a61,b2,b21,ℝl27=ℝlb61b71=VD7−a7,a71,b3,b31,ℝl28=ℝlb71b11=VD7−a1,a11,b4,b41,ℝl29=ℝla1a11=VD7−a21,a71,b2,b6,ℝl30=ℝla2a21=VD7−a31,a11,b3,b7,ℝl31=ℝla3a31=VD7−a41,a21,b4,b1,ℝl32=ℝla4a41=VD7−a51,a31,b5,b2,ℝl33=ℝla5a51=VD7−a61,a41,b6,b3,ℝl34=ℝla6a61=VD7−a71,a51,b7,b4,ℝl35=ℝla7a71=VD7−a11,a61,b1,b5,ℝl36=ℝlb1b11=VD7−a31,a71,b2,b7,ℝl37=ℝlb2b21=VD7−a41,a11,b3,b1,ℝl38=ℝlb3b31=VD7−a51,a21,b4,b2,ℝl39=ℝlb4b41=VD7−a61,a31,b5,b3,ℝl40=ℝlb5b51=VD7−a71,a41,b6,b4,ℝl41=ℝlb6b61=VD7−a11,a51,b7,b5,ℝl42=ℝlb7b71=VD7−a21,a61,b1,b6.
Since,ℝlai1bi=ℝlbiai+11=16 and ℝlai1bi≤ℝle∀e∈ED7, where 1≤i≤7.
Furthermore,∪i=17ℝlai1bi=VD7 and ℝle∩∪i=17ℝlai1bi≥ℝlai1bi∀e∈ED7. Hence, we define an upper LRF h:VD7⟶0,1 as hv=1/21∀v∈VD7. In order to show that h is minimal upper LRF consider another mapping h′:VD7⟶0,1 as hv<1/21∀v∈VD7 therefore hℝle<1 and h′<h which shows that h′ is not LRF of D3 hence LFLND7≤∑i=1281/21=4/3. Likewise for 1≤i≤7 cardinality of LRN set ℝlaiai+1 is 24 which is greater then the cardinalities of all other RLN sets. Therefore, we define a lower LRF g:VD7⟶0,1 as hv=1/24∀v∈VD7 is a maximal lower LRF hence LFLND7≥∑i=1281/24=7/6. Consequently,(10)76≤LFLND7≤43.Case 2.
Forp≥7, 1≤i≤p by Lemma 1, ℝlai1bi=5p+7/2 and ℝlai1bi≤ℝle∀e∈EDp. Furthermore, ℝle∩∪i=1pℝlai1bi≥ℝlai1bi. Hence an upper LRF h:VDp⟶0,1 is defined as hv=2/5p+7∀v∈VDp. In order to show that h is minimal upper LRF consider another function h′:VDp⟶0,1 as hv<2/5p+7∀v∈VDp therefore h′ℝle<1 and h′<h which shows that h′ is not LRF of Dp hence LFLNDp≤∑i=14p2/5p+7=8p/5p+7. Likewise the cardinality of RLN set ℝlaiai+1 is 4p−4 which is greater then the cardinalities of all other LRN sets. Therefore, we define a maximal LRF g:Dp⟶0,1 as gv=1/4p−4∀v∈VDp, hence LFLNDp≥∑i=14p1/4p−4=p/p−1.
Consequently,(11)pp−1≤LFLNDp≤8p5p+7.Lemma 2.
Suppose thatDp is a convex polytope network, with p≥6 and p≅0mod2. Then.(i)
ℝlai1bi=ℝlai+11bi=5p/2 and ∪i=14pai1bi=VDp.(ii)
ℝlai1bi≤ℝle and ℝle∩∪i=1pℝlai1bi≥ℝlai1bi∀e∈EDp.Proof.
Considerai inner, ai1,bi middle and bi1 are the outer vertices of Dp, where 1≤i≤p and p+1≅1modp.(i)
ℝlai1bi=VDp−ai+2,ai+3,ai+4,….,ap+2i/2,ap+2i−1/21,ap+2i/21,bp+2i/21,bp+2i+2/21,bp+2i+4/21,…,bp1,bp+2i/2,bp+2i+2/2,bp+2i+4/2,…,bp+i−3 and ℝlbiai+11=VDp−ai,ap+2i+2/2,ap+2i+4/2,ap+2i+6/2,….,api−1,ap+2i+2/21,ap+2i+3/21,ap+2i+5/21,….,ap+i−11,bi+3,bi+4,…,bp+2i/2,bp+2i+2/21,bp+2i+4/21,bp+2i+6/21,….,bp+i−11. Note that ∪i=14pℝlai1bi=VDp and ℝlai1bi=5p/2.(ii)
ℝlai+11bi=VDp−ai,ap+2i+2/2,ap+2i+4/2,….,ap,ap+2i+2/21,ap+2i+4/21,…,ap1,bi+11,bi+21,bi+31,…bp+2i/21,bp+2i/2,bp+2i+2/2,bp+2i+4/2,…,bp−2ℝlaiai+1=VDp−bp+2i/2,bp+2i/21,bi1,bi, ℝlbi1bi+11=VDp−ai+1,ai+11,ap+2i+2/2,ap+2i+2/21, ℝlaiai1=VDp−ai+21,ap1,bi+2,bp−1, ℝlbibi1=VDp−ai+31,ap1,bi+2,bp. The cardinalities of each LRN set of Dp is illustrated in Table 2.
It is can be observed with the help of Table2 that ℝlai1bi≤ℝle. Since ∪i=1pℝlai1bi=4p therefore ℝle∩∪i=1pℝlai1bi≥ℝlai1bi∀e∈EDp.Table 2
Cardinality of each LRN set.
LRN setComparisonℝlaiai+14p−4>ℝlai1biℝlbi1bi+114p−4>ℝlai1biℝlaiai14p−4>ℝlai1biℝlbibi14p−4>ℝlai1biTheorem 4.
LetDp be a convex polytope network, with p≥4 and p≅0mod2. Then(12)pp−1≤LFLNDp≤85.Proof.
In order to prove the result, we split into two cases:Case 3.
Forp=4, we have following LRN sets; (13)ℝl1=ℝla11b1=VD4−a2,a3,b3,b31,b41,ℝl2=ℝla21b2=VD4−a3,a4,b4,b41,b11,ℝl3=ℝla31b3=VD4−a4,a1,b1,b11,b21,ℝl4=ℝla41b4=VD4−a1,a2,b2,b21,b31,ℝl5=ℝla21b1=VD4−a1,a4,b21,b31,ℝl6=ℝla31b2=VD4−a2,a1,b31,b41,ℝl7=ℝla41b3=VD4−a3,a2,b41,b11,ℝl8=ℝla51b4=VD4−a4,a3,b11,b21,ℝl9=ℝla1a2=VD4−b1,b3,b11,b31,ℝl10=ℝla2a3=VD4−b2,b4,b21,b41,ℝl11=ℝla3a4=VD4−b3,b1,b31,b11,ℝl12=ℝla4a1=VD4−b4,b2,b41,b21,ℝl13=ℝla1a11=VD4−a21,a41,b31,b2,b3,ℝl14=ℝla2a21=VD4−a31,a11,b41,b3,b4,ℝl15=ℝla3a31=VD4−a41,a21,b11,b4,b1,ℝl16=ℝla4a41=VD4−a11,a31,b21,b1,b2,ℝl17=ℝlb1b11=VD4−a31,b2,b4,ℝl18=ℝlb2b21=VD4−a41,b3,b1,ℝl19=ℝlb3b31=VD4−a11,b4,b2,ℝl20=ℝlb4b41=VD4−a21,b1,b3,ℝl21=ℝlb11b21=VD4−a2,a4,a21,a41,ℝl22=ℝlb21b31=VD4−a3,a1,a31,a11,ℝl23=ℝlb31b41=VD4−a4,a2,a41,a21,ℝl24=ℝlb41b11=VD4−a1,a3,a11,a31.
Since,ℝlai1bi=11 and ℝlai1bi≤ℝle∀e∈ED4, where 1≤i≤4. Furthermore, ∪i=14ℝlai1bi=D4 and ℝle∩∪i=14ℝlai1bi≥ℝlai1bi∀e∈ED4. Hence, we define an upper LRF h:D4⟶0,1 as hv=1/10∀v∈VD4. In order to show that h is minimal upper LRF consider another function h′:VD4⟶0,1 as h′v<1/11∀v∈VD4 therefore h′ℝle<1 and h′<h which shows that h′ is not LRF of D4 therefore LFLND4≤∑i=1161/11=16/11. Likewise for 1≤i≤4 cardinality of LRN set ℝlbibi1 is 13 which is greater then the cardinalities of all other LRN sets. Therefore there exist a maximal lower LRF g:D4⟶0,1 and which is defined as gv=1/13∀v∈VD4 hence LFLND4≥∑i=1161/13=16/13. Consequently,(14)1613≤LFLND4≤1611.Case 4.
Forp≥6, 1≤i≤p by Lemma 2, ℝlai1bi=5p/2 and ℝlai1bi≥ℝle∀e∈EDp. Furthermore, ℝle∩∪i=1pℝlai1bi≥ℝlai1bi. Hence there exist an upper LRF h:VDp⟶0,1 and is defined as hv=2/5p∀v∈VDp. In order to show that h is minimal upper LRF consider another function h′:VDp⟶0,1 as hv<2/5p∀v∈VDp therefore h′ℝle<1 and h′<h which shows that h′ is not LRF of Dp hence LFLNDp≤∑i=14p2/5p=8/5. Likewise the cardinality of LRN set ℝlbi1bi+11 is 4p−4 which is greater or equal to the cardinalities of all other LRN sets of Dp. Hence, we define a maximal lower LRF g:VDp⟶0,1 is as gv=1/4p−4∀v∈VDp, therefore LFLNDp≥∑i=14p1/4p−4=p/p−1.
Consequently,(15)pp−1≤LFLNDp≤85.


### 3.2. LFLN of Convex PolytpoeEp

In this particular subsection, we have computed the LRN sets and LFLN of convex polytope networkEp. The VEp=VDp and EEp=EDp∪bibi+1:1≤i≤p. The order and size of Ep is 4p and 7p respectively. For more details see Figure 2.Figure 2
Convex polytope networkEp.Lemma 3.
LetEp be a convex polytope network, with p≥3 and p≅1mod2. Then.(i)
ℝlai1bi=ℝlbi+1ai1=2p+2 and ∪i=1pℝlai1bi=VEp,(ii)
ℝlai1bi≤ℝle and ℝle∩∪i=1pℝlai1bi≥ℝlai1bi∀e∈EEp.Proof.
Considerai inner, ai1,bi middle and bi1 are outer vertices of Ep, where 1≤i≤p and p+1≅1modp.(i)
ℝlai1bi=VEp−ai+1,ai+2,ai+3,….,ap+2i−1/2,ap+2i+1/21,ap+2i+3/21,ap+2i+5/21,….,ap1,bp+2i+1/21,bp+2i+3/21,….,bp,bp+2i+1/21,bp+2i+3/21,bp+2i+5/21,…,bp+i−11 and ℝlbiai+11=VEp−ai,ap+2i+3/21,ap+2i+5/21,ap+2i+7/21,….,ap+i−11,bi+11,bi+21,…,bp+2i+1/21,bp+2i+3/21,bp+2i+5/21,…,bp+i−11,bp+2i+1/2,bp+2i+3/2,bp+2i+5/2,…,bp+i−1. Note that ∪i=1pℝlai1bi=4p and ℝlai1bi=2p+2.(ii)
ℝlaiai+1=VEp−ap+2i+1/2,ap+2i+1/21,bi1,bi, ℝlbi1bi+11=VEp−ai+1,ai+11,bp+2i+1/21,bp+2i+1/2, ℝlaiai1=VEp−ai+11,ai+21,…,ap+2i+1/21,bp+2i+1/2,bp+2i+1/21,, ℝlbibi1=VEp, ℝlbibi+1=VEp−ai+1,ai+11,bp+2i+1/2,bp+2i+1/21 Now, we illustrate the cardinalities of the LRN sets in Table 3 and also compare them.
It can be observed with the help of Table3 that ℝlai1bi≤ℝle. Since ∪i=1pℝlai1bi=4p therefore ℝle∩∪i=1pℝlai1bi≥ℝlai1bi∀e∈EEp.Table 3
Cardinality of each LRN set.
RLN setComparisonℝlaiai+14p−4>ℝlai1biℝlbi1bi+114p−4>ℝlai1biℝlaiai14p−4>ℝlai1biℝlbibi14p>ℝlai1biℝlbibi4p−4>ℝlai1biTheorem 5.
LetE3 be a convex polytope network. Then(16)65≤LFLNE3≤32.Proof.
The LRN for convex polytopeE3 are(17)ℝl1=ℝla1a2=VE3−a3,a31,b1,b11,ℝl2=ℝla2a3=VE3−a1,a21,b2,b21,ℝl3=ℝla3a1=VE3−a2,a11,b3,b31ℝl4=ℝlb1b2=VE3−a2,b3,a21,b21ℝl5=ℝlb2b3=VE3−a3,b1,a31,b31,ℝl6=ℝlb3b1=VE3−a1,b2,a11,b11,ℝl7=ℝlb11b21=VE3−b31,b3,a21,a2,ℝl8=ℝlb21b31=VE3−b11,b1,a31,a3,ℝl9=ℝlb31b11=VE3−b21,b2,a11,a1,ℝl10=ℝla11b1=VE3−a2,a31,b2,b31,ℝl11=ℝla21b2=VE3−a3,a11,b3,b11,ℝl12=ℝla31b3=VE3−a1,a21,b1,b21,ℝl13=ℝla21b1=VE3−a1,b2,b21,ℝl14=ℝla31b2=VE3−a2,b3,a31;ℝl15=ℝla11b3=VE3−a3,b1,a11,ℝl16=ℝla11a1=VE3−a21,a31,,ℝl17=ℝla21a2=VE3−a31,a11,,ℝl18=ℝla31a3=VE3−a11,a21,,ℝl19=ℝlb11b1=VE3,ℝl20=ℝlb21b2=VE3,ℝl21=ℝlb31b3=VE3.
For1≤i≤3 the cardinality of each LRN set ℝlaiai+1 is 8 which is less then the other LRN sets of E3. Furthermore, ∪i=112ℝlaiai+1=VE3 and ℝle∩∪i=13ℝlaiai+1≥ℝlaiai+1∀e∈EE3.
Hence there exist an upper LRFh:VEp⟶0,1 is defined as hv=1/8∀v∈VE3. In order to show that h is minimal upper LRF consider another function h′:VE3⟶0,1 as h′v<1/10∀v∈VE3 therefore h′ℝle<1 and h′<h which shows that h′ is not LRF of E3 hence LFLNE3≤∑i=1121/8=3/2. Likewise for 1≤i≤3 cardinality of LRN set ℝlbi1bi is 12 which is greater then the cardinalities of all other LRN sets. Hence there exist a maximal lower LRF g:Dp⟶0,1 is defined as gv=1/12∀v∈VE3, therefore LFLNE3>∑i=1121/12=1. Consequently,(18)1<LFLNE3≤32.Theorem 6.
LetEp be a convex polytope network, with p≥5 and p≅1mod2. Then(19)1<LFLNEp≤2pp+1.Proof.
In order to prove the result, we split it into two casesCase 5.
Forp=5, we have the following LRN sets;(20)ℝl1=ℝla11b1=VE5−a2,a3,a41,b4,b5,a51,b41,b51,ℝl2=ℝla21b2=VE5−a3,a4,a51,b5,b1,a11,b51,b11,ℝl3=ℝla31b3=VE5−a4,a5,a11,b1,b2,a21,b11,b21,ℝl4=ℝla41b4=VE5−a5,a1,a21,b2,b3,a31,b21,b31,ℝl5=ℝla51b5=VE5−a1,a2,a31,b3,b4,a41,b31,b41,ℝl6=ℝla21b1=VE5−a1,a5,a31,a41,b2,b3,b21,b31,ℝl7=ℝla31b2=VE5−a2,a1,a41,a51,b3,b4,b31,b41,ℝl8=ℝla41b3=VE5−a3,a2,a51,a11,b4,b5,b41,b51,ℝl9=ℝla51b4=VE5−a4,a3,a11,a21,b5,b1,b51,b11,ℝl10=ℝla11b5=VE5−a5,a4,a21,a31,b1,b2,b11,b21,ℝl11=ℝla1a2=VE5−a4,a41,b1,b11,ℝl12=ℝla2a3=VE5−a5,a51,b2,b21,ℝl13=ℝla3a4=VE5−a1,a11,b3,b31,ℝl14=ℝla4a5=VE5−a2,a21,b4,b41,ℝl15=ℝla5a1=VE5−a3,a31,b5,b51,ℝl16=ℝlb11b21=VE5−a2,a21,b4,b41,ℝl17=ℝlb21b31=VE5−a3,a31,b5,b51,ℝl18=ℝlb31b41=VE5−a4,a41,b1,b11,ℝl19=ℝlb41b51=VE5−a5,a51,b2,b21,ℝl20=ℝlb51b11=VE5−a1,a11,b3,b31,ℝl21=ℝla1a11=VE5−a21,a31,b4,b41,ℝl22=ℝla2a21=VE5−a31,a41,b5,b51,ℝl23=ℝla3a31=VE5−a41,a51,b1,b11,ℝl24=ℝla4a41=VE5−a51,a11,b2,b21,ℝl25=ℝla5a51=VE5−a11,a21,b3,b31,ℝl26=ℝlb1b2=VE5−a2,a21,b4,b41,ℝl27=ℝlb2b3=VE5−a3,a31,b5,b51,ℝl28=ℝlb3b4=VE5−a4,a41,b1,b11,ℝl29=ℝlb4b1=VE5−a5,a51,b2,b21,ℝl30=ℝlb5b2=VE5−a1,a11,b3,b31,ℝl31=ℝlb1b11=VE5,ℝl32=ℝlb2b21=VE5,ℝl33=ℝlb3b31=VE5,ℝl34=ℝlb4b41=VE5,ℝl35=ℝlb5b51=VE5.
Since,ℝlai1bi=ℝlai+11bi=12 and ℝlai1bi≤ℝle∀e∈EE5, where 1≤i≤5. Furthermore, ∪i=14ℝai1bil=VE5 and ℝle∩∪i=14ℝlai1bi≥ℝlai1bi∀e∈EE5. Hence, we define an upper LRF h:VE5⟶0,1 as hv=1/12∀v∈VE5. In order to show that h is minimal upper LRF consider another function h′:VE5⟶0,1 as hv<1/12∀v∈VE5 therefore h′ℝle<1 and h′<h which shows that h′ is not LRF of Dp therefore LFLN E5≤∑i=1201/12=5/3. Likewise for 1≤i≤5 cardinality of LRN set ℝlbibi1 is 20 which is greater then the cardinalities of all other LRN sets. Hence there exist a maximal lower LRF g:VE5⟶0,1 and it is defined by gv=1/20∀v∈VE5, therefore LFLNE5>∑i=1201/20=1. Consequently,(21)1<LFLNE5≤53.Case 6.
Forp≥5, 1≤i≤p by Lemma 3, ℝlai1bi=2p+2 and ℝlai1bi≥ℝle∀e∈EEp. Furthermore, ℝle∩∪i=1pℝlai1bi≥ℝlai1bi. Therefore, we define an upper LRF h:Ep⟶0,1 as hv=1/2p+4∀v∈VEp. In order to show that h is minimal upper LRF consider another function h′:VEp⟶0,1 as h′v<1/2p+2∀v∈VEp therefore h′ℝle<1 and h′<h which shows that h′ is not LRF of Dp hence LFLNEp≤∑i=14p1/2p+2=2p/p+1. Likewise the cardinality of LRN set ℝlbibi1 is 4p which is greater then the cardinalities of all other LRN sets of Ep . Hence there exist a maximal LRF g:Ep⟶0,1 and it is defined as gv=1/4p∀v∈VEp therefore LFLNEp>∑i=14p1/4p=1. Consequently,(22)1<LFLNEp≤2pp+1.Lemma 4.
LetEp be a convex polytope network, with p≥4 and p≅0mod2. Then.(i)
ℝlai1bi=ℝlbiai+11=2p+1 and ∪i=1pℝlai1bi=VEp.(ii)
ℝlai1bi≤ℝle and ℝle∩∪i=1pℝlai1bi≥ℝlai1bi∀e∈EEp.Proof.
Considerai inner, ai1,bi middle and bi1 are outer vertices of Ep respectively, where 1≤i≤p and p+1≅1modp.(i)
ℝlai1bi=VEp−ai+1,ai+2,ai+3,….,ap+2i/2,ap+2i+2/21,ap+2i+4/21,ap+2i+6/21,….,ap+i−11,bp+2i/21,bp+2i+2/21,bp+2i+4/21,….,bp+i−11,bp+2i/2,bp+2i+2/2,bp+2i+4/2,….,bp+i−1 and ℝlbiai+11=VEp−ai,ap+2i+2/2,ap+2i+4/2,….,ap+i−1,ai+21,ai+31,…,ap+2i/21,bi+11,bi+21,…,bp+2i/21,bi+1,bi+2,…,bp+2i/2. Note that ∪i=1pℝlai1bi=4p and ℝlai1bi=2p+1.(ii)
ℝlaiai+1=VEp−bp+2i/2,bp+2i+1/21,bi1,bi, ℝlbi1bi+11=VEp−ap+2i/2,ap+2i/21, ℝlaiai1=VEp−ai+11,ai+21,…,ap1,bp+2i/2,bp+2i/21, ℝlbibi1=VEp, ℝlbibi+1=VEpai+1,ai+11. , The cardinalities of all the LRN sets are illustrated in Table 4.
Now it is clear thatℝlai1bi≤ℝle. Since ∪i=1pℝlai1bi=4p therefore ℝle∩∪i=1pℝlai1bi≥ℝlai1bi∀e∈EEp.Table 4
Cardinality of each LRN set ofEp.
LRN setComparisonℝlaiai+14p−4>ℝlai1biℝlbi1bi+114p−4>ℝlai1biℝlaiai13p>ℝlai1biℝlbibi14p>ℝlai1biℝlbibi+14p−2>ℝlai1biTheorem 7.
LetEp be a convex polytope network, where p≥4 and p≅0mod2. Then(23)1<LFLNEp≤4p2p+1.Proof.
In order to prove the result, we split it into two casesCase 7.
Forp=4, we have following LRN sets;(24)ℝl1=ℝla11b1=VE4−a2,a3,a41,b3,b4,b31,b41,ℝl2=ℝla21b2=VE4−a3,a4,a11,b4,b1,b41,b51,ℝl3=ℝla31b3=VE4−a4,a1,a21,b1,b2,b11,b11,ℝl4=ℝla41b4=VE4−a1,a2,a31,b2,b3,b21,b21,ℝl5=ℝla21b1=VE4−a1,a4,a31,b2,b3,b21,b31,ℝl6=ℝla31b2=VE4−a2,a1,a41,b3,b4,b31,b41,ℝl7=ℝla41b3=VE4−a3,a2,a11,b4,b1,b41,b11,ℝl8=ℝla11b4=VE4−a4,a3,a21,b1,b2,b11,b21,ℝl9=ℝla1a2=VE4−b31,b11,b1,b3,ℝl10=ℝla2a3=VE4−b41,b21,b2,b4,ℝl11=ℝla3a4=VE4−b11,b31,b3,b1,ℝl12=ℝla4a1=VE4−b21,b41,b4,b2,ℝl13=ℝla1a11=VE4−a21,a31,a41,b3,ℝl14=ℝla2a21=VE4−a31,a41,a11,b4,ℝl15=ℝla3a31=VE4−a41,a11,a21,b1,ℝl16=ℝla4a41=VE4−a11,a21,a31,b2,ℝl17=ℝlb11b21=VE4−a2,a4,a21,a41,ℝl18=ℝlb21b31=VE4−a3,a1,a31,a11,ℝl19=ℝlb31b41=VE4−a4,a2,a41,a21,ℝl20=ℝlb41b11=VE4−a1,a3,a11,a31,ℝl21=ℝlb1b2=VE4−a2,a4,a21,a41,ℝl22=ℝlb2b3=VE4−a3,a1,a31,a11,ℝl23=ℝlb3b4=VE4−a4,a2,a41,a21,ℝl24=ℝlb4b1=VE4−a1,a3,a11,a31,ℝl25=ℝlb1b11=VE4,ℝl26=ℝlb2b21=VE4,ℝl27=ℝlb3b31=VE4,ℝl28=ℝlb4b41=VE4.
Since,ℝlai1bi=ℝlai+11bi=9 and ℝlai1bi≤ℝle∀e∈EE4, where 1≤i≤4. Furthermore, ∪i=14ℝlai1bi=VE4 and ℝle∩∪i=14ℝlai1bi≥ℝlai1bi∀e∈EE4. Hence, we define an upper LRF h:VE4⟶0,1 as hv=1/9∀v∈VE4. In order to show that h is minimal upper LRF consider another function h′:VE4⟶0,1 as hv<1/9∀v∈VE4 therefore h′ℝle<1 and h′<h which shows that h′ is not LRF of E4 therefore LFLNE4≤∑i=1161/9=16/9. Likewise for 1≤i≤3 cardinality of LRN set ℝlbibi1 is 20 which is greater then the cardinalities of all other LRN sets. Hence there exist a maximal lower LRF g:VE4⟶0,1 is defined by gv=1/20∀v∈VE4, therefore LFLNE4>∑i=1201/20=1. Consequently,(25)1<LFLNE4≤169.Case 8.
Forp≥4, 1≤i≤p by Lemma 4, ℝlai1bi=2p+1 and ℝlai1bi≥ℝle∀e∈EEp. Furthermore, ℝle∩∪i=1pℝlai1bi≥ℝlai1bi. Hence, we define an upper LRF h:Ep⟶0,1 as hv=1/2p+1∀v∈VEp. In order to show that h is minimal upper LRF consider another function h′:VDp⟶0,1 as hv<1/2p+1∀v∈VEp this implies h′ℝle<1 and h′<h which shows that h′ is not LRF of Ep therefore LFLNEp≤∑i=14p1/2p+1=4p/2p+1. Likewise the cardinality of LRN set ℝlbibi1 is 4p which is greater then the cardinalities of all the other LRN sets. Therefore there exist a maximal lower LRF g:Ep⟶0,1 and it is defined as gv=1/4p∀v∈VEp therefore LFLNEp>∑i=14p1/4p=1. Consequently,(26)1<LFLNEp≤4p2p+1.


## 3.1. LFLN of Convex Polytpoe NetworkDp

In this particular subsection, we have computed resolving local neighbourhood sets and LFLN ofDp in the form of exact values and sharp lower and upper bounds.The convex polytope network is introduced by Bača [41] and LN of Dp is 3 is proved in [10]. The vertex set VDp consists of inner ai:1≤i≤p, middle ai1:1≤i≤p, bi:1≤i≤p and outer vertices bi1:1≤i≤p. The edge set Dp is defined as EDp=aiai+1:1≤i≤p∪ai1ai:1≤i≤p∪ai1bi:1≤i≤p∪bi1bi:1≤i≤p∪bi1bi+11:1≤i≤p∪ai+11bi:1≤i≤p. Furthermore, the order and size of Dp are 4p and 6p respectively and for complete details see Figure 1.Figure 1
Convex polytope networkDp.Lemma 1.
LetDp be a convex polytope network, with p≥5 and p≅1mod2. Then.(i)
ℝlai1bi=ℝlbiai+11=5p+7/2 and ∪i=1pℝlai1bi=VDp.(ii)
ℝlai1bi≤ℝle and ℝle∩∪i=1pℝlai1bi≥ℝlai1bi∀e∈EDp.Proof.
Considerai inner, ai1,bi middle and bi1 are outer vertices of Dp, where 1≤i≤p and p+1≅1modp.(i)
ℝlai1bi=VDp−ai+1,ai+2,ai+3,….,ap+2i−1/2,ap+2i−1/21,bp+2i+1/21,bp+2i+3/21,bp+2i+1/51,…,bp+i−11 and ℝlbiai+11=VDp−ai,ap+2i+3/2,ap+2i+5/2,ap+2i+7/2,….,ap+i−1,bi+11,bi+21,…,bp+2i+1/51. Note that ∪i=1pℝlai1bi=VDp and ℝlai1bi=5p+7/2.(ii)
ℝlaiai+1=VDp−ap+2i+1/2,ap+2i+1/21,bi1,bi, ℝlbi1bi+11=VDp−ai+1,ai+11,bp+2i+1/21,bp+2i+1/2, ℝlaiai1=VDp−ai+21,ap1,bi+2,bp and ℝlbibi1=VDp−ai+31,ap1,bi+2,bp.
The cardinalities of all the LRN sets ofDp are illustrated in Table 1.
From Table1, we note that ℝlai1bi≤ℝle. Since ∪i=1pℝlai1bi=4p therefore ℝle∩∪i=1pℝlai1bi≥ℝlai1bi∀e∈EDp.Table 1
The cardinalities of all the LRN sets ofDp.
RLN setComparisonℝlaiai+14p−4>ℝlai1biℝlbi1bi+114p−4>ℝlai1biℝlaiai14p−4>ℝlai1biℝlbibi14p−4>ℝlai1biTheorem 1.
LetD3 be a convex polytope network. Then(3)65≤LFLND3≤32.Proof.
The LRN sets of convex polytope networkD3 are: ℝl1=ℝla1a2=VD3−a3,a31,b1,b11,(4)ℝl2=ℝla2a3=VD3−a1,a21,b2,b21ℝl3=ℝla3a1=VD3−a2,a11,b3,b31ℝl4=ℝla11b1=VD3−a2,a31,b2,b31ℝl5=ℝla21b2=VD3−a3,a11,b3,b11ℝl6=ℝla31b3=VD3−a1,a21,b1,b21ℝl7=ℝlb11b21=VD3−b31,b3,a21,a2ℝl8=ℝlb21b31=VD3−b11,b1,a31,a3ℝl9=ℝlb31b11=VD3−b21,b2,a11,a1ℝl10=ℝlb11b1=VD3−b2,b3,a31ℝl11=ℝlb21b2=VD3−b3,b1,a11ℝl12=ℝlb31b3=VD3−b1,b2,a21ℝl13=ℝla11a1=VD3−a21,a31ℝl14=ℝla21a2=VD3−a31,a11ℝl15=ℝla31a3=VD3−a11,a21ℝl16=ℝla21b1=VD3−a1,a21ℝl17=ℝla31b2=VD3−a2,a31ℝl18=ℝla11b3=VD3−a3,a11.
Since,ℝlaiai+1=ℝlbi1bi+11=ℝlbi1bi=ℝlai1ai=8 which is less then the other LRN sets, where 1≤i≤3. Furthermore, ∪i=13ℝlaiai+1=VD3 and ℝle∩∪i=13ℝlaiai+1≥ℝlaiai+1∀e∈ED3.
Therefore, we define an upper LRFh:D3⟶0,1 as hv=1/8∀v∈VD3. In order to show that h is minimal LRF consider another upper LRF h′:VD3⟶0,1 as hv<1/8∀v∈VD3 therefore hℝle<1 and h′<h which shows that h′ is not LRF of D3 hence LFLND3≤∑i=1121/8=3/2. Likewise for 1≤i≤3 cardinality of LRN set ℝlai1bi is 10 which is greater then the cardinalities of all other LRN sets of D3. Therefore, we define a maximal lower LRF g:Dp⟶0,1 as gv=1/10∀v∈VD3. Hence LFLND3≥∑i=1121/10=6/5. Consequently,(5)65≤LFLND3≤35.Theorem 2.
LetD5 be a convex polytope network. Then(6)LFLND5=54.Proof.
The LRN sets are given by:ℝl1=ℝla1a2=VD5−a4,a41,b1,b11,(7)ℝl2=ℝla2a3=VD5−a5,a51,b2,b21ℝl3=ℝla3a4=VD5−a1,a11,b3,b31ℝl4=ℝla4a5=VD5−a2,a21,b4,b41ℝl5=ℝla5a1=VD5−a3,a31,b5,b51ℝl6=ℝlb1b11=VD5−a31,a51,b2,b5ℝl7=ℝlb2b21=VD5−a41,a11,b3,b1ℝl8=ℝlb3b31=VD5−a51,a21,b4,b2ℝl9=ℝlb4b41=VD5−a11,a31,b5,b1ℝl10=ℝlb5b51=VD5−a21,a41,b1,b2ℝl11=ℝlb11b21=VD5−a21,a2,b4,b41ℝl12=ℝlb21b31=VD5−a31,a3,b5,b51ℝl13=ℝlb31b41=VD5−a41,a4,b1,b11ℝl14=ℝlb41b51=VD5−a51,a5,b2,b21ℝl15=ℝlb51b11=VD5−a11,a1,b3,b31ℝl16=ℝla1a11=VD5−a21,a51,b2,b4ℝl17=ℝla2a21=VD5−a31,a11,b3,b5ℝl18=ℝla3a31=VD5−a41,a21,b4,b1ℝl19=ℝla4a41=VD5−a51,a31,b5,b2ℝl20=ℝla5a51=VD5−a11,a41,b1,b3ℝl21=ℝla11b1=VD5−a2,a3,b41,b51ℝl22=ℝla21b2=VD5−a3,a4,b51,b11ℝl23=ℝla31b3=VD5−a4,a5,b11,b21ℝl24=ℝla41b4=VD5−a5,a1,b21,b31ℝl25=ℝla51b5=VD5−a1,a2,b31,b41ℝl26=ℝla21b1=VD5−a1,a5,b21,b31ℝl27=ℝla31b2=VD5−a2,a1,b31,b41ℝl28=ℝla41b3=VD5−a3,a2,b41,b51ℝl29=ℝla51b4=VD5−a4,a3,b51,b11ℝl30=ℝla11b5=VD5−a5,a4,b11,b21.
It is clear that the cardinality of each RLN set ofD5 is 16. Therefore, we define a constant function h:VD5⟶0,1 as hv=1/16∀v∈VD5. Hence LFLND5=∑i=1201/16=5/4.Theorem 3.
LetDp be a convex polytope network, with p≥7 and p≅1mod2. Then(8)pp−1≤LFLNDp≤8p5p+7.Proof.
To prove the result, we split it into two casesCase 1.
Forp=7 , we have following LRN sets;ℝl1=ℝla11b1=VD7−a2,a3,a4,a41,b51,b61,b71,(9)ℝl2=ℝla21b2=VD7−a3,a4,a5,a51,b61,b71,b11ℝl3=ℝla31b3=VD7−a4,a5,a6,a61,b71,b11,b21ℝl4=ℝla41b4=VD7−a5,a6,a7,a71,b11,b21,b31ℝl5=ℝla51b5=VD7−a6,a7,a1,a11,b21,b31,b41ℝl6=ℝla61b6=VD7−a7,a1,a2,a21,b31,b41,b51ℝl7=ℝla71b7=VD7−a1,a2,a3,a31,b41,b51,b61ℝl8=ℝla21b1=VD7−a1,a6,a7,b21,b31,b41,b51ℝl9=ℝla31b2=VD7−a2,a7,a1,b31,b41,b51,b61ℝl10=ℝla41b3=VD7−a3,a1,a2,b41,b51,b61,b71ℝl11=ℝla51b4=VD7−a4,a2,a3,b51,b61,b71,b11ℝl12=ℝla61b5=VD7−a5,a3,a4,b61,b71,b11,b21ℝl13=ℝla71b6=VD7−a6,a4,a5,b71,b11,b21,b31ℝl14=ℝla11b7=VD7−a7,a5,a6,b11,b21,b31,b41,ℝl15=ℝla1a2=VD7−a5,a51,b1,b11ℝl16=ℝla2a3=VD7−a6,a61,b2,b21,ℝl17=ℝla3a4=VD7−a7,a71,b3,b31,ℝl18=ℝla4a5=VD7−a1,a11,b4,b41,ℝl19=ℝla5a6=VD7−a2,a21,b5,b51,ℝl20=ℝla6a7=VD7−a3,a31,b6,b61,ℝl21=ℝla7a1=VD7−a4,a41,b7,b71,ℝl22=ℝlb11b21=VD7−a2,a21,b5,b51ℝl23=ℝlb21b31=VD7−a3,a31,b6,b61,ℝl24=ℝlb31b41=VD7−a4,a41,b7,b71,ℝl25=ℝlb41b51=VD7−a5,a51,b1,b11,ℝl26=ℝlb51b61=VD7−a6,a61,b2,b21,ℝl27=ℝlb61b71=VD7−a7,a71,b3,b31,ℝl28=ℝlb71b11=VD7−a1,a11,b4,b41,ℝl29=ℝla1a11=VD7−a21,a71,b2,b6,ℝl30=ℝla2a21=VD7−a31,a11,b3,b7,ℝl31=ℝla3a31=VD7−a41,a21,b4,b1,ℝl32=ℝla4a41=VD7−a51,a31,b5,b2,ℝl33=ℝla5a51=VD7−a61,a41,b6,b3,ℝl34=ℝla6a61=VD7−a71,a51,b7,b4,ℝl35=ℝla7a71=VD7−a11,a61,b1,b5,ℝl36=ℝlb1b11=VD7−a31,a71,b2,b7,ℝl37=ℝlb2b21=VD7−a41,a11,b3,b1,ℝl38=ℝlb3b31=VD7−a51,a21,b4,b2,ℝl39=ℝlb4b41=VD7−a61,a31,b5,b3,ℝl40=ℝlb5b51=VD7−a71,a41,b6,b4,ℝl41=ℝlb6b61=VD7−a11,a51,b7,b5,ℝl42=ℝlb7b71=VD7−a21,a61,b1,b6.
Since,ℝlai1bi=ℝlbiai+11=16 and ℝlai1bi≤ℝle∀e∈ED7, where 1≤i≤7.
Furthermore,∪i=17ℝlai1bi=VD7 and ℝle∩∪i=17ℝlai1bi≥ℝlai1bi∀e∈ED7. Hence, we define an upper LRF h:VD7⟶0,1 as hv=1/21∀v∈VD7. In order to show that h is minimal upper LRF consider another mapping h′:VD7⟶0,1 as hv<1/21∀v∈VD7 therefore hℝle<1 and h′<h which shows that h′ is not LRF of D3 hence LFLND7≤∑i=1281/21=4/3. Likewise for 1≤i≤7 cardinality of LRN set ℝlaiai+1 is 24 which is greater then the cardinalities of all other RLN sets. Therefore, we define a lower LRF g:VD7⟶0,1 as hv=1/24∀v∈VD7 is a maximal lower LRF hence LFLND7≥∑i=1281/24=7/6. Consequently,(10)76≤LFLND7≤43.Case 2.
Forp≥7, 1≤i≤p by Lemma 1, ℝlai1bi=5p+7/2 and ℝlai1bi≤ℝle∀e∈EDp. Furthermore, ℝle∩∪i=1pℝlai1bi≥ℝlai1bi. Hence an upper LRF h:VDp⟶0,1 is defined as hv=2/5p+7∀v∈VDp. In order to show that h is minimal upper LRF consider another function h′:VDp⟶0,1 as hv<2/5p+7∀v∈VDp therefore h′ℝle<1 and h′<h which shows that h′ is not LRF of Dp hence LFLNDp≤∑i=14p2/5p+7=8p/5p+7. Likewise the cardinality of RLN set ℝlaiai+1 is 4p−4 which is greater then the cardinalities of all other LRN sets. Therefore, we define a maximal LRF g:Dp⟶0,1 as gv=1/4p−4∀v∈VDp, hence LFLNDp≥∑i=14p1/4p−4=p/p−1.
Consequently,(11)pp−1≤LFLNDp≤8p5p+7.Lemma 2.
Suppose thatDp is a convex polytope network, with p≥6 and p≅0mod2. Then.(i)
ℝlai1bi=ℝlai+11bi=5p/2 and ∪i=14pai1bi=VDp.(ii)
ℝlai1bi≤ℝle and ℝle∩∪i=1pℝlai1bi≥ℝlai1bi∀e∈EDp.Proof.
Considerai inner, ai1,bi middle and bi1 are the outer vertices of Dp, where 1≤i≤p and p+1≅1modp.(i)
ℝlai1bi=VDp−ai+2,ai+3,ai+4,….,ap+2i/2,ap+2i−1/21,ap+2i/21,bp+2i/21,bp+2i+2/21,bp+2i+4/21,…,bp1,bp+2i/2,bp+2i+2/2,bp+2i+4/2,…,bp+i−3 and ℝlbiai+11=VDp−ai,ap+2i+2/2,ap+2i+4/2,ap+2i+6/2,….,api−1,ap+2i+2/21,ap+2i+3/21,ap+2i+5/21,….,ap+i−11,bi+3,bi+4,…,bp+2i/2,bp+2i+2/21,bp+2i+4/21,bp+2i+6/21,….,bp+i−11. Note that ∪i=14pℝlai1bi=VDp and ℝlai1bi=5p/2.(ii)
ℝlai+11bi=VDp−ai,ap+2i+2/2,ap+2i+4/2,….,ap,ap+2i+2/21,ap+2i+4/21,…,ap1,bi+11,bi+21,bi+31,…bp+2i/21,bp+2i/2,bp+2i+2/2,bp+2i+4/2,…,bp−2ℝlaiai+1=VDp−bp+2i/2,bp+2i/21,bi1,bi, ℝlbi1bi+11=VDp−ai+1,ai+11,ap+2i+2/2,ap+2i+2/21, ℝlaiai1=VDp−ai+21,ap1,bi+2,bp−1, ℝlbibi1=VDp−ai+31,ap1,bi+2,bp. The cardinalities of each LRN set of Dp is illustrated in Table 2.
It is can be observed with the help of Table2 that ℝlai1bi≤ℝle. Since ∪i=1pℝlai1bi=4p therefore ℝle∩∪i=1pℝlai1bi≥ℝlai1bi∀e∈EDp.Table 2
Cardinality of each LRN set.
LRN setComparisonℝlaiai+14p−4>ℝlai1biℝlbi1bi+114p−4>ℝlai1biℝlaiai14p−4>ℝlai1biℝlbibi14p−4>ℝlai1biTheorem 4.
LetDp be a convex polytope network, with p≥4 and p≅0mod2. Then(12)pp−1≤LFLNDp≤85.Proof.
In order to prove the result, we split into two cases:Case 3.
Forp=4, we have following LRN sets; (13)ℝl1=ℝla11b1=VD4−a2,a3,b3,b31,b41,ℝl2=ℝla21b2=VD4−a3,a4,b4,b41,b11,ℝl3=ℝla31b3=VD4−a4,a1,b1,b11,b21,ℝl4=ℝla41b4=VD4−a1,a2,b2,b21,b31,ℝl5=ℝla21b1=VD4−a1,a4,b21,b31,ℝl6=ℝla31b2=VD4−a2,a1,b31,b41,ℝl7=ℝla41b3=VD4−a3,a2,b41,b11,ℝl8=ℝla51b4=VD4−a4,a3,b11,b21,ℝl9=ℝla1a2=VD4−b1,b3,b11,b31,ℝl10=ℝla2a3=VD4−b2,b4,b21,b41,ℝl11=ℝla3a4=VD4−b3,b1,b31,b11,ℝl12=ℝla4a1=VD4−b4,b2,b41,b21,ℝl13=ℝla1a11=VD4−a21,a41,b31,b2,b3,ℝl14=ℝla2a21=VD4−a31,a11,b41,b3,b4,ℝl15=ℝla3a31=VD4−a41,a21,b11,b4,b1,ℝl16=ℝla4a41=VD4−a11,a31,b21,b1,b2,ℝl17=ℝlb1b11=VD4−a31,b2,b4,ℝl18=ℝlb2b21=VD4−a41,b3,b1,ℝl19=ℝlb3b31=VD4−a11,b4,b2,ℝl20=ℝlb4b41=VD4−a21,b1,b3,ℝl21=ℝlb11b21=VD4−a2,a4,a21,a41,ℝl22=ℝlb21b31=VD4−a3,a1,a31,a11,ℝl23=ℝlb31b41=VD4−a4,a2,a41,a21,ℝl24=ℝlb41b11=VD4−a1,a3,a11,a31.
Since,ℝlai1bi=11 and ℝlai1bi≤ℝle∀e∈ED4, where 1≤i≤4. Furthermore, ∪i=14ℝlai1bi=D4 and ℝle∩∪i=14ℝlai1bi≥ℝlai1bi∀e∈ED4. Hence, we define an upper LRF h:D4⟶0,1 as hv=1/10∀v∈VD4. In order to show that h is minimal upper LRF consider another function h′:VD4⟶0,1 as h′v<1/11∀v∈VD4 therefore h′ℝle<1 and h′<h which shows that h′ is not LRF of D4 therefore LFLND4≤∑i=1161/11=16/11. Likewise for 1≤i≤4 cardinality of LRN set ℝlbibi1 is 13 which is greater then the cardinalities of all other LRN sets. Therefore there exist a maximal lower LRF g:D4⟶0,1 and which is defined as gv=1/13∀v∈VD4 hence LFLND4≥∑i=1161/13=16/13. Consequently,(14)1613≤LFLND4≤1611.Case 4.
Forp≥6, 1≤i≤p by Lemma 2, ℝlai1bi=5p/2 and ℝlai1bi≥ℝle∀e∈EDp. Furthermore, ℝle∩∪i=1pℝlai1bi≥ℝlai1bi. Hence there exist an upper LRF h:VDp⟶0,1 and is defined as hv=2/5p∀v∈VDp. In order to show that h is minimal upper LRF consider another function h′:VDp⟶0,1 as hv<2/5p∀v∈VDp therefore h′ℝle<1 and h′<h which shows that h′ is not LRF of Dp hence LFLNDp≤∑i=14p2/5p=8/5. Likewise the cardinality of LRN set ℝlbi1bi+11 is 4p−4 which is greater or equal to the cardinalities of all other LRN sets of Dp. Hence, we define a maximal lower LRF g:VDp⟶0,1 is as gv=1/4p−4∀v∈VDp, therefore LFLNDp≥∑i=14p1/4p−4=p/p−1.
Consequently,(15)pp−1≤LFLNDp≤85.


## 3.2. LFLN of Convex PolytpoeEp

In this particular subsection, we have computed the LRN sets and LFLN of convex polytope networkEp. The VEp=VDp and EEp=EDp∪bibi+1:1≤i≤p. The order and size of Ep is 4p and 7p respectively. For more details see Figure 2.Figure 2
Convex polytope networkEp.Lemma 3.
LetEp be a convex polytope network, with p≥3 and p≅1mod2. Then.(i)
ℝlai1bi=ℝlbi+1ai1=2p+2 and ∪i=1pℝlai1bi=VEp,(ii)
ℝlai1bi≤ℝle and ℝle∩∪i=1pℝlai1bi≥ℝlai1bi∀e∈EEp.Proof.
Considerai inner, ai1,bi middle and bi1 are outer vertices of Ep, where 1≤i≤p and p+1≅1modp.(i)
ℝlai1bi=VEp−ai+1,ai+2,ai+3,….,ap+2i−1/2,ap+2i+1/21,ap+2i+3/21,ap+2i+5/21,….,ap1,bp+2i+1/21,bp+2i+3/21,….,bp,bp+2i+1/21,bp+2i+3/21,bp+2i+5/21,…,bp+i−11 and ℝlbiai+11=VEp−ai,ap+2i+3/21,ap+2i+5/21,ap+2i+7/21,….,ap+i−11,bi+11,bi+21,…,bp+2i+1/21,bp+2i+3/21,bp+2i+5/21,…,bp+i−11,bp+2i+1/2,bp+2i+3/2,bp+2i+5/2,…,bp+i−1. Note that ∪i=1pℝlai1bi=4p and ℝlai1bi=2p+2.(ii)
ℝlaiai+1=VEp−ap+2i+1/2,ap+2i+1/21,bi1,bi, ℝlbi1bi+11=VEp−ai+1,ai+11,bp+2i+1/21,bp+2i+1/2, ℝlaiai1=VEp−ai+11,ai+21,…,ap+2i+1/21,bp+2i+1/2,bp+2i+1/21,, ℝlbibi1=VEp, ℝlbibi+1=VEp−ai+1,ai+11,bp+2i+1/2,bp+2i+1/21 Now, we illustrate the cardinalities of the LRN sets in Table 3 and also compare them.
It can be observed with the help of Table3 that ℝlai1bi≤ℝle. Since ∪i=1pℝlai1bi=4p therefore ℝle∩∪i=1pℝlai1bi≥ℝlai1bi∀e∈EEp.Table 3
Cardinality of each LRN set.
RLN setComparisonℝlaiai+14p−4>ℝlai1biℝlbi1bi+114p−4>ℝlai1biℝlaiai14p−4>ℝlai1biℝlbibi14p>ℝlai1biℝlbibi4p−4>ℝlai1biTheorem 5.
LetE3 be a convex polytope network. Then(16)65≤LFLNE3≤32.Proof.
The LRN for convex polytopeE3 are(17)ℝl1=ℝla1a2=VE3−a3,a31,b1,b11,ℝl2=ℝla2a3=VE3−a1,a21,b2,b21,ℝl3=ℝla3a1=VE3−a2,a11,b3,b31ℝl4=ℝlb1b2=VE3−a2,b3,a21,b21ℝl5=ℝlb2b3=VE3−a3,b1,a31,b31,ℝl6=ℝlb3b1=VE3−a1,b2,a11,b11,ℝl7=ℝlb11b21=VE3−b31,b3,a21,a2,ℝl8=ℝlb21b31=VE3−b11,b1,a31,a3,ℝl9=ℝlb31b11=VE3−b21,b2,a11,a1,ℝl10=ℝla11b1=VE3−a2,a31,b2,b31,ℝl11=ℝla21b2=VE3−a3,a11,b3,b11,ℝl12=ℝla31b3=VE3−a1,a21,b1,b21,ℝl13=ℝla21b1=VE3−a1,b2,b21,ℝl14=ℝla31b2=VE3−a2,b3,a31;ℝl15=ℝla11b3=VE3−a3,b1,a11,ℝl16=ℝla11a1=VE3−a21,a31,,ℝl17=ℝla21a2=VE3−a31,a11,,ℝl18=ℝla31a3=VE3−a11,a21,,ℝl19=ℝlb11b1=VE3,ℝl20=ℝlb21b2=VE3,ℝl21=ℝlb31b3=VE3.
For1≤i≤3 the cardinality of each LRN set ℝlaiai+1 is 8 which is less then the other LRN sets of E3. Furthermore, ∪i=112ℝlaiai+1=VE3 and ℝle∩∪i=13ℝlaiai+1≥ℝlaiai+1∀e∈EE3.
Hence there exist an upper LRFh:VEp⟶0,1 is defined as hv=1/8∀v∈VE3. In order to show that h is minimal upper LRF consider another function h′:VE3⟶0,1 as h′v<1/10∀v∈VE3 therefore h′ℝle<1 and h′<h which shows that h′ is not LRF of E3 hence LFLNE3≤∑i=1121/8=3/2. Likewise for 1≤i≤3 cardinality of LRN set ℝlbi1bi is 12 which is greater then the cardinalities of all other LRN sets. Hence there exist a maximal lower LRF g:Dp⟶0,1 is defined as gv=1/12∀v∈VE3, therefore LFLNE3>∑i=1121/12=1. Consequently,(18)1<LFLNE3≤32.Theorem 6.
LetEp be a convex polytope network, with p≥5 and p≅1mod2. Then(19)1<LFLNEp≤2pp+1.Proof.
In order to prove the result, we split it into two casesCase 5.
Forp=5, we have the following LRN sets;(20)ℝl1=ℝla11b1=VE5−a2,a3,a41,b4,b5,a51,b41,b51,ℝl2=ℝla21b2=VE5−a3,a4,a51,b5,b1,a11,b51,b11,ℝl3=ℝla31b3=VE5−a4,a5,a11,b1,b2,a21,b11,b21,ℝl4=ℝla41b4=VE5−a5,a1,a21,b2,b3,a31,b21,b31,ℝl5=ℝla51b5=VE5−a1,a2,a31,b3,b4,a41,b31,b41,ℝl6=ℝla21b1=VE5−a1,a5,a31,a41,b2,b3,b21,b31,ℝl7=ℝla31b2=VE5−a2,a1,a41,a51,b3,b4,b31,b41,ℝl8=ℝla41b3=VE5−a3,a2,a51,a11,b4,b5,b41,b51,ℝl9=ℝla51b4=VE5−a4,a3,a11,a21,b5,b1,b51,b11,ℝl10=ℝla11b5=VE5−a5,a4,a21,a31,b1,b2,b11,b21,ℝl11=ℝla1a2=VE5−a4,a41,b1,b11,ℝl12=ℝla2a3=VE5−a5,a51,b2,b21,ℝl13=ℝla3a4=VE5−a1,a11,b3,b31,ℝl14=ℝla4a5=VE5−a2,a21,b4,b41,ℝl15=ℝla5a1=VE5−a3,a31,b5,b51,ℝl16=ℝlb11b21=VE5−a2,a21,b4,b41,ℝl17=ℝlb21b31=VE5−a3,a31,b5,b51,ℝl18=ℝlb31b41=VE5−a4,a41,b1,b11,ℝl19=ℝlb41b51=VE5−a5,a51,b2,b21,ℝl20=ℝlb51b11=VE5−a1,a11,b3,b31,ℝl21=ℝla1a11=VE5−a21,a31,b4,b41,ℝl22=ℝla2a21=VE5−a31,a41,b5,b51,ℝl23=ℝla3a31=VE5−a41,a51,b1,b11,ℝl24=ℝla4a41=VE5−a51,a11,b2,b21,ℝl25=ℝla5a51=VE5−a11,a21,b3,b31,ℝl26=ℝlb1b2=VE5−a2,a21,b4,b41,ℝl27=ℝlb2b3=VE5−a3,a31,b5,b51,ℝl28=ℝlb3b4=VE5−a4,a41,b1,b11,ℝl29=ℝlb4b1=VE5−a5,a51,b2,b21,ℝl30=ℝlb5b2=VE5−a1,a11,b3,b31,ℝl31=ℝlb1b11=VE5,ℝl32=ℝlb2b21=VE5,ℝl33=ℝlb3b31=VE5,ℝl34=ℝlb4b41=VE5,ℝl35=ℝlb5b51=VE5.
Since,ℝlai1bi=ℝlai+11bi=12 and ℝlai1bi≤ℝle∀e∈EE5, where 1≤i≤5. Furthermore, ∪i=14ℝai1bil=VE5 and ℝle∩∪i=14ℝlai1bi≥ℝlai1bi∀e∈EE5. Hence, we define an upper LRF h:VE5⟶0,1 as hv=1/12∀v∈VE5. In order to show that h is minimal upper LRF consider another function h′:VE5⟶0,1 as hv<1/12∀v∈VE5 therefore h′ℝle<1 and h′<h which shows that h′ is not LRF of Dp therefore LFLN E5≤∑i=1201/12=5/3. Likewise for 1≤i≤5 cardinality of LRN set ℝlbibi1 is 20 which is greater then the cardinalities of all other LRN sets. Hence there exist a maximal lower LRF g:VE5⟶0,1 and it is defined by gv=1/20∀v∈VE5, therefore LFLNE5>∑i=1201/20=1. Consequently,(21)1<LFLNE5≤53.Case 6.
Forp≥5, 1≤i≤p by Lemma 3, ℝlai1bi=2p+2 and ℝlai1bi≥ℝle∀e∈EEp. Furthermore, ℝle∩∪i=1pℝlai1bi≥ℝlai1bi. Therefore, we define an upper LRF h:Ep⟶0,1 as hv=1/2p+4∀v∈VEp. In order to show that h is minimal upper LRF consider another function h′:VEp⟶0,1 as h′v<1/2p+2∀v∈VEp therefore h′ℝle<1 and h′<h which shows that h′ is not LRF of Dp hence LFLNEp≤∑i=14p1/2p+2=2p/p+1. Likewise the cardinality of LRN set ℝlbibi1 is 4p which is greater then the cardinalities of all other LRN sets of Ep . Hence there exist a maximal LRF g:Ep⟶0,1 and it is defined as gv=1/4p∀v∈VEp therefore LFLNEp>∑i=14p1/4p=1. Consequently,(22)1<LFLNEp≤2pp+1.Lemma 4.
LetEp be a convex polytope network, with p≥4 and p≅0mod2. Then.(i)
ℝlai1bi=ℝlbiai+11=2p+1 and ∪i=1pℝlai1bi=VEp.(ii)
ℝlai1bi≤ℝle and ℝle∩∪i=1pℝlai1bi≥ℝlai1bi∀e∈EEp.Proof.
Considerai inner, ai1,bi middle and bi1 are outer vertices of Ep respectively, where 1≤i≤p and p+1≅1modp.(i)
ℝlai1bi=VEp−ai+1,ai+2,ai+3,….,ap+2i/2,ap+2i+2/21,ap+2i+4/21,ap+2i+6/21,….,ap+i−11,bp+2i/21,bp+2i+2/21,bp+2i+4/21,….,bp+i−11,bp+2i/2,bp+2i+2/2,bp+2i+4/2,….,bp+i−1 and ℝlbiai+11=VEp−ai,ap+2i+2/2,ap+2i+4/2,….,ap+i−1,ai+21,ai+31,…,ap+2i/21,bi+11,bi+21,…,bp+2i/21,bi+1,bi+2,…,bp+2i/2. Note that ∪i=1pℝlai1bi=4p and ℝlai1bi=2p+1.(ii)
ℝlaiai+1=VEp−bp+2i/2,bp+2i+1/21,bi1,bi, ℝlbi1bi+11=VEp−ap+2i/2,ap+2i/21, ℝlaiai1=VEp−ai+11,ai+21,…,ap1,bp+2i/2,bp+2i/21, ℝlbibi1=VEp, ℝlbibi+1=VEpai+1,ai+11. , The cardinalities of all the LRN sets are illustrated in Table 4.
Now it is clear thatℝlai1bi≤ℝle. Since ∪i=1pℝlai1bi=4p therefore ℝle∩∪i=1pℝlai1bi≥ℝlai1bi∀e∈EEp.Table 4
Cardinality of each LRN set ofEp.
LRN setComparisonℝlaiai+14p−4>ℝlai1biℝlbi1bi+114p−4>ℝlai1biℝlaiai13p>ℝlai1biℝlbibi14p>ℝlai1biℝlbibi+14p−2>ℝlai1biTheorem 7.
LetEp be a convex polytope network, where p≥4 and p≅0mod2. Then(23)1<LFLNEp≤4p2p+1.Proof.
In order to prove the result, we split it into two casesCase 7.
Forp=4, we have following LRN sets;(24)ℝl1=ℝla11b1=VE4−a2,a3,a41,b3,b4,b31,b41,ℝl2=ℝla21b2=VE4−a3,a4,a11,b4,b1,b41,b51,ℝl3=ℝla31b3=VE4−a4,a1,a21,b1,b2,b11,b11,ℝl4=ℝla41b4=VE4−a1,a2,a31,b2,b3,b21,b21,ℝl5=ℝla21b1=VE4−a1,a4,a31,b2,b3,b21,b31,ℝl6=ℝla31b2=VE4−a2,a1,a41,b3,b4,b31,b41,ℝl7=ℝla41b3=VE4−a3,a2,a11,b4,b1,b41,b11,ℝl8=ℝla11b4=VE4−a4,a3,a21,b1,b2,b11,b21,ℝl9=ℝla1a2=VE4−b31,b11,b1,b3,ℝl10=ℝla2a3=VE4−b41,b21,b2,b4,ℝl11=ℝla3a4=VE4−b11,b31,b3,b1,ℝl12=ℝla4a1=VE4−b21,b41,b4,b2,ℝl13=ℝla1a11=VE4−a21,a31,a41,b3,ℝl14=ℝla2a21=VE4−a31,a41,a11,b4,ℝl15=ℝla3a31=VE4−a41,a11,a21,b1,ℝl16=ℝla4a41=VE4−a11,a21,a31,b2,ℝl17=ℝlb11b21=VE4−a2,a4,a21,a41,ℝl18=ℝlb21b31=VE4−a3,a1,a31,a11,ℝl19=ℝlb31b41=VE4−a4,a2,a41,a21,ℝl20=ℝlb41b11=VE4−a1,a3,a11,a31,ℝl21=ℝlb1b2=VE4−a2,a4,a21,a41,ℝl22=ℝlb2b3=VE4−a3,a1,a31,a11,ℝl23=ℝlb3b4=VE4−a4,a2,a41,a21,ℝl24=ℝlb4b1=VE4−a1,a3,a11,a31,ℝl25=ℝlb1b11=VE4,ℝl26=ℝlb2b21=VE4,ℝl27=ℝlb3b31=VE4,ℝl28=ℝlb4b41=VE4.
Since,ℝlai1bi=ℝlai+11bi=9 and ℝlai1bi≤ℝle∀e∈EE4, where 1≤i≤4. Furthermore, ∪i=14ℝlai1bi=VE4 and ℝle∩∪i=14ℝlai1bi≥ℝlai1bi∀e∈EE4. Hence, we define an upper LRF h:VE4⟶0,1 as hv=1/9∀v∈VE4. In order to show that h is minimal upper LRF consider another function h′:VE4⟶0,1 as hv<1/9∀v∈VE4 therefore h′ℝle<1 and h′<h which shows that h′ is not LRF of E4 therefore LFLNE4≤∑i=1161/9=16/9. Likewise for 1≤i≤3 cardinality of LRN set ℝlbibi1 is 20 which is greater then the cardinalities of all other LRN sets. Hence there exist a maximal lower LRF g:VE4⟶0,1 is defined by gv=1/20∀v∈VE4, therefore LFLNE4>∑i=1201/20=1. Consequently,(25)1<LFLNE4≤169.Case 8.
Forp≥4, 1≤i≤p by Lemma 4, ℝlai1bi=2p+1 and ℝlai1bi≥ℝle∀e∈EEp. Furthermore, ℝle∩∪i=1pℝlai1bi≥ℝlai1bi. Hence, we define an upper LRF h:Ep⟶0,1 as hv=1/2p+1∀v∈VEp. In order to show that h is minimal upper LRF consider another function h′:VDp⟶0,1 as hv<1/2p+1∀v∈VEp this implies h′ℝle<1 and h′<h which shows that h′ is not LRF of Ep therefore LFLNEp≤∑i=14p1/2p+1=4p/2p+1. Likewise the cardinality of LRN set ℝlbibi1 is 4p which is greater then the cardinalities of all the other LRN sets. Therefore there exist a maximal lower LRF g:Ep⟶0,1 and it is defined as gv=1/4p∀v∈VEp therefore LFLNEp>∑i=14p1/4p=1. Consequently,(26)1<LFLNEp≤4p2p+1.


## 4. Conclusion

In this dissertation, we studied the LFLN of different families of convex polytope networks (Dp, Ep) and after establishing the bounds of LFLN of both convex polytope networks, we conclude that both of them posses boundedness when p⟶∞.Exact value of LFLN in one case is,(i)
•LFLND5=5/4.(ii)
Boundedness of LFLN ofDp and Ep illustrated in Table 5.Table 5
Boundedness of convex polytopes (Dp, Ep) via LFLN.
NetworkLFLNLower bound of LFLN whenp⟶∞Upper bound of LFLN whenp⟶∞CommentDpp/p−1≤LFLNDp≤8p/5p+718/5BoundedDpp/p−1≤LFLNDp≤8/518/5BoundedEp1<LFLNEp≤2p/p+112BoundedEp1<LFLNEp≤4p/2p+112BoundedNow, we close our discussion with the following open problem, characterize all the classes of convex polytopes networks those attain exact value of local fractional locating number.



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