Local Fractional Locating Number of Convex Polytope Networks

e 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.


Introduction
Slater [1] introduced the methodology to compute the locating set of a connected network. He de ned 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 di erent term called as metric dimension. ey also brie y 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 di erent 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]. ere 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 speci c conditions [4] and Currie and Olllermann used the idea of fractional locating number (FLN) to nd solution of speci c IPP as well [27]. e 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 speci c conditions [28]. Later on Arguman et al. characterized all those networks have FLN exactly |V(G)|/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] [35,36]. For the study of LFLN of generalized gear, sunlet and convex polytope networks see [37][38][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. e manuscript is organised as Section 2 contains preliminaries and Sections 3 and 4 have main results and conclusion respectively.

Preliminaries
A network G is an order pair (V(G), E(G))), where V(G) is the vertex set and E(G) is the edge set. A walk is a nite 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. e distance between two vertices a and b is donated by d(a, b) is de ned as the length of the shortest path between a and b. For further preliminary results of networking theory see [40]. w 3 ), . . . , d(x, w p )). If distinct vertices of G have unique representation with respect to W then W is known as locating/resolving set. e minimum cardinality of W is called locating number (LN) of G that is de ned as LN(G) min |W|: W is the resolving set of G . (1) For an edge ab ∈ E(G) the local resolving neighbourhood set (LRN) is the collection of all vertices of G which resolve an edge ab and it is donated by

Main Results
is section is devoted to the main results in which, we have examined the LFLN of cretin family convex polytope networks D p and E p and it has been proved that these polytope networks remain bounded under LFLN when their order approaches to in nity.  [10]. e vertex set V(D) p consists of inner ( a i :  order and size of D p are 4p and 6p respectively and for complete details see Figure 1.
Proof. Consider a i inner, a 1 i , b i middle and b 1 i are outer vertices of D p , where 1 ≤ i ≤ p and p + 1 � 1(modp). Table 1.
From Table 1, we note that |R l (a 1 □ Theorem 1. Let D 3 be a convex polytope network. en Proof. e LRN sets of convex polytope network D 3 are:

RLN set Comparison
Mathematical Problems in Engineering 3 erefore, we define an upper LRF h: is 10 which is greater then the cardinalities of all other LRN sets of D 3 . erefore, we define a maximal lower LRF g: Hence Proof. e LRN sets are given by: It is clear that the cardinality of each RLN set of D 5 is 16. erefore, we define a constant function h: Proof. To prove the result, we split it into two cases □ Case 1. For p � 7 , we have following LRN sets; Mathematical Problems in Engineering 5 7 . In order to show that h is minimal upper LRF consider another mapping h ′ : 7 therefore h(R l (e)) < 1 and is 24 which is greater then the cardinalities of all other RLN sets. erefore, we define a lower LRF g: 7 is a maximal lower LRF hence LFLN(D) 7 ≥ 28 i�1 1/24 � 7/6. Consequently, is defined as h(v) � 2/5p + 7∀v ∈ V(D p ). In order to show that h is minimal upper LRF consider another function h ′ : V(D p ) ⟶ [0, 1] as h(v) < 2/5p + 7∀v ∈ V(D) p therefore h ′ (R l (e)) < 1 and |h| ′ < |h| which shows that h ′ is not Likewise the cardinality of RLN set R l (a i a i+1 ) is 4p − 4 which is greater then the cardinalities of all other LRN sets. erefore, we define a maximal LRF g: Consequently, Lemma 2. Suppose that D p is a convex polytope network, with p ≥ 6 and p � 0(mod2). en.
It is can be observed with the help of Table 2 □ Theorem 4. Let D p be a convex polytope network, with p ≥ 4 and p � 0(mod2). en Proof. In order to prove the result, we split into two cases: □ Case 3. For p � 4, we have following LRN sets; Mathematical Problems in Engineering Since, 4 . In order to show that h is minimal upper LRF consider another function h ′ : Hence there exist an upper LRF h: V(D p ) ⟶ [0, 1] and is defined as h(v) � 2/5p∀v ∈ V(D p ). In order to show that h is minimal upper LRF consider another function h ′ : ) is 4p − 4 which is greater or equal to the cardinalities of all other LRN sets of D p . Hence, we define a maximal lower LRF g:

LFLN of Convex Polytpoe E p .
In this particular subsection, we have computed the LRN sets and LFLN of convex e order and size of E p is 4p and 7p respectively. For more details see Figure 2.
Proof. Consider a i inner, a 1 i , b i middle and b 1 i are outer vertices of E p , where 1 ≤ i ≤ p and p + 1 � 1(modp).
Now, we illustrate the cardinalities of the LRN sets in Table 3 and also compare them.
It can be observed with the help of Table 3 that □ Theorem 5. Let E 3 be a convex polytope network. en Proof. e LRN for convex polytope E 3 are For 1 ≤ i ≤ 3 the cardinality of each LRN set R l (a i a i+1 ) is 8 which is less then the other LRN sets of E 3 . Furthermore, Hence there exist an upper LRF h: . In order to show that h is minimal upper LRF consider another function is 12 which is greater then the cardinalities of all other LRN sets. Hence there exist a maximal lower LRF g: □ Theorem 6. Let E p be a convex polytope network, with p ≥ 5 and p 1(mod2). en Proof. In order to prove the result, we split it into two cases □ Case 5. For p 5, we have the following LRN sets;