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.
1. Introduction
A network is a mathematical structure consisting of two set of vertices and set of edges , where and are called order and size of , respectively. In , the length of the shortest path between any two vertices is called distance which is usually denoted by . 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). Consider ; then, becomes an ordered set of vertices bearing some ordering imposed by us. For any , the distance of from all the elements of in -tuple metric form is given by . The set becomes a resolving set if, for any pair of distinct vertices , we have . The resolving set bearing minimum number of vertices in forms the metric basis of , and its cardinality represents the MD of denoted by [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, , , and 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 network , for , a vertex resolves in if . For any pair of vertices , the resolving neighbourhood set (RNs) of is given by .
Suppose that, in a connected network, , bearing as its order. A function is called the upper resolving function (URF) of if in , where .
A URF is called the minimal upper resolving function (MURF) if is another URF such that and for at least one . Similarly, a function defined as is called lower resolving function (LRF) if in . An LRF is called the maximal lower resolving function (MLRF) if is another LRF such that and for at least one . The FMD of a connected network is defined as , where or .
3.1. Convex Polytopes
The network of convex polytope type I bearing 3-sided faces, 4-sided faces, 5-sided faces, and a pair of -sided faces is obtained by the combination of the network of convex polytope and prism network [28]. The sets and are and , respectively. In the same manner, the cycles induced by , , , , and are called by inner, interior, exterior, and outer cycle, respectively. Figure 1 illustrates .
The network of convex polytope type II is created out of the network of by adding new edges . It comprises of 3 3-sided faces, 4-sided faces, 5-sided faces, and a pair of -sided faces. Similarly, the cycles induced by , , , , and are called by inner, interior, exterior, and outer cycle, respectively. Figure 2 illustrates .
The sets and are given by and , 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 that is a connected network and is the resolving neighbourhood set for any . For , if , then for each resolving neighbourhood set of .
Lemma 1. Let be a connected network and be the resolving neighbourhood set. Then,where and .
Proof. Here, we define a mapping such that for . By Proposition 1, for , we haveThis shows the fact that is a lower resolving function (LRF). In order to show that is maximal LRF, suppose that there exists another LRF such that , where for at least one . For all such that , we haveHence, . This shows that is not LRF, and consequently, is maximal LRF. Let be another maximal LRF of . Then,Now, we assume the following three cases: (a) , (b) , and (c) for some . Case (a): if . For such that , we have . This shows that is not LRF. Hence, this case does not hold. Case (b): suppose that . Then, Consequently, Case (c): if for some , suppose that and . We observe that ; otherwise, for , which implies that is not a LRF. Consider As , hence, Consequently, Therefore, from all the above case, we have 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. Lemmas 2 and 3 concern with RNs of and , respectively. Similarly, Theorems 1 and 2 give the upper and lower bounds of FMD of the aforementioned networks.
Lemma 2. Let be a type I convex polytope, with and . For , , and , and ; then,(a), , , and (b) and (c) and with (d), where and (e), where (f) and (g) and (h) and (i) and
Proof. (a)The RNs of and are and . We note that , and , and ,(b)The RNs of and are and , respectively. It is clear from the above that and .(c)The RNs of , and are , , , respectively. Clearly, . Since , then ,(d)The RN’s of , and are given by , , , where . We can see that and .(e)The RNs of , , , and are , , , and , and , respectively. We can see that and .(g)The resolving neighbourhood of is . We can see that and .(h)The resolving neighbourhoods of and are , , , , , , , and , respectively. We can see that and .(i)The RN of is . Clearly, and .
Theorem 1. If with and , then .
Proof. Case I. The RNs are given as follows. In the same way, from Lemma 2, we can see that . Similarly, from Lemma 2, we can see that . Tables 1 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 ; this implies and . Now, we define a function such that . As for of are pairwise overlapping, hence, is another minimal resolving function of such that . As a result, . Similarly, Table 7 shows the RNs with maximum cardinality of ; hence, by Lemma 1, . Therefore,Case II. We have seen from Lemma 2 that the RNs with minimum cardinality of are and and . Let and . Now, we define a mapping such that We can see that is a RF for with because . On the contrary, assume that there is another resolving function , such that , for at least one . As a consequence, , where is a RN of with minimum cardinality . This implies that is not a resolving function which is contradiction. Therefore, is a minimal resolving function that attains minimum for . Since all are having pairwise nonempty intersection, so there is another minimal resolving function of of such that . Hence, assigning to the vertices of in and calculating the summation of all the weights, we obtain Also, the RN with maximum cardinality of is . Let and ; thus, from Lemma 2, we have . Therefore, we conclude the following:
Lemma 3. Let be a type II convex polytope, where and . For , , , , and , then(a), , , and (b) and (c) and ,(d) and , where (e) and with , where (f) and , where (g) and , where (h) and , where (i) and (j) and (k) and (l) and (m) and (n) and
Proof. (a)The RNs of and are and , respectively. We note that and . Also, .(b)The RNs of and are , , and , respectively. Clearly, . Since , then , .(c)The RNs of and are , ,, , and . Clearly, . Since , then .(d)The RN’s of and are , , , , and , respectively. We can see that, for , and ,(e)The RNs of , and are , , , , , , and , respectively. We can see that, for , and .(f)The RNs of and are , , and , respectively. Clearly, for and .(g)The RNs of and is and , respectively. Clearly, and .(h)The RNs of and are and , respectively. Clearly, and .(i)The RN of is . Clearly, and .(j)The RN of is as we can see that and .(k)The RNs of and are and , as we can see that and .(l)The RN of is . Clearly, and .(m)The RNs of and are and , respectively. Clearly, and .(n)The RNs of and are , as both are equal to ; therefore, .
Theorem 2. If with and , then .
Proof. Case I. The RNs are given as follows. From Lemma 3, we see that . In Tables 8–14, 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 this implies and , where . Now, we define a function such that , as for of are pairwise overlapping; hence, is another minimal resolving function of such that . As a result, . On the contrary, Table 4 shows the RNs with maximum cardinality of ; hence, by Lemma 1, . Therefore,Case II. We have seen from Lemma 3 that the RNs with minimum cardinality of are and and . Let and . Now, we define a mapping such that We can see that is a RF for with because . On the contrary, assume that there is another resolving function , such that , for at least one . As a consequence, , where is a RN of with minimum cardinality . This implies that is not a resolving function which is contradiction. Therefore, is a minimal resolving function that attains minimum for . Since all are having pairwise nonempty intersection, so there is another minimal resolving function of of such that . Hence, assigning to the vertices of in and calculating the summation of all the weights, we obtain Also, the RNs with maximum cardinality of are and . Let and ; thus, from Lemma 1, we have . Therefore, we conclude the following:
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 II and (iii)Table 16 shows the summary of main results and Table 17 gives the values of FMDs as they tend to
6.1. Open Problem
To characterize the networks with lower bound of FMD greater than 1 is still an open problem.
Data Availability
The data used to support the findings of the study are included within the article. However, more details of the data can be obtained from the corresponding author upon request.
Conflicts of Interest
The authors have no conflicts of interest.