Abstract

Metric dimension is an effective tool to study different distance-based problems in the field of telecommunication, robotics, computer networking, integer programming, chemistry, and electrical networking. In this paper, we study the latest form of metric dimension called fractional metric dimension of some connected networks such as circular diagonal ladder, double sun flower, and double path networks.

1. Introduction

The resolving or locating sets were first introduced by Slater, and he called order of the minimum resolving set as location number [1]. By following the idea of locating sets, Melter and Harary defined the term metric dimension (MD). Moreover, they gave a characterization of the MD of trees [2]. Chartrand et al. computed MD of the path and unicyclic networks, and they also characterized all those connected networks of order having MD 1, , and . Furthermore, they proposed the solution of integer programming problem (IPP) with the help of MD under certain conditions [3]. Constant MD of cycle-related networks was computed by Murtaza et al. [4]. It was also observed that problem to find MD of a connected network is NP-hard [5]. Locating sets have applications in network discovery [6], coin weighting, robot navigation, pattern recognition [7], and joints in networks [8]. Some interesting results of different invariants of MD can be seen in [9–12], and recently, Dalal et al. computed edge MD of some Toeplitz networks [13].

Currie and Oellermann found integral as well as nonintegral solutions of IPP by using the idea of fractional metric dimension (FMD) [14]. Fahr et al. presented an optimal solution of the IPP by using concept of FMD [15]. Later Arguman and Matthew calculated FMD of some important networks and also defined specific computational criteria to compute FMD of some connected networks [16, 17]. FMD of vertex transitive and hierarchical product of networks was computed by Feng et al. Moreover, they also established bounds of FMD of Cartesian product of networks [18,19]. For the study of FMD of corona product, lexicographic product, and unicyclic and generalized sunlet networks, see [20–22]. Later on Al Khalidi et al. designed computational criteria to compute bounds of FMD of connected networks [23]. Furthermore, Javaid et al. characterized all those connected networks with FMD being exactly 1 [24] and bounds of FMD for metal organic compounds calculated by Moshin et al. [25]. Recently, in 2021, Hassan et al. computed the L-F metric dimension of generalized gear networks in the form of exact value and bounds. Moreover, they also proved that all these networks remain unbounded when their order approaches to infinity [26]. In this dissertation, we have computed exact value of FMD of connected networks such as double sunflower network, circular diagonal network, and double path network.

The article is organized as follows: Section 2 consists of the preliminaries, Section 3 deals with the main results, and Section 4 contains the conclusion.

2. Preliminaries

Let be a simple undirected network where is the vertex set and is the edge set, respectively. A network whose vertices are ordered in a way that two vertices are adjacent if and only if they are consecutive in the list is called path network. For any two vertices , the distance between any two vertices and denoted by is the number of edges between them. A network is connected if there exists a path between any two vertices. For further study about these preliminary concepts, see [27].

For and , a representation of the vertex with respect to is . If each possesses unique representation with , then set becomes resolving set having elements of and the order of minimum resolving set is called MD of defined as .

A vertex resolves pair of vertices if distance from to is not equal to distance from to . For a pair , the resolving neighbourhood set is defined as . A function is called resolving function if for each of , where . Then, FMD is defined as min\{ is minimal resolving function of \}.

3. Main Results

In this particular section, we present main result to compute exact value of FMD of different connected networks such as circular diagonal ladder networks, double sunflower networks, and double path networks.

3.1. Fractional Metric Dimension of Double Sunflower Network

For , the double sunflower is a network of order and size , which is obtained from the sunflower network by adding a new vertex on each edge and adding edges for each i (see Figure 1) [28].

Figure 1 

Double sunflower network .

Lemma 1. Let with be a double sunflower network. Then,(a) and .(b) and , where denotes all other possible resolving neighbourhood sets of .

Proof. Let , , and be the vertices of , where , , and we have the following.(a) with and .(b), , , , , , , and . , and . , and . , and . , and . , and . . , , .It can be observed from Table 1 that and , where are the other RN sets.

Theorem 1. Let with be a double sunflower network. Then,

Proof. For different vertices of , we have following cases.

Case 1. For , the possible RN sets are shown in Table 2.
From above RN sets, the RN sets , and are the other RN sets, where . Furthermore, and . Hence, there exists a constant function defined by , , which shows that is a resolving function. In order to show that is a minimal resolving function, consider , where , , and hence which means is not a resolving function. Consequently,

Case 2. For and , by Lemma 1, and and , . Hence, there exists a function defined by :such that which shows that is a resolving function. In order to show that is a minimal resolving function, consider that there exists another resolving function such that , and hence which means is not a resolving function. Consequently,

3.2. Fractional Metric Dimension of Circular Diagonal Ladder Network

A circular diagonal ladder of size and order is obtained from a prism network by adding some double crossing edges and (see Figure 2) (for more information about these networks, see [28]).

Figure 2 

Circular diagonal ladder .

Lemma 2. If with is a circular diagonal ladder network, then(a) and .(b) and , where are all the other possible resolving neighbourhood sets.

Proof. Consider inner and outer vertices of , respectively, where , , and we have(a) with and .(b), , , , , , , , , , .It can be observed from Table 3 that , .

Theorem 2. Let with be a circular diagonal ladder network. Then,

Proof. For different vertices of , we have the following cases.

Case 1. In this case, we compute FMD of with the help of Table 4.
For , and where are the other RN sets of . Furthermore, . Hence, we define a function such that for each and ; therefore, is a RF. To show that is a minimal RF, we take another RF such that ; then, . Consequently,

Case 2. For , and where are the other RN sets of . Furthermore, . Hence, we define a function such that for each and ; therefore, is a RF. To show that is a minimal RF, we take another RF such that ; then, which shows that is a minimal resolving function. Consequently,

3.3. FMD of Double Path Network

In this section, our aim is to compute FMD of double path network. The double path network is obtained from path network by taking two copies of by taking every vertex in one copy with the open neighbourhood of the correspondence vertex of its second copy. Furthermore, and (for details, see Figure 3) [29].

Figure 3 

Double path network .

Lemma 3. Let with be a double path network. Then,(a) and .(b) and , where are all the possible resolving neighbourhood sets of .

Proof. Let , be the vertices of , where , , , , and .(a)Since , and .(b), , , . Now with the help of Table 5, we compare the cardinalities of each RN set.From Table 5, it is observed that . Since , .

Theorem 3. Let be a double path network with . Then,

Proof. For different vertices of , we have the following cases.

Case 1. In this case, we compute all the RN sets of and their cardinalities with the help of Table 6.
Since and of , and . Furthermore, the RN sets are not pairwise disjoint. Hence, we define a function such that for each and ; therefore, is a RF. To show that is a minimal RF, we take another RF such that ; then, . Hence, is a minimal RF. Consequently,