# Studies of Connected Networks via Fractional Metric Dimension

Authors: Hassan Zafar; Muhammad Javaid; Ebenezer Bonyah
Journal: Journal of Mathematics
 (2022)

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## 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.


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## Body

## 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 q having MD 1, q−1, and q−2. 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: Section2 consists of the preliminaries, Section 3 deals with the main results, and Section 4 contains the conclusion.


## 2. Preliminaries

LetT be a simple undirected network where VT=a1,a2,a3,…,aq is the vertex set and ET=e1,e2,e3,…,eq 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 a,b⊆VT, the distance between any two vertices a and b denoted by da,b 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].Forℤ=x1,x2,x3,…,xq⊆VT and y∈VT, a representation of the vertex y with respect to ℤ is ry|ℤ=ry,x1,ry,x2,ry,x3,…,ry,xq. If each y∈VT possesses unique representation with ℤ, then set ℤ becomes resolving set having q elements of T and the order of minimum resolving set is called MD of T defined as dimT=minℤ:ℤis resolving set ofT.A vertexc∈VT resolves pair of vertices a,b if distance from a to c is not equal to distance from c to b. For a pair a,b⊆VT, the resolving neighbourhood set is defined as ℛa,b=c∈VT:da,c≠db,c. A function ϕ:VT⟶0,1 is called resolving function if ϕℛa,b≥1 for each ℛa,b of T, where ϕℛa,b=∑c∈ℛa,bϕx. Then, FMD is defined as dimfT= min\{|ϕ|:ϕ is minimal resolving function of T\}.


## 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

For1≤i≤q, the double sunflower DSDFq is a network of order 3q and size 5q, which is obtained from the sunflower network by adding a new vertex ci on each edge aiai+1 and adding edges bici for each i (see Figure 1) [28].Figure 1
Double sunflower networkDSFq.Lemma 1.
LetDSFq with q≥3 be a double sunflower network. Then,(a)
ℛbici=2 and ∪i=1qℛbici=3q.(b)
ℛbici<ℛx,y and ℛx,y∩∪i=1qℛbici>ℛbici, where ℛx,y denotes all other possible resolving neighbourhood sets of DSFq.Proof.
Letai, bi, and ci be the vertices of DSFq, where 1≤i≤q, q+1≅1modq, and we have the following.(a)
ℛbici=bi,ci with ℛbici=2 and ∪i=1qℛbici=3q.(b)
ℛaibi=VDSFn−ci, ℛbiai+1=VDSFq−ci, ℛaici=VDSFq−bi, ℛciai+1=VDSFq−bi, ℛai,ci+1=VDSFq, ℛai,bi+1=VDSFq, ℛbi,bz=VDSFq−ai+z+1/2, i≅1mod2 and i<z. ℛbi,bu=VDSFq−ci+u/2, i≅1mod2 and i<u. ℛbi,cz=VDSFq−ai+z+1/2, i≅0mod2 and i<z. ℛbi,cu=VDSFq−ci+u/2, i≅1mod2 and i<z. ℛbi,cz=VDSFq−ai+u+1/2, i≅0mod2 and i<z. ℛbi,cu=VDSFq−ai+u+1/2, i≅0mod2 and i<z. ℛbi,at=VDSFq. ℛbi,aq=VDSFq, ℛai,bi+1=VDSFq, ℛaq,ci=VDSFq.
It can be observed from Table1 that ℛbici<ℛx,y and |ℛx,y∩∪i=1qℛbici>ℛbici, where ℛx,y are the other RN sets.Table 1
Order of each RN set ofDSFq.
OrderComparisonℛaibi=3q−12<3q−1ℛbiai+1=3q−12<3q−1ℛaici=3q−12<3q−1ℛbiai+1=3q−12<3q−1ℛai,ci+1=3q2<3qℛai,ci+1=3q2<3qℛai,bi+1=3q2<3qℛbi,aq=3q2<3qℛai,ci+1=3q2<3qℛbi,bz=3q−12<3q−1ℛaq,ci=3q2<3qℛbi,bu=3q−12<3q−1ℛbi,bu=3q−12<3q−1ℛbi,cz=3q−12<3q−1ℛbi,at=3q2<3qTheorem 1.
LetDSFq with q≥3 be a double sunflower network. Then, dimFrDSFq=q.Proof.
For different vertices ofDSFq, we have following cases.Case 1.
Forq=3, the possible RN sets are shown in Table 2.
From above RN sets, the RN setsℛbici=2, ℛbici<ℛx,y and ℛx,y are the other RN sets, where 1≤i≤3. Furthermore, ∪i=13ℛbici=6 and ℛx,y∪i=13ℛbici≥2. Hence, there exists a constant function Ψ:VDSF3⟶0,1 defined by Ψv=1/2, ∀v∈∪i=13ℛbici, which shows that Ψ is a resolving function. In order to show that Ψ is a minimal resolving function, consider Ψ′:VDSF3⟶0,1, where Ψ′v, <Ψv, and hence Ψ′ℛbici<1 which means Ψ′ is not a resolving function. Consequently,(1)dimFrDSF3=∑i=1612=3.Table 2
RN sets and the elements ofDSF3.
RN setElementsℛ1=ℛa1,a2VDSF3−a3,b1,c1,ℛ2=ℛa1,a3VDSF3−a2,b3,c3,ℛ3=ℛa1,c2VDSF3,ℛ4=ℛa1,b2VDSF3,ℛ5=ℛb1,b2VDSF3−a2,b3,c3,ℛ6=ℛb1,b3VDSF3−a3,b1,c1,ℛ7=ℛb1,c2VDSF3−a2,b3,c3,ℛ8=ℛb1,c3VDSF3−a1,b2,c2,ℛ9=ℛb1,a3VDSF3,ℛ10=ℛb1,b3VDSF3−a1,b2,c2ℛ11=ℛb1,b3VDSF3−a1,b2,c2,ℛ12=ℛb2,b3VDSF3−a3,b1,c1,ℛ13=ℛb2,c1VDSF3−a2,b3,c3,ℛ14=ℛa2,a3VDSF3−a1,b2,c2,ℛ15=ℛa2,c3VDSF3−a1,b2,c2,ℛ16=ℛa2,b3VDSF3,ℛ17=ℛa3,c1VDSF3,ℛ18=ℛb3,c2VDSF3−a3,b1,c2,ℛ19=ℛb3,c1VDSF3−a1,b2,c2,ℛ20=ℛc1,c3VDSF3−a1,b2,c2,ℛ21=ℛc1,c2VDSF3−a2,b3,c3,ℛ22=ℛa1b1VDSF3−c1,ℛ23=ℛa2b2VDSF3−c2,ℛ24=ℛa3b3VDSF3−c3,ℛ25=ℛb1a2VDSF3−c1,ℛ26=ℛb2a3VDSF3−c2,ℛ27=ℛb3a1VDSF3−c3,ℛ28=ℛa1c1VDSF3−b1,ℛ29=ℛa2c2VDSF3−b2,ℛ30=ℛa3c3VDSF3−b3,ℛ31=ℛc1a2VDSF3−b1,ℛ32=ℛc2a3VDSF3−b2,ℛ33=ℛc3a1VDSF3−b3,ℛ34=ℛb1c1b1,c1,ℛ35=ℛb2c2b2,c2,ℛ36=ℛb3c3b3,c3.Case 2.
For1≤i≤q and q≥5, by Lemma 1, ℛbici=2, and ∪i=1qℛbici=2q and ℛx.y∩∪i=1qℛbici≥2, ∀x,y∈VDSFq. Hence, there exists a function Ψ:VDSFq⟶0,1 defined by Ψv=1/2:(2)Ψv=12,∀v∈∪i=1nℛbici,0,otherwsie,such that Ψℛx,y≥1 which shows that Ψ is a resolving function. In order to show that Ψ is a minimal resolving function, consider that there exists another resolving function Ψ′:VDSFq⟶0,1 such that Ψ′v<Ψv, and hence Ψ′ℛx,y≥1 which means Ψ is not a resolving function. Consequently,(3)dimFrDSFq=∑i=12q12=q.


### 3.2. Fractional Metric Dimension of Circular Diagonal Ladder Network

A circular diagonal ladderCDLq of size 5q and order 2q is obtained from a prism network Dq by adding some double crossing edges aibi+1 and ai+1bi (see Figure 2) (for more information about these networks, see [28]).Figure 2
Circular diagonal ladderCDLq.Lemma 2.
IfCDLq with q≥4 is a circular diagonal ladder network, then(a)
ℛaibi=2 and ∪i=1qℛaibi=2q.(b)
ℛaibi<ℛx,y and ℛx,y∩∪i=1qℛaibi≥ℛaibi, where ℛx,y are all the other possible resolving neighbourhood sets.Proof.
Consider innerai and outer bi vertices of CDLq, respectively, where q+1≅1modq,u≅1mod2, z≅0mod2, and we have(a)
ℛaibi=ai,bi with ℛaibi=2 and ∪i=1qℛaibi=2q.(b)
ℛbibi+1=VCDLq−ai,ai+1, ℛaiai+1=VCDLq−bi,bi+1, ℛbiai−1=VCDLq−ai−1,bi+1, ℛbiai+1=VCDLq−bi+1,ai, ℛai,ai+2=VCDLn−ai+1,bi+1,an+2i+2/2,bn+2i+2/2, ℛbi,bt=VCDLq, ℛau,au+z=VCDLq−a2u+z/2,b2u+z/2,a2u+q/2,b2u+q/2, ℛbu,bu+z=VCDLq−a2u+z/2,b2u+z/2,a2u+q/2,b2u+q/2, ℛau,az=VCDLq, ℛa1,aq−1=VCDLq−aq,bq,aq/2,bq/2, ℛb1,bq−1=VCDLq−aq,bq,aq/2,bq/2.
It can be observed from Table3 that ℛaibi<ℛx,y, ∀x,y∈VCDLq.Table 3
RN sets and the elements ofCDLq.
Order of each RN set ofCDLqComparisonℛaiai+1=2q−22<2q−2ℛbibi+1=2q−22<2n−2ℛbiai−1=2q−22<2q−2ℛbiai+1=2q−22<2q−2ℛau,au+z=2q−42<2q−4ℛbu,bu+z=2q−12<2q−4ℛau,az=2q2<2qℛa1,aq−1=2q−42<2q−4ℛb1,bq−1=2q−42<2q−4Theorem 2.
LetCDLq with q≥4 be a circular diagonal ladder network. Then,(4)dimFrCDLq=q.Proof.
For different vertices ofCDLq, we have the following cases.Case 1.
In this case, we compute FMD ofCDLq with the help of Table 4.
For1≤i≤4, ℛaibi=2 and ℛaibi<ℛx,y where ℛx,y are the other RN sets of CDL4. Furthermore, ∪i=14ℛaibi=VCDL4. Hence, we define a function Ψ:VCDL4⟶0,1 such that Ψx=1/2 for each v∈VCDL4 and Ψx,y≥1∀x,yVCDL4; therefore, Ψ is a RF. To show that Ψ is a minimal RF, we take another RF such that Ψ′x,y<Ψx,y; then, Ψ′x,y<1. Consequently,(5)dimFrCDL4=∑i=1812=4Table 4
Cardinality of each RN set.
CardinalityComparisonℛ1=ℛb1b2VCDL4−a1,a2,ℛ2=ℛb2b3VCDL4−a2,a3,ℛ3=ℛb3b4VCDL4−a3,a4,ℛ4=ℛb4b1VCDL4−a4,a1,ℛ5=ℛa1a2VCDL4−b1,b2,ℛ6=ℛa2a3VCDL4−b2,b3,ℛ7=ℛa3a4VCDL4−a1,a2,ℛ8=ℛa4a1VCDL4−b4,b1,ℛ9=ℛa1b2VCDL4−a2,b1,ℛ10=ℛa2b3VCDL4−a3,b2,ℛ11=ℛa3b4VCDL4−a4,b3,ℛ12=ℛa4b1VCDL4−a4,b3,ℛ13=ℛa1b4VCDL4−a4,b1,ℛ14=ℛb1a2VCDL4−b2,a1,ℛ15=ℛb2,a3VCDL4−b3,a2,ℛ16=ℛb3,a4VCDL4−b4,a3,ℛ17=ℛb4,a1VCDL4−b1,a4,ℛ18=ℛa1b1a1,b1,ℛ19=ℛa2b2a2,b2,ℛ20=ℛa3b3a3,b3,ℛ21=ℛa4b4a4,b4,ℛ22=ℛa1,a3a1,a3,b1,b3,ℛ23=ℛa1,b3a1,a3,b1,b3,ℛ24=ℛb1,a3a1,a3,b1,b3,ℛ25=ℛb1,b3a1,a3,b1,b3,ℛ26=ℛa2,a4a2,a4,b2,b4,ℛ27=ℛa2,b4a2,a4,b2,b4.Case 2.
Forq≥5, ℛaibi=2 and ℛaibi<ℛx,y where ℛx,y are the other RN sets of CDLq. Furthermore, ∪i=12qℛaibi=VCDLq. Hence, we define a function Ψ:VCDLq⟶0,1 such that Ψx=1/2 for each v∈VCDLq and Ψx,y≥1∀x,y∈VCDL4; therefore, Ψ is a RF. To show that Ψ is a minimal RF, we take another RF such that Ψ′x,y<Ψ′x,y; then, Ψ′x,y<1 which shows that Ψ is a minimal resolving function. Consequently,(6)dimFrCDLq=∑i=12q12=q.


### 3.3. FMD of Double Path Network

In this section, our aim is to compute FMD of double path network. The double path networkPq2 is obtained from path network Pq by taking two copies of Pq by taking every vertex ai in one copy with the open neighbourhood Nai of the correspondence vertex of its second copy. Furthermore, VPq2=2q and EPq2=2q+6 (for details, see Figure 3) [29].Figure 3
Double path networkPq2.Lemma 3.
LetPq2 with q≥3 be a double path network. Then,(a)
ℛaibi=2 and ∪i=1qℛai,bi=VPq2.(b)
ℛai,bi<ℛx,y and ℛx,y∩∪i=1qℛai,bi≥2, where ℛx,y are all the possible resolving neighbourhood sets of Pq2.Proof.
Letai, bi be the vertices of Pq2, where 1≤i≤q, q≅1mod2, j≅0mod2, p≅0mod2, and q+1≅1modq.(a)
Sinceℛai,bi=ai,bi, ℛai,bi=2 and ∪i=1qℛaibi=VPq2=2q.(b)
ℛai,bq=VPq2−bi+q/2,ai+q/2, ℛai,aq=VPq2−bi+q/2,ai+q/2, ℛbi,bq=VPq2−bi+q/2,ai+q/2, ℛaj,aq=VPq2−aj+p/2,aj+p/2,ℛbj,bp=VPq2−bj+p/2,bj+p/2,ℛaj,bp=VPq2−aj+p/2,bj+p/2. Now with the help of Table 5, we compare the cardinalities of each RN set.
From Table5, it is observed that ℛai,bi<ℛx,y. Since ∪i=1nRai,bi=VPq2, ℛx,y∩∪i=1qℛai,bi≥2.Table 5
The possible resolving neighbourhood sets ofPq2.
ℛai,aq=2q−22<2q−2ℛai,bq=2q−22<2q−2ℛbi,bq=2q−22<2q−2ℛaj,ap=2q−22<2q−2ℛbj,bp=2q−22<2q−2Theorem 3.
LetPq2 be a double path network with q≥3. Then,(7)dimFrPq2=q.Proof.
For different vertices ofPq2, we have the following cases.Case 1.
In this case, we compute all the RN sets ofPq2 and their cardinalities with the help of Table 6.
Sinceℛe=ℛa1,b1=ℛa2,b2=ℛa3,b3=ℛa1,a3=ℛb1,b3=2 and ℛe<ℛx,y of P32, ∪i=13ℛei=6 and ℛx,y∩∪ℛe≥2. Furthermore, the RN sets ℛe are not pairwise disjoint. Hence, we define a function Ψ:VP32⟶0,1 such that Ψv=1/2 for each v∈VP32 and Ψx,y≥1∀x,yVP32; therefore, Ψ is a RF. To show that Ψ is a minimal RF, we take another RF such that Ψ′x,y<Ψx,y; then, Ψx,y<1. Hence, Ψ is a minimal RF. Consequently,(8)dimFrP32=∑i=1612=3.Table 6
Cardinality of each RN set.
CardinalityComparisonRN sets ofPq2Elementsℛ1=ℛa1,b1a1,b1,ℛ2=ℛa3,b3a3,b3,ℛ3=ℛa1,a2VP32,ℛ3=ℛa1,a3a1,a3,ℛ5=ℛa1b2VP32,ℛ6=ℛa1,b3VP32−b2,a3,ℛ7=ℛa2,b1VP32,ℛ8=ℛa2,b2a2,b2,ℛ9=ℛa2,b3VP32,ℛ10=ℛa2,a3VP32,ℛ11=ℛa3,b1a3,b1,ℛ12=ℛa3,b2VP32,ℛ13=ℛb1,b3b1,b3,ℛ14=ℛb1b2VP32,ℛ15=ℛb2b3VP32.Case 2.
Forq≥4, by Lemma 3, ℛai,bi=2 and ∪i=1qℛai,bi=VPq2 where the elements of ℛa,bi are pairwise disjoint. Hence, we define a constant function Ψ:VPq2⟶0,1 such that Ψv=1/2 for each v∈VP32 and Ψx,y≥1∀x,yVP32; therefore, Ψ is a RF. To show that Ψ is a minimal RF, we take another RF such that Ψ′x,y<Ψx,y; then, Ψ′x,y<1 which shows that Ψ is a minimal resolving function, ∀x,y∈VP32. Consequently,(9)dimFrPq2=∑i=12q12=q.


## 3.1. Fractional Metric Dimension of Double Sunflower Network

For1≤i≤q, the double sunflower DSDFq is a network of order 3q and size 5q, which is obtained from the sunflower network by adding a new vertex ci on each edge aiai+1 and adding edges bici for each i (see Figure 1) [28].Figure 1
Double sunflower networkDSFq.Lemma 1.
LetDSFq with q≥3 be a double sunflower network. Then,(a)
ℛbici=2 and ∪i=1qℛbici=3q.(b)
ℛbici<ℛx,y and ℛx,y∩∪i=1qℛbici>ℛbici, where ℛx,y denotes all other possible resolving neighbourhood sets of DSFq.Proof.
Letai, bi, and ci be the vertices of DSFq, where 1≤i≤q, q+1≅1modq, and we have the following.(a)
ℛbici=bi,ci with ℛbici=2 and ∪i=1qℛbici=3q.(b)
ℛaibi=VDSFn−ci, ℛbiai+1=VDSFq−ci, ℛaici=VDSFq−bi, ℛciai+1=VDSFq−bi, ℛai,ci+1=VDSFq, ℛai,bi+1=VDSFq, ℛbi,bz=VDSFq−ai+z+1/2, i≅1mod2 and i<z. ℛbi,bu=VDSFq−ci+u/2, i≅1mod2 and i<u. ℛbi,cz=VDSFq−ai+z+1/2, i≅0mod2 and i<z. ℛbi,cu=VDSFq−ci+u/2, i≅1mod2 and i<z. ℛbi,cz=VDSFq−ai+u+1/2, i≅0mod2 and i<z. ℛbi,cu=VDSFq−ai+u+1/2, i≅0mod2 and i<z. ℛbi,at=VDSFq. ℛbi,aq=VDSFq, ℛai,bi+1=VDSFq, ℛaq,ci=VDSFq.
It can be observed from Table1 that ℛbici<ℛx,y and |ℛx,y∩∪i=1qℛbici>ℛbici, where ℛx,y are the other RN sets.Table 1
Order of each RN set ofDSFq.
OrderComparisonℛaibi=3q−12<3q−1ℛbiai+1=3q−12<3q−1ℛaici=3q−12<3q−1ℛbiai+1=3q−12<3q−1ℛai,ci+1=3q2<3qℛai,ci+1=3q2<3qℛai,bi+1=3q2<3qℛbi,aq=3q2<3qℛai,ci+1=3q2<3qℛbi,bz=3q−12<3q−1ℛaq,ci=3q2<3qℛbi,bu=3q−12<3q−1ℛbi,bu=3q−12<3q−1ℛbi,cz=3q−12<3q−1ℛbi,at=3q2<3qTheorem 1.
LetDSFq with q≥3 be a double sunflower network. Then, dimFrDSFq=q.Proof.
For different vertices ofDSFq, we have following cases.Case 1.
Forq=3, the possible RN sets are shown in Table 2.
From above RN sets, the RN setsℛbici=2, ℛbici<ℛx,y and ℛx,y are the other RN sets, where 1≤i≤3. Furthermore, ∪i=13ℛbici=6 and ℛx,y∪i=13ℛbici≥2. Hence, there exists a constant function Ψ:VDSF3⟶0,1 defined by Ψv=1/2, ∀v∈∪i=13ℛbici, which shows that Ψ is a resolving function. In order to show that Ψ is a minimal resolving function, consider Ψ′:VDSF3⟶0,1, where Ψ′v, <Ψv, and hence Ψ′ℛbici<1 which means Ψ′ is not a resolving function. Consequently,(1)dimFrDSF3=∑i=1612=3.Table 2
RN sets and the elements ofDSF3.
RN setElementsℛ1=ℛa1,a2VDSF3−a3,b1,c1,ℛ2=ℛa1,a3VDSF3−a2,b3,c3,ℛ3=ℛa1,c2VDSF3,ℛ4=ℛa1,b2VDSF3,ℛ5=ℛb1,b2VDSF3−a2,b3,c3,ℛ6=ℛb1,b3VDSF3−a3,b1,c1,ℛ7=ℛb1,c2VDSF3−a2,b3,c3,ℛ8=ℛb1,c3VDSF3−a1,b2,c2,ℛ9=ℛb1,a3VDSF3,ℛ10=ℛb1,b3VDSF3−a1,b2,c2ℛ11=ℛb1,b3VDSF3−a1,b2,c2,ℛ12=ℛb2,b3VDSF3−a3,b1,c1,ℛ13=ℛb2,c1VDSF3−a2,b3,c3,ℛ14=ℛa2,a3VDSF3−a1,b2,c2,ℛ15=ℛa2,c3VDSF3−a1,b2,c2,ℛ16=ℛa2,b3VDSF3,ℛ17=ℛa3,c1VDSF3,ℛ18=ℛb3,c2VDSF3−a3,b1,c2,ℛ19=ℛb3,c1VDSF3−a1,b2,c2,ℛ20=ℛc1,c3VDSF3−a1,b2,c2,ℛ21=ℛc1,c2VDSF3−a2,b3,c3,ℛ22=ℛa1b1VDSF3−c1,ℛ23=ℛa2b2VDSF3−c2,ℛ24=ℛa3b3VDSF3−c3,ℛ25=ℛb1a2VDSF3−c1,ℛ26=ℛb2a3VDSF3−c2,ℛ27=ℛb3a1VDSF3−c3,ℛ28=ℛa1c1VDSF3−b1,ℛ29=ℛa2c2VDSF3−b2,ℛ30=ℛa3c3VDSF3−b3,ℛ31=ℛc1a2VDSF3−b1,ℛ32=ℛc2a3VDSF3−b2,ℛ33=ℛc3a1VDSF3−b3,ℛ34=ℛb1c1b1,c1,ℛ35=ℛb2c2b2,c2,ℛ36=ℛb3c3b3,c3.Case 2.
For1≤i≤q and q≥5, by Lemma 1, ℛbici=2, and ∪i=1qℛbici=2q and ℛx.y∩∪i=1qℛbici≥2, ∀x,y∈VDSFq. Hence, there exists a function Ψ:VDSFq⟶0,1 defined by Ψv=1/2:(2)Ψv=12,∀v∈∪i=1nℛbici,0,otherwsie,such that Ψℛx,y≥1 which shows that Ψ is a resolving function. In order to show that Ψ is a minimal resolving function, consider that there exists another resolving function Ψ′:VDSFq⟶0,1 such that Ψ′v<Ψv, and hence Ψ′ℛx,y≥1 which means Ψ is not a resolving function. Consequently,(3)dimFrDSFq=∑i=12q12=q.


## 3.2. Fractional Metric Dimension of Circular Diagonal Ladder Network

A circular diagonal ladderCDLq of size 5q and order 2q is obtained from a prism network Dq by adding some double crossing edges aibi+1 and ai+1bi (see Figure 2) (for more information about these networks, see [28]).Figure 2
Circular diagonal ladderCDLq.Lemma 2.
IfCDLq with q≥4 is a circular diagonal ladder network, then(a)
ℛaibi=2 and ∪i=1qℛaibi=2q.(b)
ℛaibi<ℛx,y and ℛx,y∩∪i=1qℛaibi≥ℛaibi, where ℛx,y are all the other possible resolving neighbourhood sets.Proof.
Consider innerai and outer bi vertices of CDLq, respectively, where q+1≅1modq,u≅1mod2, z≅0mod2, and we have(a)
ℛaibi=ai,bi with ℛaibi=2 and ∪i=1qℛaibi=2q.(b)
ℛbibi+1=VCDLq−ai,ai+1, ℛaiai+1=VCDLq−bi,bi+1, ℛbiai−1=VCDLq−ai−1,bi+1, ℛbiai+1=VCDLq−bi+1,ai, ℛai,ai+2=VCDLn−ai+1,bi+1,an+2i+2/2,bn+2i+2/2, ℛbi,bt=VCDLq, ℛau,au+z=VCDLq−a2u+z/2,b2u+z/2,a2u+q/2,b2u+q/2, ℛbu,bu+z=VCDLq−a2u+z/2,b2u+z/2,a2u+q/2,b2u+q/2, ℛau,az=VCDLq, ℛa1,aq−1=VCDLq−aq,bq,aq/2,bq/2, ℛb1,bq−1=VCDLq−aq,bq,aq/2,bq/2.
It can be observed from Table3 that ℛaibi<ℛx,y, ∀x,y∈VCDLq.Table 3
RN sets and the elements ofCDLq.
Order of each RN set ofCDLqComparisonℛaiai+1=2q−22<2q−2ℛbibi+1=2q−22<2n−2ℛbiai−1=2q−22<2q−2ℛbiai+1=2q−22<2q−2ℛau,au+z=2q−42<2q−4ℛbu,bu+z=2q−12<2q−4ℛau,az=2q2<2qℛa1,aq−1=2q−42<2q−4ℛb1,bq−1=2q−42<2q−4Theorem 2.
LetCDLq with q≥4 be a circular diagonal ladder network. Then,(4)dimFrCDLq=q.Proof.
For different vertices ofCDLq, we have the following cases.Case 1.
In this case, we compute FMD ofCDLq with the help of Table 4.
For1≤i≤4, ℛaibi=2 and ℛaibi<ℛx,y where ℛx,y are the other RN sets of CDL4. Furthermore, ∪i=14ℛaibi=VCDL4. Hence, we define a function Ψ:VCDL4⟶0,1 such that Ψx=1/2 for each v∈VCDL4 and Ψx,y≥1∀x,yVCDL4; therefore, Ψ is a RF. To show that Ψ is a minimal RF, we take another RF such that Ψ′x,y<Ψx,y; then, Ψ′x,y<1. Consequently,(5)dimFrCDL4=∑i=1812=4Table 4
Cardinality of each RN set.
CardinalityComparisonℛ1=ℛb1b2VCDL4−a1,a2,ℛ2=ℛb2b3VCDL4−a2,a3,ℛ3=ℛb3b4VCDL4−a3,a4,ℛ4=ℛb4b1VCDL4−a4,a1,ℛ5=ℛa1a2VCDL4−b1,b2,ℛ6=ℛa2a3VCDL4−b2,b3,ℛ7=ℛa3a4VCDL4−a1,a2,ℛ8=ℛa4a1VCDL4−b4,b1,ℛ9=ℛa1b2VCDL4−a2,b1,ℛ10=ℛa2b3VCDL4−a3,b2,ℛ11=ℛa3b4VCDL4−a4,b3,ℛ12=ℛa4b1VCDL4−a4,b3,ℛ13=ℛa1b4VCDL4−a4,b1,ℛ14=ℛb1a2VCDL4−b2,a1,ℛ15=ℛb2,a3VCDL4−b3,a2,ℛ16=ℛb3,a4VCDL4−b4,a3,ℛ17=ℛb4,a1VCDL4−b1,a4,ℛ18=ℛa1b1a1,b1,ℛ19=ℛa2b2a2,b2,ℛ20=ℛa3b3a3,b3,ℛ21=ℛa4b4a4,b4,ℛ22=ℛa1,a3a1,a3,b1,b3,ℛ23=ℛa1,b3a1,a3,b1,b3,ℛ24=ℛb1,a3a1,a3,b1,b3,ℛ25=ℛb1,b3a1,a3,b1,b3,ℛ26=ℛa2,a4a2,a4,b2,b4,ℛ27=ℛa2,b4a2,a4,b2,b4.Case 2.
Forq≥5, ℛaibi=2 and ℛaibi<ℛx,y where ℛx,y are the other RN sets of CDLq. Furthermore, ∪i=12qℛaibi=VCDLq. Hence, we define a function Ψ:VCDLq⟶0,1 such that Ψx=1/2 for each v∈VCDLq and Ψx,y≥1∀x,y∈VCDL4; therefore, Ψ is a RF. To show that Ψ is a minimal RF, we take another RF such that Ψ′x,y<Ψ′x,y; then, Ψ′x,y<1 which shows that Ψ is a minimal resolving function. Consequently,(6)dimFrCDLq=∑i=12q12=q.


## 3.3. FMD of Double Path Network

In this section, our aim is to compute FMD of double path network. The double path networkPq2 is obtained from path network Pq by taking two copies of Pq by taking every vertex ai in one copy with the open neighbourhood Nai of the correspondence vertex of its second copy. Furthermore, VPq2=2q and EPq2=2q+6 (for details, see Figure 3) [29].Figure 3
Double path networkPq2.Lemma 3.
LetPq2 with q≥3 be a double path network. Then,(a)
ℛaibi=2 and ∪i=1qℛai,bi=VPq2.(b)
ℛai,bi<ℛx,y and ℛx,y∩∪i=1qℛai,bi≥2, where ℛx,y are all the possible resolving neighbourhood sets of Pq2.Proof.
Letai, bi be the vertices of Pq2, where 1≤i≤q, q≅1mod2, j≅0mod2, p≅0mod2, and q+1≅1modq.(a)
Sinceℛai,bi=ai,bi, ℛai,bi=2 and ∪i=1qℛaibi=VPq2=2q.(b)
ℛai,bq=VPq2−bi+q/2,ai+q/2, ℛai,aq=VPq2−bi+q/2,ai+q/2, ℛbi,bq=VPq2−bi+q/2,ai+q/2, ℛaj,aq=VPq2−aj+p/2,aj+p/2,ℛbj,bp=VPq2−bj+p/2,bj+p/2,ℛaj,bp=VPq2−aj+p/2,bj+p/2. Now with the help of Table 5, we compare the cardinalities of each RN set.
From Table5, it is observed that ℛai,bi<ℛx,y. Since ∪i=1nRai,bi=VPq2, ℛx,y∩∪i=1qℛai,bi≥2.Table 5
The possible resolving neighbourhood sets ofPq2.
ℛai,aq=2q−22<2q−2ℛai,bq=2q−22<2q−2ℛbi,bq=2q−22<2q−2ℛaj,ap=2q−22<2q−2ℛbj,bp=2q−22<2q−2Theorem 3.
LetPq2 be a double path network with q≥3. Then,(7)dimFrPq2=q.Proof.
For different vertices ofPq2, we have the following cases.Case 1.
In this case, we compute all the RN sets ofPq2 and their cardinalities with the help of Table 6.
Sinceℛe=ℛa1,b1=ℛa2,b2=ℛa3,b3=ℛa1,a3=ℛb1,b3=2 and ℛe<ℛx,y of P32, ∪i=13ℛei=6 and ℛx,y∩∪ℛe≥2. Furthermore, the RN sets ℛe are not pairwise disjoint. Hence, we define a function Ψ:VP32⟶0,1 such that Ψv=1/2 for each v∈VP32 and Ψx,y≥1∀x,yVP32; therefore, Ψ is a RF. To show that Ψ is a minimal RF, we take another RF such that Ψ′x,y<Ψx,y; then, Ψx,y<1. Hence, Ψ is a minimal RF. Consequently,(8)dimFrP32=∑i=1612=3.Table 6
Cardinality of each RN set.
CardinalityComparisonRN sets ofPq2Elementsℛ1=ℛa1,b1a1,b1,ℛ2=ℛa3,b3a3,b3,ℛ3=ℛa1,a2VP32,ℛ3=ℛa1,a3a1,a3,ℛ5=ℛa1b2VP32,ℛ6=ℛa1,b3VP32−b2,a3,ℛ7=ℛa2,b1VP32,ℛ8=ℛa2,b2a2,b2,ℛ9=ℛa2,b3VP32,ℛ10=ℛa2,a3VP32,ℛ11=ℛa3,b1a3,b1,ℛ12=ℛa3,b2VP32,ℛ13=ℛb1,b3b1,b3,ℛ14=ℛb1b2VP32,ℛ15=ℛb2b3VP32.Case 2.
Forq≥4, by Lemma 3, ℛai,bi=2 and ∪i=1qℛai,bi=VPq2 where the elements of ℛa,bi are pairwise disjoint. Hence, we define a constant function Ψ:VPq2⟶0,1 such that Ψv=1/2 for each v∈VP32 and Ψx,y≥1∀x,yVP32; therefore, Ψ is a RF. To show that Ψ is a minimal RF, we take another RF such that Ψ′x,y<Ψx,y; then, Ψ′x,y<1 which shows that Ψ is a minimal resolving function, ∀x,y∈VP32. Consequently,(9)dimFrPq2=∑i=12q12=q.


## 4. Conclusion

This paper deals with the latest invariant of metric dimension called fractional metric dimension, and we have computed exact value of fractional metric dimension of different connected networks like double sunflower network, circular diagonal ladder network, and double path network. Furthermore, it is also proved that the fractional metric dimension of these networks depends on their order. Now, we close our discussion with the following open problem.Characterization of the connected works attaining the exact value of fractional metric dimension is still an open problem under the condition that the cardinality of the minimum resolving neighborhood sets is greater than 2.



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