Abstract
Distance-based numeric parameters play a pivotal role in studying the structural aspects of networks which include connectivity, accessibility, centrality, clustering modularity, complexity, vulnerability, and robustness. Several tools like these also help to resolve the issues faced by the different branches of computer science and chemistry, namely, navigation, image processing, biometry, drug discovery, and similarities in chemical compounds. For this purpose, in this article, we are considering a family of networks that exhibits rotationally symmetric behaviour known as circular ladders consisting of triangular, quadrangular, and pentagonal faced ladders. We evaluate their upper bounds of fractional metric dimensions of the aforementioned networks.
1. Introduction
A network comprises of two collections of distinct objects, which are set containing nodes/vertices and set of objects that forms connection among those nodes/vertices , where . The cardinality of constitutes the order of denoted by , and size of is the cardinality of denoted by . For , represents the distance of shortest path between and . In order to have an in depth study of graph theory’s strata, please refer to [1–3].
Day by day technological advancements are restructuring our lives by replacing manpower with robots, devices, and machinery. In the same instance, the necessity of restructuring is also making it inevitable for us to utilize these machineries which are keeping in view cost effectiveness and serving the large amount of ques in production line, health services, and public dealing areas with minimum number of them. Here, distance-based parameters come to fulfill the aforementioned objectives. Consider , then becomes an ordered set of vertices whenever some ordering is being imposed on them. The -tuple metric form of with respect to is represented by , where for shows the distance of from . changes into resolving set if for any pair of distinct vertices such that . The resolving set having minimum number of vertices in presents the metric dimension (MD) of denoted by [4, 5].
For any connected network, Slater [6, 7] laid the foundation of resolving set by calling it as locating set. After studying these terminologies at their own, Harary and Melter named it as the MD of a network. Since then, a number of researchers considered different families of networks for the computation of MD. The results regarding the families of networks with constant and bounded MD can be found in [8]. Chartrand et al. proved the results regarding the metric dimensions of path and cycle in [4]. Similarly, , and as networks bearing constant, MD has been shown in [8]. Moreover, in [9], the MD of the generalized Petersen network is proved to be bounded. Also the MD of the Jahangir network and infinite regular networks has been studied in [10, 11], respectively.
Chartrand et al. [4] employed MD to acquire the solution of integer programming problem (IPP). Later on, Currie and Oellermann [12] established the concept of fractional metric dimension (FMD) and acquired the higher accuracy solution. Fehr et al. [13] utilized FMD to obtain the optimal solution of a certain linear programming relaxation problem. Arumugam and Mathew [14] uncovered some peculiar features of FMD. Afterwards, a bunch of results appeared related to the FMD of several networks that are the resultants of graph products, namely, Cartesian, hierarchial, corona, and lexicographic comb products (see [14–18]). Furthermore, the FMD of the generalized Jahangir network and permutation network can be found in [19, 20], respectively. In a recent time, Raza et al. [21] evaluated the FMD of metal organic networks.
Aisyah et al. defined the notion of local fractional metric dimension (LFMD) and calculated for corona product of two networks [22]. Similarly, Liu et al. [23] calculated the LFMD of rotationally symmetric and planar networks. More recently, Javaid et al. evaluated the sharp extremal values of LFMD of connected networks [24]. Results regarding the LFMD of cycle related networks can be found in [25]. In this article, we have considered a family of network bearing rotational symmetry known by circular ladders consisting of different triangular, quadrangular, and pentagonal faces. This article propels in the following manners: Section 1 is introduction, Section 2 covers the role of MD in Robotics and Chemistry, and Section 3 is of preliminaries. In Section 4, we will discuss the idea behind the creation of the circular ladders under consideration. Section 5, contains the main results, and Section 6 closes this article with some conclusions and future directions.
2. Applications
2.1. Robotics
The ever increasing demand of networking enlarged the study of distance-based dimensions. All such tools are used to allocate an appropriate region to an interpolar for its effective usage [6, 26]. The placement of robots at different sites such as restaurants, production units, and public health facilities come under study by using one such tool in [27]. The effective rectification of example and picture handling and handling of informational structures using the aforementioned parameters has been given in [28]. The parameter like these has not only made it possible to serve restaurants, factories, and public service areas with a minimum number of robots but also truncated the delay in their serving. To find more applications on this topic, we refer to [29, 30].
2.2. Chemistry
A chemical compound in graph theoretic form is regarded as a molecular graph having nodes as atoms and links between them as bonds between them [4]. In this picturesque form of a compound along with distance-based parameters give chemists an opportunity to remove discrepancies in a chemical structure and to find the sites showing similar properties in them. All such strategies can be found in [5, 6, 28, 31].
3. Preliminaries
A vertex is said to resolve a pair in if . For , then the resolving neighbourhood (RN) of the pair is given by .
Assume a connected network bearing as its order. A function is called resolving function (RF) of if each RN set , . An RF of is known as minimal RF if any function such that and for at least one , which is not an RF of . The FMD of the network is given by , where [14].
4. Construction of Networks
Here, we give the construction of networks covered in this article. A cycle is a network bearing the regularity of 2, 3 vertices, and 2 faces known as triangle. On the other hand, is a network having the regularity of 2 with 4 vertices and 2 faces. A 3-faced quadrangle is obtained after joining any pair of nonadjacent vertices of it. Figure 1(a) shows the planar quadrangles having 2 and 3 faces. A network bearing the regularity of 2, 5 nodes, and 2 faces is known as pentagon. The 3-faced pentagons are formed by joining any two nonadjacent nodes of 2-faced pentagon, and 4-faced pentagons are formed by forming links between any two pairs of nonadjacent nodes of 3-faced pentagon. Figure 1(b) illustrates some of the possible 2-faced, 3-faced, and 4-faced planar pentagons.
(a)
(b)
4.1. Triangular Circular Ladder
The triangular circular ladder (TCL) is created by joining each edge of a cycle with the corresponding nodes , where . Thus, each step of TCL consists of 2-faced triangle and its vertex and edge sets are as follows: , with and respectively. Figure 2 shows TCL.
4.2. Quadrangular Circular Ladders
For , the quadrangular circular ladder (QCL) (prism) is a cubic network which is the resultant of the Cartesian product , with and . Its sets of vertices and edges are, respectively, given by the following: , .
The two possible QCLs and having each step of 3-faced pentagons are formed by forming edges either or in quadrangular circular ladder . The sets of vertices and edges of these networks are, respectively, given as follows: and and and with and .
Figure 3 illustrates , , and . It can be observed from Figure 3 that .
(a)
(b)
(c)
4.3. Pentagonal Circular Ladders
The each step of a pentagonal circular ladder (PCL) consists of 2-faced pentagon, and it is created by inserting a new vertex between the vertices and of , where and are given by and with and , respectively.
Figure 4 illustrates .
The three possible PCLs , each step of which is having 3-faced pentagons are formed by creating edges either , or (shown by magenta colour in Figure 5) in pentagonal circular ladder . Figure 5 illustrates , and .
(a)
(b)
(c)
As seen from Figure 5, . The sets and are given as follows: and , and , and with and .
The three possible PCLs and whose each step is having 4-faced pentagons are created by forming edge (shown by cyan colour in Figure 6) in , edge in , edges in , or edges and in the plane network . Figure 6(a) illustrates , 6(b) illustrates , and 6(c) illustrates , respectively.
(a)
(b)
(c)
It can be seen from Figure 6 that .
The sets of vertices and edges of these networks are , , , , with and , respectively.
5. Main Results
5.1. Triangular Circular Ladder
Lemma 1. Let be TCL with and . For , , and , we have(a) and , where .(b) for , where and with .
Proof. (a)The RNs of and are and . We note that , and .(b)The RNs of for are , , , , and , respectively.Clearly, . Since , then .
Theorem 1. If with and . Then, .
Proof. Case 1. When . The RNs are given as follows. We have seen from the above that Table 1 shows the RNs with maximum cardinality that is 10 and Table 2 shows the RNs with minimum cardinality that is 7, which is less than the cardinalities of all other RNs for that is 10. Moreover, which implies and , where . Now, we define a function such that . As we can see, for of are pairwise overlapping; hence, another minimal resolving function of such that . As a result, . Case 2. . According to Lemma 1 the RNs with minimum cardinality are and having the cardinality of and , where . Let and . Now, we define a function such that . We can see that is a resolving function for because . Suppose that there is another resolving function , such that , for at least one . As a result, , where is an RN of having the minimum cardinality of . This shows that is not a resolving function which is contradiction. Hence, is a minimal resolving function that achieves minimum for . As a matter of fact for of are pairwise overlapping; hence, another minimal resolving function of such that . As a result, we arrive at the following: .
5.2. Quadrangular Circular Ladder
Lemma 2. Let be the 2-faced QCL with and . For , , and . Then,(a) and .(b)For , , and , where .(c)For , , and , where .
Proof. (a)The RNs of and are given by and , respectively. We note that , and .(b)The RNs of for are as follows: , and respectively. Clearly, . Since , then , it can be easily seen that and .(c)The RNs of , , and are . We can see that and .
Theorem 2. If with and . Then, .
Proof. Case 1. When . The RNs are given as follows. We can see that Tables 3 and 4 show the RNs with maximum cardinalities, that is, 12 and 8, respectively, whereas, Table 5 shows the RNs with minimum cardinality, that is, 6, and which implies and , where . Here, we define a function such that . Also for of are pairwise overlapping; hence, another minimal resolving function of such that . As a result, . Case 2. When . We have seen from Lemma 2 that the RNs with minimum cardinality of are and , respectively. Let and . Then, we define a function such that We see that is a resolving function for because . Now, suppose on contrary that there is another resolving function , such that , for at least one . As a result, , where is an RN of having minimum cardinality . This shows that is not a resolving function which is a contradiction to our supposition. Therefore, is a minimal resolving function that achieves minimum for . Since all the RNs of are equal, hence another minimal resolving function of such that . Assigning to all the vertices of and calculating their sum, we get: .
Lemma 3. Let be the 3-faced QCL with and . For , , and . Then(a) with , , and .(b) and with and .
Proof. (a)The RNs of and are and respectively. We note that and .(b)The RNs of and are given by: , , , , , , , . Clearly, . Since then .
Theorem 3. Let with and . Then, .
Proof. Case 1. When . The RNs are given as follows. Tables 6 and 7 given above shows the RNs with cardinalities of 10 and 6 respectively. Clearly, the table shows the RNs bearing the minimum cardinality of 6 and which implies and , where . Now, we define a function such that . Moreover, for of are pairwise overlapping; thus, is another minimal resolving function of such that . As a result, . Case 2. It is clear from Lemma 3 that the RNs bearing the minimum cardinality of in are and respectively, with . Let and . Then we define a mapping such that We see that is a resolving function for because . Now suppose on contrary that there is another resolving function , such that , for at least one . As a result, , where is a RN of having minimum cardinality . This shows that is not a resolving function which is a contradiction to our supposition. Therefore, is a minimal resolving function that achieves minimum for . Since all the RNs of are equal, hence another minimal resolving function of such that \enleadertwodots Assigning to all the vertices of and calculating their sum, we get .
Corollary 1. Let with and . Then, .
Proof. The result is evident as .
5.3. Pentagonal Circular Ladders
Lemma 4. Let be the 2-faced QCL with and . For , , and . Then,(a) with , and .(b) with and .(c) and with .(d) and .(e) and .
Proof. (a)The RNs of and are and , respectively. We can see that and and .(b)The RNs of and are and mod }, respectively. Since for , thus .(c)The RNs of and are , , and , respectively. Clearly, for . Since , then .(d)The RN of is . , , . Clearly, for and .(e)The RNs of and are and , respectively. We can see that and .
Theorem 4. If with and . Then, .
Proof. Case 1. (i) When . The RNs are given as follows. We can see that Tables 8 and 9 show the RNs with maximum cardinality of 16 and 15, respectively, whereas both Tables 10 and 11 show the RNs with minimum cardinality of 14. Moreover, which implies and . Now, we define a mapping such that . It is observed that for of are pairwise overlapping; hence, another minimal resolving function of such that . As a result, .(ii)When . The RNs are given as follows. We can see that Tables 12–14 shows the RNs with maximum cardinalities that is 22 and 20, respectively, whereas Table 15 shows the RNs with minimum cardinality of 19. Moreover, which implies and . Now, we define a mapping such that . It is observed that for of are pairwise overlapping; hence, another minimal resolving function of such that . As a result, .(iii)For . We have seen that whereas . In generic form, we can write it as . Case 2. When . The RNs are given as follows. We can see that Tables 16–18 show the RNs with maximum cardinalities, that is, 28 and 26 and 25, respectively, whereas Table 19 shows the RNs with minimum cardinality of 23. Moreover, which implies and . Now, we define a mapping such that . It is observed that for of are pairwise overlapping; hence, another minimal resolving function of such that . As a result, . Case 3. . From Lemma 4, we can see that the RNs with minimum cardinality of are and , respectively. Moreover, for . Let and . Then, we define a mapping such that . We note that is a resolving function for because . Now, assume that there is another resolving function , such that , for at least one . As a result, , where is a RN of bearing minimum cardinality . This shows that is not a resolving function a contradiction. Therefore, is a minimal resolving function for that achieves minimum . As a matter of fact for of are pairwise overlapping; hence, another minimal resolving function of such that . Therefore, assigning to all the vertices of and calculating their sum, we get: .
Lemma 5. Let be the 3-faced QCL with and . For , , and . Then,(a), , and .(b) and .(c) and with and .(d) and .(e) and .
Proof. (a)The RNs of and are and . We note that and .(b)The RNs of and are and respectively. It is clear from the above that and .(c)The RNs of and are , , , , , , Clearly, . Since , then .(d)The RNs of and are and Clearly, . Since , then .(e)The RN of is . Clearly, and .
Theorem 5. If with and . Then, .
Proof. Case 1. When . The RNs are given as follows. Tables 20–22 show the RNs with maximum cardinality of 18, 16, and 14, respectively. On the other hand, Table 23 bears RNs with the minimum cardinality of 10. Also, it is observed that which implies and . Now, we define a function such that . As for of are pairwise overlapping; hence, another minimal resolving function of such that . As a result, . Case 2. . As Lemma 5 clears the fact that among the RNs of , those that have the minimum cardinality of are and , respectively. Also, , where . Let and . Then, we define a function such that . It is seen that is a resolving function for because . Now, suppose on contrary that there is another resolving function , such that , for at least one . As a result, , where is an RN of having minimum cardinality . This shows that is not a resolving function, a contradiction. Thus, is a minimal resolving function for which attains minimum . Also, for of are pairwise overlapping; hence, another minimal resolving function of such that . Therefore, assigning to all the vertices of and calculating their sum, we get .
Lemma 6. Let be the 3-faced QCL with and . For , , and . Then,(a), and .(b) and with and .(c).(d).(e) and .(f) and .(g) and .
Proof. (a)The RN of is . We note that and .(b)The RNs of , , and are , , , , , , , , , Clearly, . Since , then .(c)The RN of , , , and are , , , , Clearly, . Since , then .(d)The RNs of , , , and are , , , , Clearly, . Since , then .(e)The RNs of and are . Clearly, and .(f)The RN of is given by . Clearly, and .(g)The RN of is given by . Clearly, and .
Theorem 6. If with and . Then, .
Proof. Case 1. When The RNs are given as follows. We note that the cardinalities of RNs as shown in Tables 24–26 are 16, 14, and 13, respectively. Similarly, the cardinality of RNs given in Table 27 is 11 which is less than the least among all the RNs. Moreover, which implies and . Now, we define a function such that . Furthermore, for of are pairwise overlapping; hence, another minimal resolving function of such that . As a result, . Case 2. We have seen from Lemma 6 for all and. Let and . Then, we define a function such that . We see that is a resolving function for because . Now, suppose on contrary that there is another resolving function , such that , for at least one . As a result, , where is an RN of having the minimum cardinality of . This shows that is not a resolving function, a contradiction. Thus, is a minimal resolving function for that achieves minimum . Moreover, for of are pairwise overlapping; hence, another minimal resolving function of such that . Therefore, assigning to all the vertices of and calculating their sum, we get .
Corollary 2. If with and . Then, .
Proof. The above assertion is valid as .
Lemma 7. Let be the 4-faced QCL with and . For , , and . Then,(a) with , , and ,(b) and .(c) with and .(d) and .(e) and .(f) and .
Proof. (a)The RNs of and are and , respectively. We note that and .(b)The RNs of and are , and , respectively. We can see that . Since , then , where .(c)The RNs of and are , for , , , Clearly, . Since , then .(d)The RNs of and are , , and , respectively. We can see from these sets that and .(e)The RNs of and are , , , , respectively. We note that and .(f)The RN of is . Clearly, and .
Theorem 7. If with and . Then, .
Proof. Case 1. When . The RNs are given as follows. In Tables 28–31, the cardinalities of RNs are 16, 15, 14, and 13, respectively. In the same manner, Table 32 is having RNs withholding the cardinality of 10. Moreover, which implies and . Now, we define a function such that . We have seen that for of are pairwise overlapping; hence, another minimal resolving function of such that . As a result, . Case 2. . It can be seen from Lemma 7 that among all the RNs of , for all and. Let and . Then, we define a function such that . It is found that is a resolving function for because . On contrary, assume that there is another resolving function , such that , for at least one . As a result, , where is an RN of having the minimum cardinality of . This shows that is not a resolving function, a contradiction. Thus, is a minimal resolving function for that achieves minimum . Moreover, for of are pairwise overlapping; hence, another minimal resolving function of such that . Therefore, assigning to all the vertices of and calculating their sum, we get .
Corollary 3. If with and . Then, .
Proof. This can be derived easily as .
Lemma 8. Let be the 4-faced QCL with and . For , , and . Then,(a), , and .(b) with and .(c) and .(d) and .(e) and .(f) and .(g) and .(h) and .
Proof. (a)The RNs of and are and , respectively. Clearly, for . We note that and .(b)The RNs of , and are , , and , respectively. We can see that and .(c)The RNs of and are , , and where , respectively. Clearly, and .(d)The RNs of and are and , respectively. Clearly, and .(e)The RN of is . Clearly, and .(f)The RN of is . Clearly, and .(g)The RN of is . Clearly, and .(h)The RN of is . Clearly, and .
Theorem 8. If with and . Then, .
Proof. Case 1. When . The RNs are given as follows. Tables 33–37 show the RNs of with cardinalities 16, 15, 14, 13, and 10, respectively. Moreover, in Table 37, which implies and = , where . Now, we define a function such that . Furthermore, for are pairwise overlapping; hence, another minimal resolving function of such that . As a result, . Case 2. . As Lemma 8 clears the fact that for all and. Let and . Then, we define a function such that . We find that is a resolving function for because . Assume on contrary that there is another resolving function , such that , for at least one . As a result, , where is an RN of having the minimum cardinality of . This shows that is not a resolving function, a contradiction. Thus, is a minimal resolving function for that achieves minimum . Also for of are pairwise overlapping; hence, another minimal resolving function of such that . Therefore, assigning to all the vertices of and calculating their sum, we get .
6. Conclusion
In this article,(i)we have found the upper bounds of FMD of symmetric networks called by TCL (), QCLs (), and PCLs (, and ).(ii)Table 38 shows the summary of main results, and Table 39 gives the values of FMDs as they tend to .(iii)the networks having the maximum FMD of 2 are and .(iv)in contrast, is the network bearing the minimum FMD of .(v)the obtained results are the generalization of [23].(vi)to find the extremal values of FMD of asymmetric networks is still an open problem.(vii)moreover, the evaluation of the FMD and LFMD of connected networks such as convex polytopes is an open problem as well.
Data Availability
The data used to support the findings of this study are included within this paper and are available from the corresponding author upon request.
Conflicts of Interest
The authors have no conflicts of interest.