Computing Bounds of Fractional Metric Dimension of Metal Organic Graphs

Raza, Mohsin; Alrowaili, Dalal Awadh; Javaid, Muhammad; Shabbir, Khurram

doi:https://doi.org/10.1155/2021/5539569

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Volume 2021 | Article ID 5539569 | https://doi.org/10.1155/2021/5539569

Computing Bounds of Fractional Metric Dimension of Metal Organic Graphs

Mohsin Raza,¹Dalal Awadh Alrowaili,²Muhammad Javaid,¹and Khurram Shabbir³

Academic Editor: Kashif Ali

Received14 Jan 2021

Revised10 Feb 2021

Accepted27 Feb 2021

Published12 Mar 2021

Abstract

Metal organic graphs are hollow structures of metal atoms that are connected by ligands, where metal atoms are represented by the vertices and ligands are referred as edges. A vertex resolves the vertices and of a graph if . For a pair of vertices of , is called its resolving neighbourhood set. For each pair of vertices and in , if , then from to the interval is called resolving function. Moreover, for two functions and , is called minimal if and for at least one . The fractional metric dimension (FMD) of is denoted by and defined as , where . If we take a pair of vertices of as an edge of , then it becomes local fractional metric dimension (LFMD) . In this paper, local fractional and fractional metric dimensions of are computed for in the terms of upper bounds. Moreover, it is obtained that metal organic is one of the graphs that has the same local and fractional metric dimension.

1. Introduction

For a connected graph , a vertex is said to resolve a pair of vertices of if . A set is called a resolving set of if each pair of vertices of is resolved by some vertex in S. The metric dimension of is denoted by and is defined as

For a pair of vertices of , the resolving neighborhood is defined as . A resolving function is a real-valued function such that for each distinct pair of vertices of , where . A resolving function is called minimal if any function such that and for at least one is not a resolving function of . The fractional metric dimension (FMD) of is denoted by and defined aswhere . Now, if we take a pair of vertices of as an edge of , then the aforesaid defined resolving neighborhood , minimal resolving function , and FMD become local resolving neighborhood , local minimal resolving function, and local fractional metric dimension , respectively.

First of all, Harary and Melter [1] defined the concept of metric dimension to study the substructures of chemical compounds having similar properties which are used in pharmaceutical industries for the drug discoveries. Later on, Chartrand et al. [2] and Currie & Oellermann [3, 4] improved the solution of IPP with the help of the procedure of metric dimension. Moreover, it is used in navigation system, image processing, and robotic problems [5]. For various results of metric dimension on different graphs, refer to [6–9].

Fehr et al. [10] introduced the concept of fractional metric dimension (FMD), and they obtained the optimal solution of a certain linear programming relaxation problem with the help of FMD. Arumugam and Mathew [11] present various properties of FMD. The FMD of metal organic framework (MOF) is computed in [12], where MOF is obtained from the cycle of odd order. Moreover, different classes of graphs such as product-based graphs and Hamming, Johnson, and permutation graphs are studied with the help of FMD [13–17]. Liu et al. [18] computed the FMD of generalized Jahangir graph. Recently, Aisyah et al. defined the concept of local fractional metric dimension (LFMD) and computed it for the corona product of graphs [19]. Liu et al. [20] computed the LFMD of rotationally symmetric networks. Javaid et al. [21] calculated the sharp bounds of LFMD of connected networks.

Metal organic graph consists of metal atoms, where atoms are linked with thes help of organic ligands which act like a linker. Therefore, has led to a new world of remarkable applications and it has a large surface area that allows these chemicals compounds to absorb huge quantity of several gases such as carbon dioxide hydrogen and methane acting as a gas storage chemical compound. It is also utilized for environmental protection and cleaning energy with the help of capturing carbon dioxide. Being small density, high surface structure flexibility, and tuneable pore functionality, metal organic frameworks also play an important role in liquid-phase separation that is industrial step with critical roles in petrochemical, chemical, nuclear, and pharmaceutical industries. These frame works are also used in heterogeneous catalyst, drugs delivery, and sensing conductivity [22–25].

In this paper, upper bounds for LFMD and FMD of the metal organic graphs are calculated, where MOGs are obtained with the help of the cycles of even order. Moreover, the unboundedness of the obtained results is also discussed. Rest of the paper is organized as follows: Section 1 includes the introduction. Construction of MOG is discussed in Section 2. LFMD of metal organic graphs is added in Section 3. FMD of is calculated in Section 4. Conclusion is presented in Section 5.

2. Construction of Metal Organic Graphs

In this section, we describe the construction of metal organic graphs. Let for be a metal organic graph with vertex set and edge set , where . Figure 1 shows for .

(a)

(b)

(c)

3. LFMD of Metal Organic Graphs

In this section, local resolving neighbourhood sets of metal organic graphs are discussed in Lemmas 1 and 2 and local fractional metric dimension is calculated in Theorem 1.

Lemma 1. Let for and be metal organic graph, then . For , , , . Moreover, and .

Proof. The local resolving neighborhood of metal organic graphs, for , , , . with and , and we have .

Lemma 2. Let for and be a metal organic graph with . Then, the following holds:(a)For , , , and .(b)For , , and .(c)For , and .(d)For , and .

Proof. (a)The local resolving neighborhood for , , , , with , . Therefore, .(b)The local resolving neighborhood for , , , with and . Therefore, we have .(c)The local resolving neighborhood for , , , with and . Therefore, we have .(d)The local resolving neighborhood for , , , with and . Therefore, we have .

Theorem 1. Let for and be the metal organic graphs, then .

Proof. In view of Lemmas 1 and 2 for for , , , , and .
We have for all . Moreover, the local resolving neighbourhood of minimum cardinality is not disjoint. Therefore, local fractional metric of is given as follows:For and , we haveHence, .

4. FMD of Metal Organic Graphs

In this section, the resolving neighbourhood sets of metal organic graphs are calculated in Lemmas 3–8. Bounds of FMD are computed in Theorems 2 and 3.

Lemma 3. Let for and be metal organic graph, then . For , , , . Moreover, and .

Proof. The resolving neighborhood sets of metal organic graph for , , , , with and , and we have .

Lemma 4. Let for and be metal organic graphs, then for , , , and :(a).(b).(c).(d).(e).

Proof. (a)The resolving neighborhood for , , , with and . Therefore, we have .(b)The resolving neighborhood for , , . When , When , with , , and . and . Therefore, we have .(c)The resolving neighborhood for , , . When , When , with , , and . and . Therefore, we have .(d)The resolving neighborhood for , , . When , When , with and . Therefore, we have .(e)The resolving neighborhood for , , . with and . Therefore, we have .

Lemma 5. Let for and be metal organic graph. Then, for , , , and :(a).(b).(c).(d).

Proof. (a)The resolving neighborhood for , , . When , When , with , , and . and . Therefore, we have .(b)The resolving neighborhood for , , . with and . Therefore, we have .(c)The resolving neighborhood for , , , with and . Therefore, we have .(d)The resolving neighborhood for , , . with and . Therefore, we have .

Lemma 6. Let for and be metal organic graph. Then, the following holds.(a)For , , , , and .

Proof. The resolving neighborhood for , .
When , for ,When , for ,with and . Therefore, we have .

Corollary 1. (i)For , , , where and .(ii)For , , where .

Lemma 7. Let for and be metal organic graph. Then, for , , , and .

Proof. The resolving neighborhood for , , .
When ,When ,with and . Therefore, we have .

Lemma 8. Let for and be metal organic graph. Then, for and .(a).(b).(c).(d).(e).(f).(g).(h).(i).(j).(k). Proof:(a)The resolving neighborhood for . When , When , with and , when . , when . Therefore, we have .(b)The resolving neighborhood for , : with and . Therefore, we have .(c)The resolving neighborhood for , . When , with and . Therefore, we have .(d)The resolving neighborhood for , : with and . Therefore, we have .(e)The resolving neighborhood for , : with and . Therefore, we have .(f)The resolving neighborhood for , : with and . Therefore, we have .(g)The resolving neighborhood for , : with and . Therefore, we have .(h)The resolving neighborhood for , : with and . Therefore, we have .(i)The resolving neighborhood for , , with and . Therefore, we have .(j)The resolving neighborhood for , : with and . Therefore, we have .(k)The resolving neighborhood for , . When , When , with and , when , when . Therefore, we have .

Corollary 2. For , , where and .

Theorem 2. The FMD of metal organic graph for is

Proof. Case 1: when , then the RNs are as follows. Since, for , the cardinality of each RN is 6, as given in Table 1, which is less than the cardinalities of all other RNs of , as given in Table 2, where . Moreover, ; this implies that and . Consequently, . Case 2: when , as shown in Figure 1, the RNs are as follows. Since, for , the cardinality of each RN is 8, as given in Table 3, which is less than the cardinalities of all other RNs of , as given in Table 4, where . Moreover, ; this implies that and . Consequently, . Case 3: when , as shown in Figure 1, then the RNs are as follows.Since, for , the cardinality of each RN is 8, as given in Table 5, which is less than the cardinalities of all other RNs of , as given in Table 6, where . Moreover, ; this implies that and .
Consequently, .

Theorem 3. Let for and be the metal organic graph. Then, .

Proof. In view of Lemmas 3–8 for , , , , and . Also, we have for all . Moreover, the local resolving neighbourhood of minimum cardinality is not disjoint. Therefore, the fractional metric of is given as follows:For and , we haveHence, .

5. Conclusion

In this section, we conclude the obtained results as follows:(i)The FMD of for is obtained as given in Table 7.(ii)We note that as we increase in for , the FMD also increases.(iii)This is one of the important graphs that has same FMD and LFMD having unique resolving and local resolving neighbourhood sets.(iv)The problem is still open to characterize the graphs with same FMD and LFMD.

Data Availability

The data used to support the finding of this study are included within the article. Additional data can be obtained from the corresponding author upon request.

Disclosure

There is no funding source.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

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Copyright © 2021 Mohsin Raza et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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