# Computing Bounds of Fractional Metric Dimension of Metal Organic Graphs

Authors: Mohsin Raza; Dalal Awadh Alrowaili; Muhammad Javaid; Khurram Shabbir
Journal: Journal of Chemistry
 (2021)

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## 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 vertexx resolves the vertices u and v of a graph G if du,x≠dv,x. For a pair u,v of vertices of G, Ru,v=x∈VG:dx,u≠dx,v is called its resolving neighbourhood set. For each pair of vertices u and v in VG, if fRu,v≥1, then f from VG to the interval 0,1 is called resolving function. Moreover, for two functions f and g, f is called minimal if f≤g and fv≠gv for at least one v∈VG. The fractional metric dimension (FMD) of G is denoted by dimfG and defined as dimfG=ming:gis a minimal resolving function ofG, where g=∑v∈VGgv. If we take a pair of vertices u,v of G as an edge e=uv of G, then it becomes local fractional metric dimension (LFMD) dimlfG. In this paper, local fractional and fractional metric dimensions of MOGn are computed for n≅1mod2 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.


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## Body

## 1. Introduction

For a connected graphG, a vertex x∈VG is said to resolve a pair u,v of vertices of G if dx,u≠dx,v. A set S⊆VG is called a resolving set of G if each pair of vertices of G is resolved by some vertex in S. The metric dimension of G is denoted by dimG and is defined as(1)dimG=minS:Sis a resolving set ofG.For a pairu,v of vertices of G, the resolving neighborhood Ru,v is defined as Ru,v=w∈VG:dw,u≠dw,v. A resolving function is a real-valued function g:VG⟶0,1 such that gRu,v≥1 for each distinct pair of vertices of G, where gRu,v=∑x∈Ru,vgx. A resolving function g is called minimal if any function f:VG⟶0,1 such that f≤g and fv≠gv for at least one v∈V is not a resolving function of G. The fractional metric dimension (FMD) of G is denoted by dimfG and defined as(2)dimfG=ming:gis a minimal resolving function ofG,where g=≤∑v∈VGgv. Now, if we take a pair of vertices u,v of G as an edge e=uv of G, then the aforesaid defined resolving neighborhood Ru,v, minimal resolving function g, and FMD dimfG become local resolving neighborhood LRuv, local minimal resolving function, and local fractional metric dimension dimlfG, 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 graphMOG consists of metal atoms, where atoms are linked with thes help of organic ligands which act like a linker. Therefore, MOG 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: Section1 includes the introduction. Construction of MOG is discussed in Section 2. LFMD of metal organic graphs is added in Section 3. FMD of MOG 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. LetMOGn for n≥3 be a metal organic graph with vertex set VMOGn=ui:1≤i≤n∪vj:1≤j≤2n and edge set EMOGn=uiui+1:1≤i≤n−1∪vjvj+1:1≤j≤2n−1∪unu1,v2nv1∪uivj,uivj+1:1≤i≤n,1≤j≤2n, where VMOGn=EMOGn=3n. Figure 1 shows MOGn for n∈5,7,9.Figure 1
MOGn for n=5 (a), n=7 (b), and n=9 (c).
(a)(b)(c)


## 3. LFMD of Metal Organic Graphs

In this section, local resolving neighbourhood sets of metal organic graphs are discussed in Lemmas1 and 2 and local fractional metric dimension is calculated in Theorem 1.Lemma 1.
LetMOGn for n≡1mod2 and n≥5 be metal organic graph, then LRet=LRet=vivj=8. For 1≤k≤n, j=i+1, i∈2k−1, 1≤t≤n. Moreover, ∪LRet=vm:1≤m≤2n and ∪LRet=α=2n.Proof.
The local resolving neighborhood of metal organic graphs, for1≤k≤n, j=i+1, i∈2k−1, 1≤t≤n. LRvivj=vl:2k−1≤l≤k−3,2k−4≤l≤2k−2 with LRet=8 and ∪t=1nLRet=vs:1≤s≤2n, and we have ∪t=1nLRet=2n.Lemma 2.
LetMOGn for n≡1mod2 and n≥9 be a metal organic graph with 1≤t≤n. Then, the following holds:(a)
For1≤k≤n, j=i+1, i∈2k, LRet<LRvivj and LRvivj∩∪t=1nLRet≥LRet.(b)
For1≤i≤n−1, j=i+1, LRet<LRuiuj and LRuiuj∩∪t=1nLRet≥LRet.(c)
For1≤i≤n, j=2i−1,2i,LRet<LRuiuj and LRuiuj∩∪t=1nLRet≥LRet.(d)
For1≤i≤n, j=2i,LRet<LRuiuj and LRuiuj∩∪t=1nLRet≥LRet.Proof.
(a)
The local resolving neighborhood for1≤k≤n, j=i+1, i∈2k, 1≤t≤n,(3)LRvivj=up:1≤p≤n,wherep≠n+12+k=m,vq:1≤q≤2n,whereq≠2m,2m−1,withLRvivj=3n−3>8=LRet, LRvivj∩∪t=1nLRet=vq:1≤q≤2n,q≠2m,2m−1. Therefore, LRvivj∩∪t=1nLRet=2n−2>LRet.(b)
The local resolving neighborhood for1≤i≤n−1, j=i+1, 1≤t≤n,(4)LRuiuj=up:1≤p≤n,wherep≠n+i+j2+k=m,vq:1≤q≤2n,whereq≠2m,2m−1,withLRuiuj=3n−3>8=LRet and LRuiuj∩∪t=1nLRet=vq:1≤q≤2n,q≠2m,2m−1. Therefore, we have LRuiuj∩∪t=1nLRet=2n−2>LRet.(c)
The local resolving neighborhood for1≤i≤n, j=2i−1, 1≤t≤n,(5)LRuivj=up:1≤p≤n;vq:1≤q≤2nwhereq≠j+1,2,−2,−3,withLRuiuj=3n−4>8=LRet and LRuiuj∩∪t=1nLRet=vq:q≠j+1,2,−2,−3. Therefore, we have LRuiuj∩∪t=1nLRet=2n−4>LRet.(d)
The local resolving neighborhood for1≤i≤n, j=2i−1, 1≤t≤n,(6)LRuivj=up:1≤p≤n;vq:1≤q≤2n,whereq≠j+2,3,−1,−2,withLRuiuj=3n−4>8=LRet and LRuiuj∩∪t=1nLRet=vq:q≠j+2,3,−1,−2. Therefore, we have LRuiuj∩∪t=1nLRet=2n−4>LRet.Theorem 1.
LetMOGn for n≡1mod2 and n≥5 be the metal organic graphs, then dimlfMOGn≤n/4.Proof.
In view of Lemmas1 and 2 for for 1≤k≤n, j=i+1, i∈2k−1, 1≤t≤n, LRet=LRvivj=8 and X=∪t=1nLRet=2n.
We haveRxy≤Ret for all xy∈EMOGn. Moreover, the local resolving neighbourhood of minimum cardinality is not disjoint. Therefore, local fractional metric of MOGn is given as follows:(7)dimlfMOGn≤∑t=1X1LRet.
ForX=2n and LRet=8, we have(8)dimlfMOGn≤∑t=12n18.
Hence,dimlfMOGn≤n/4.


## 4. FMD of Metal Organic Graphs

In this section, the resolving neighbourhood sets of metal organic graphs are calculated in Lemmas3–8. Bounds of FMD are computed in Theorems 2 and 3.Lemma 3.
LetMOGn for n≡1mod2 and n≥9 be metal organic graph, then Ret=Ret=vi,vj=8. For 1≤k≤n, j=i+1, i∈2k−1, 1≤t≤n. Moreover, ∪t=1nRet=vm:1≤m≤2n and ∪t=1nRet=α=2n.Proof.
The resolving neighborhood sets of metal organic graph for1≤k≤n, j=i+1, i∈2k−1, 1≤t≤n, Rvi,vj=vl:2k−1≤l≤k−3,2k−4≤l≤2k−2 with Ret=8 and ∪t=1nRet=vs:1≤s≤2n, and we have ∪t=1nRet=2n.Lemma 4.
LetMOGn for n≡1mod2 and n≥9 be metal organic graphs, then for 1≤k≤n, i∈2k−1, 1≤t≤n, Ret<Rvi,vj and Rvi,vj∩∪t=1nRet≥Ret:(a)
j∈i+1.(b)
j∈i+2,i+6.(c)
j∈i+3,i+7.(d)
j∈i+4,i+8.(e)
j∈i+5.Proof.
(a)
The resolving neighborhood for1≤k≤n, i∈2k−1j∈i+1, 1≤t≤n,(9)Rvi,vj=up:1≤p≤n,wherep≠n+12+k=m,vq:1≤q≤2n,whereq≠2m,2m−1,withRvi,vj=3n−3>8=Ret and Rvi,vj∩∪t=1nRet=vq:1≤q≤2n:q≠2m,2m−1. Therefore, we have Rvi,vj∩∪t=1nRet=2n−2>Ret.(b)
The resolving neighborhood for1≤k≤n, i∈2k−1j∈i+2,i+6, 1≤t≤n.Whenj∈i+2,(10)Rvi,vj=up:1≤p≤n,wherep≠n+2i+12=m,vq:1≤q≤2n,whereq≠2m,2m−1,i+1.Whenj∈i+6,(11)Rvi,vj=up:1≤p≤n,wherep≠n+2i+32=m,vq:1≤q≤2n,whereq≠2m,2m−1,i+3,withRvi,vj=3n−4>8=Ret, Rvi,vj∩∪t=1nRet=vq:1≤q≤2n:q≠2m,2m−1,i+1, and j∈i+2. Rvi,vj∩∪t=1nRet=vq:1≤q≤2n:q≠2m,2m−1,i+3 and j∈i+6. Therefore, we have Rvi,vj∩∪t=1nRet=2n−3>Ret.(c)
The resolving neighborhood for1≤k≤n, i∈2k−1j∈i+3,i+7, 1≤t≤n.Whenj∈i+3,(12)Rvi,vj=up:1≤p≤n,wherep≠n+2i+12=m,vq:1≤q≤2n,whereq≠2m,2m−1.Whenj∈i+7,(13)Rvi,vj=up:1≤p≤n,wherep≠n+2i+32=m,vq:1≤q≤2n,whereq≠2m,2m−1,withRvi,vj=3n−3>8=Ret, Rvi,vj∩∪t=1nRet=vq:1≤q≤2n:q≠2m,2m−1, and j∈i+3. Rvi,vj∩∪t=1nRet=vq:1≤q≤2n:q≠2m,2m−1 and j∈i+7. Therefore, we have Rvi,vj∩∪t=1nRet=2n−2>Ret.(d)
The resolving neighborhood for1≤k≤n, i∈2k−1j∈i+4,i+8, 1≤t≤n.Whenj∈i+4,(14)Rvi,vj=up:1≤p≤n,wherep≠j−12,vq:1≤q≤2n,whereq≠i+j2.Whenj∈i+8,(15)Rvi,vj=up:1≤p≤n,wherep≠j−32,vq:1≤q≤2n,whereq≠i+j2,withRvi,vj=3n−2>8=Ret and Rvi,vj∩∪t=1nRet=vq:1≤q≤2n:q≠i+j/2. Therefore, we have Rvi,vj∩∪t=1nRet=2n−1>Ret.(e)
The resolving neighborhood for1≤k≤n, i∈2k−1j∈i+5, 1≤t≤n.(16)Rvi,vj=up:1≤p≤n,wherep≠j−22,vq:1≤q≤2n,withRvi,vj=3n−1>8=Ret and Rvi,vj∩∪t=1nRet=vq:1≤q≤2n. Therefore, we have Rvi,vj∩∪t=1nRet=2n>Ret.Lemma 5.
LetMOGn for n≡1mod2 and n≥9 be metal organic graph. Then, for 1≤k≤n, i∈2k, 1≤t≤n, Ret<Rvi,vj and Rvi,vj∩∪t=1nRet≥Ret:(a)
j∈i+2,i+6.(b)
j∈i+3,i+7.(c)
j∈i+4,i+8.(d)
j∈i+5.Proof.
(a)
The resolving neighborhood for1≤k≤n, i∈2kj∈i+2,i+6, 1≤t≤n.Whenj∈i+2,(17)Rvi,vj=up:1≤p≤n,wherep≠n+i+12=m,vq:1≤q≤2n,whereq≠2m,2m−1,i+1.Whenj∈i+6,(18)Rvi,vj=up:1≤p≤n,wherep≠n+i+32=m,vq:1≤q≤2n,whereq≠2m,2m−1,i+3,withRvi,vj=3n−4>8=Ret, Rvi,vj∩∪t=1nRet=vq:1≤q≤2n:q≠2m,2m−1,i+1, and j∈i+2. Rvi,vj∩∪t=1nRet=vq:1≤q≤2n:q≠2m,2m−1,i+3 and j∈i+6. Therefore, we have Rvi,vj∩∪t=1nRet=2n−3>Ret.(b)
The resolving neighborhood for1≤k≤n, i∈2kj∈i+3,i+7, 1≤t≤n.(19)Rvi,vj=up:1≤p≤n,wherep≠j+i+14,vq:1≤q≤2n,withRvi,vj=3n−1>8=Ret and Rvi,vj∩∪t=1nRet=vq:1≤q≤2n. Therefore, we have Rvi,vj∩∪t=1nRet=2n>Ret.(c)
The resolving neighborhood for1≤k≤n, i∈2kj∈i+4,i+8, 1≤t≤n,(20)Rvi,vj=up:1≤p≤n,wherep≠i+j4,vq:1≤q≤2n:q≠i+j2,withRvi,vj=3n−2>8=Ret and Rvi,vj∩∪t=1nRet=vq:1≤q≤2n:q≠i+j/2. Therefore, we have Rvi,vj∩∪t=1nRet=2n−1>Ret.(d)
The resolving neighborhood for1≤k≤n, i∈2kj∈i+5, 1≤t≤n.(21)Rvi,vj=up:1≤p≤n,wherep≠n+i+32=m,vq:1≤q≤2n,whereq≠2m,2m−1,withRvi,vj=3n−3>8=Ret and Rvi,vj∩∪t=1nRet=vq:1≤q≤2n:q≠2m,2m−1. Therefore, we have Rvi,vj∩∪t=1nRet=2n−2>Ret.Lemma 6.
LetMOGn for n≡1mod2 and n≥9 be metal organic graph. Then, the following holds.(a)
For1≤k≤n, j=i+1, i∈2k, 1≤t≤n, Ret<Rvivj and Rvi,vj∩∪t=1nRet=≥Ret.Proof.
The resolving neighborhood for1≤i≤n, j∈9+i+k:1≤k≤2n−18.
Whenk=4t−3,4t−2, for 1≤t≤n−9/2,(22)Rvi,vj=up:1≤p≤n,wherep≠n+i+62=m,vq:1≤q≤2n,whereq≠2m,2m−1.
Whenk=4t−1,4t, for 1≤t≤n−9/2,(23)Rvi,vj=up:1≤p≤n,wherep≠i+72=m,vq:1≤q≤2n,whereq≠2m,2m−1,with Rvi,vj=3n−3>8=Ret and Rvi,vj∩∪t=1nRet=vq:1≤q≤2n:q≠2m,2m−1. Therefore, we have Rvi,vj∩∪t=1nRet=2n−2>Ret.Corollary 1.
(i)
For1≤k≤n, i∈2k−1, Rvi,vj=Rvi,vm, where j∈i+s:2≤s≤8 and m∈i−s:2≤s≤8.(ii)
For1≤k≤n, i∈2kRvi,vj=Rvi,vj, where j∈i+s,i−s:2≤s≤7.Lemma 7.
LetMOGn for n≡1mod2 and n≥9 be metal organic graph. Then, for 1≤i≤n,1≤k≤n−1, j=i+k, 1≤t≤n, Ret<Ruiuj and Rui,uj∩∪t=1nRet=≥Ret.Proof.
The resolving neighborhood for1≤i≤n, 1≤k≤n−1, j=i+k.
Whenk≅1mod2,(24)Rui,uj=up:1≤p≤n,wherep≠n+i+j2=m,vq:1≤q≤2n,whereq≠2m,2m−1.
Whenk≅0mod2,(25)Rui,uj=up:1≤p≤n,wherep≠i+j2=m,vq:1≤q≤2n,whereq≠2m,2m−1,with Rui,uj=3n−3>8=Ret and Rui,uj∩∪t=1nRet=vq:1≤q≤2n,q≠2m,2m−1. Therefore, we have Rui,uj∩∪t=1nRet=2n−2>Ret.Lemma 8.
LetMOGn for n≡1mod2 and n≥9 be metal organic graph. Then, for 1≤i≤n,Ret<Rui,vj and Rui,uj∩∪t=1nRet≥Ret.(a)
j∈2i−1,2i.(b)
j∈2i+1.(c)
j∈2i+2.(d)
j∈2i+3,2i+4.(e)
j∈2i+5.(f)
j∈2i+6.(g)
j∈2i+7.(h)
j∈2i+8.(i)
j∈2i+9.(j)
j∈2i+10.(k)
j∈2i+10+k:1≤k≤2n−22.Proof:(a)
The resolving neighborhood for1≤i≤n.Whenj∈2i−1,(26)Rui,vj=up:1≤p≤n,vq:1≤q≤2n:q≠j+1,j+2,j−2,j−3.Whenj∈2i,(27)Rui,vj=up:1≤p≤n,vq:1≤q≤2n,where,q≠j+2,j+3,j−1,j−1,withRui,vj=3n−4>8=Ret and Rui,uj∩∪t=1nRet=vq:1≤q≤2n:q≠j+1,j+2,j−2,j−3, when j∈2i−1. Rui,vj∩∪t=1nRet=vq:1≤q≤2n:q≠j+2,j+3,j−1,j−1, when j∈2i. Therefore, we have Rui,uj∩∪t=1nRet=2n−4>Ret.(b)
The resolving neighborhood for1≤i≤n, j∈2i+1:(28)Rui,vj=up:1≤p≤n,wherep≠mandj−i≤m≤n+12,vq:1≤q≤2n,whereq≠j−1,swherej+3≤s≤n+j−2,withRui,vj=3n−16>8=Ret and Rui,vj∩∪t=1nRet=vq:1≤q≤2n:q≠j−1,s,where,j+3≤s≤n+j−2. Therefore, we have Rui,vj∩∪t=1nRet=2n−7>Ret.(c)
The resolving neighborhood for1≤i≤n, j∈2i+2.Whenj∈i+3,(29)Rui,vj=up:1≤p≤n,wherep≠m,andj−i−1≤mn+12,vq:1≤q≤2n,whereq≠sand2j−i−1≤s≤n+j−4,withRui,vj=3n−13>8=Ret and Rui,vj∩∪t=1nRet=vq:1≤q≤2n:q≠s,2j−i−1≤s≤n+j−4. Therefore, we have Rui,vj∩∪t=1nRet=2n−8>Ret.(d)
The resolving neighborhood for1≤i≤n, j∈2i+3,2i+4:(30)Rui,vj=up:1≤p≤n,wherep≠n+12=s,vq:1≤q≤2n,whereq≠j−2,q≠2s,2s−1,withRui,vj=3n−4>8=Ret and Rui,vj∩∪t=1nRet=vq:1≤q≤2n:q≠j−2,q≠2s,2s−1. Therefore, we have Rui,vj∩∪t=1nRet=2n−3>Ret.(e)
The resolving neighborhood for1≤i≤n, j∈2i+5:(31)Rui,vj=up:1≤p≤n,wherep≠j−12,vq:1≤q≤2n,withRui,vj=3n−1>8=Ret and Rui,vj∩∪t=1nRet=vq:1≤q≤2n. Therefore, we have Rui,vj∩∪t=1nRet=2n>Ret.(f)
The resolving neighborhood for1≤i≤n, j∈2i+6:(32)Rui,vj=up:1≤p≤n,wherep≠j−12,vq:1≤q≤2n,whereq≠j−3,withRui,vj=3n−2>8=Ret and Rui,vj∩∪t=1nRet=vq:1≤q≤2n:q≠j−3. Therefore, we have Rui,vj∩∪t=1nRet=2n−1>Ret.(g)
The resolving neighborhood for1≤i≤n, j∈2i+3,2i+4:(33)Rui,vj=up:1≤p≤n,wherep≠n+2i+32=s,vq:1≤q≤2n,whereq≠j−3,q≠2s,2s−1,withRui,vj=3n−4>8=Ret and Rui,vj∩∪t=1nRet=vq:1≤q≤2n:q≠j−3,q≠2s,2s−1. Therefore, we have Rui,vj∩∪t=1nRet=2n−3>Ret.(h)
The resolving neighborhood for1≤i≤n, j∈2i+8:(34)Rui,vj=up:1≤p≤n,wherep≠n+2i+32=s,vq:1≤q≤2n,whereq≠2s,2s−1,withRui,vj=3n−3>8=Ret and Rui,vj∩∪t=1nRet=vq:1≤q≤2n:q≠2s,2s−1. Therefore, we have Rui,vj∩∪t=1nRet=2n−2>Ret.(i)
The resolving neighborhood for1≤i≤n, j∈2i+9,(35)Rui,vj=up,1≤p≤n,wherep≠j−32=s,vq,1≤q≤2n,whereq≠j−4,withRui,vj=3n−2>8=Ret and Rui,vj∩∪t=1nRet=vq:1≤q≤2n:q≠j−4. Therefore, we have Rui,vj∩∪t=1nRet=2n−1>Ret.(j)
The resolving neighborhood for1≤i≤n, j∈2i+9:(36)Rui,vj=up:1≤p≤n,wherep≠j−42=s,vq:1≤q≤2n,whereq≠2s,2s−1,withRui,vj=3n−3>8=Ret and Rui,vj∩∪t=1nRet=vq:1≤q≤2n:q≠2s,2s−1. Therefore, we have Rui,vj∩∪t=1nRet=2n−2>Ret.(k)
The resolving neighborhood for1≤i≤n, j∈10+2i+k:1≤k≤2n−22.Whenk≅1mod4,2mod4,(37)Rui,vj=up:1≤p≤n,wherep≠n+72=m,vq:1≤q≤2n,whereq≠2m,2m−1.Whenk≅0mod4,3mod4,(38)Rui,vj=up:1≤p≤n,wherep≠i+42=s,vq:1≤q≤2n,whereq≠2s,2s−1,withRvi,vj=3n−3>8=Ret and Rui,vj∩∪t=1nRet=vq:1≤q≤2n,q≠2m,2m−1, when k≅1mod4,2mod4andRui,vj∩∪t=1nRet=vq:1≤q≤2n,q≠2s,2s−1, when k≅1mod4,2mod4. Therefore, we have Rui,vj∩∪t=1nRet=2n−2>Ret.Corollary 2.
For1≤i≤n, Rui,vj=Rui,vm, where j∈2i+s:−1≤s≤10 and m∈2i+s:−2≤s≤−9.Table 1
FMD of metal organic graphs.
Resolving sets(n=3)ElementsRe1=Ru1,u2VMOG3−u3,v5,v6Re2=Ru1,u3VMOG3−u2,v3,v4Re3=Ru1,v1VMOG3−v2,v3,v5Re4=Ru1,v2VMOG3−v1,v4,v6Re5=Ru1,v3VMOG3−u2,v2,v5Re6=Ru1,v6VMOG3−u3,v1,v4Re7=Ru2,u3VMOG3−u1,v1,v2Re8=Ru2,v2VMOG3−u1,v3,v6Re9=Ru2,v3VMOG3−v1,v4,v5Re10=Ru2,v4VMOG3−v2,v3,v6Re11=Ru2,v5VMOG3−u3,v1,v4Re12=Ru3,v1VMOG3−u1,v3,v6Re13=Ru3,v4VMOG3−u2,v2,v5Re14=Ru3,v5VMOG3−v1,v3,v6Re15=Ru3,v6VMOG3−v2,v4,v5Re16=Rv1,v2VMOG3−u1,u2,u3Re17=Rv1,v3VMOG3−u3,v2,v5Re18=Rv1,v5VMOG3−u2,v3,v6Re19=Rv2,v4VMOG3−u3,v3,v6Re20=Rv2,v6VMOG3−u2,v1,v4Re21=Rv3,v4VMOG3−u1,u2,u3Re22=Rv3,v5VMOG3−u1,v1,v4Re23=Rv4,v6VMOG3−u1,v4,v5Re24=Rv5,v6VMOG3−u1,u2,u3Table 2
FMD of metal organic graphs.
Resolving sets(n=3)ElementsR1=Ru1,v4VMOG3−u2,v6R2=Ru1,v5VMOG3−u3,v3R3=Ru2,v1VMOG3−u1,v5R4=Ru2,v6VMOG3−u3,v2R5=Ru3,v2VMOG3−u1,v4R6=Ru3,v3VMOG3−u2,v1R7=Rv1,v4VMOG3−u3R8=Rv1,v6VMOG3−u5R9=Rv2,v3VMOG3−u3R11=Rv3,v6VMOG3−u1R12=Rv4,v5VMOG3−u1Theorem 2.
The FMD of metal organic graphMOGn for 3≤n≤7,n≅1mod2 is(39)dimfMOGn=32ifn=3,54ifn=5,74ifn=7.Table 3
FMD of metal organic graphs.
Resolving sets(n=5)ElementsRe1=Rv1,v2VMOG5−u1,u2,u3,u4,u5,v6,v7Re1=Rv3,v4VMOG5−u1,u2,u3,u4,u5,v8,v9Re1=Rv5,v6VMOG5−u1,u2,u3,u4,u5,v1,v10Re1=Rv7,v8VMOG5−u1,u2,u3,u4,u5,v2,v3Re1=Rv9,v10VMOG5−u1,u2,u3,u4,u5,v4,v5Table 4
FMD of metal organic graphs.
Resolving sets(n=5)ElementsR1=Ru1,u2VMOG5−u4,v7,v8R2=Ru1,u3VMOG5−u2,v3,v4R3=Ru1,u4VMOG5−u5,v9,v10R4=Ru1,u5VMOG5−u3,v5,v06R5=Ru1,v1VMOG5−v2,v3,v8,v09R6=Ru1,v2VMOG5−v1,v4,v5,v10R7=Ru1,v3VMOG5−u2,u3,v2,v6R8=Ru1,v4VMOG5−u2,u3,v7R9=Ru1,v5VMOG5−u4,v3,v8R10=Ru1,v6VMOG5−u4,v4R11=Ru1,v7VMOG5−u3,v9R12=Ru1,v8VMOG5−u3,v5,v10R13=Ru1,v9VMOG5−u4,u5,v6R14=Ru1,v10VMOG5−u4,u5,v1,v7R15=Ru2,u3VMOG5−u5,v9,v10R16=Ru2,u4VMOG5−u3,v5,v6R17=Ru2,u5VMOG5−u1,v1,v2R18=Ru2,v1VMOG5−u1,u5,v8R19=Ru2,v2VMOG5−u1,u5,v9R20=Ru2,v3VMOG5−v1,v4,v5,v10R21=Ru2,v4VMOG5−v2,v3,v6,v7R22=Ru2,v5VMOG5−u3,u4,v4,v8R23=Ru2,v6VMOG5−u3,u4,v9R24=Ru2,v7VMOG5−u5,v5,v10R25=Ru2,v8VMOG5−u5,v6R26=Ru2,v9VMOG5−u4,v1R27=Ru2,v10VMOG5−u4,v2,v7R28=Ru3,u4VMOG5−u1,v1,v2R29=Ru3,u5VMOG5−u4,v7,v8R30=Ru3,v1VMOG5−u5,v3R31=Ru3,v2VMOG5−u5,v4,v9R32=Ru3,v3VMOG5−u1,u2,v10R33=Ru3,v4VMOG5−u1,u2,v1,v5R34=Ru3,v5VMOG5−v2,v3,v6,v7R35=Ru3,v6VMOG5−v4,v5,v8,v9R36=Ru3,v7VMOG5−u4,u5,v6,v10R37=Ru3,v8VMOG5−u4,u5,v1R38=Ru3,v9VMOG5−u1,v2,v7R39=Ru3,v10VMOG5−u1,v8R40=Ru4,u5VMOG5−u2,v3,v4R41=Ru4,v1VMOG5−u2,v4,v9R42=Ru4,v2VMOG5−u2,v10R43=Ru4,v3VMOG5−u1,v5R44=Ru4,v4VMOG5−u1,v1,v6R45=Ru4,v5VMOG5−u2,u3,v2R46=Ru4,v6VMOG5−u2,u3,v2,v7R47=Ru4,v7VMOG5−v4,v5,v8,v9R48=Ru4,v8VMOG5−v1,v6,v7,v10R49=Ru4,v9VMOG5−u1,u5,v2,v8R50=Ru4,v10VMOG5−u1,u5,v3R51=Ru5,v1VMOG5−u1,u2,v4,v10R52=Ru5,v2VMOG5−u1,u2,v5R53=Ru5,v3VMOG5−u3,v1,v6R54=Ru5,v4VMOG5−u3,v2R55=Ru5,v5VMOG5−u2,v7R56=Ru5,v6VMOG5−u2,v3,v8R57=Ru5,v7VMOG5−u3,u4,v4R58=Ru5,v8VMOG5−u3,u4,v5,v9R59=Ru5,v9VMOG5−v1,v6,v7,v9R60=Ru5,v10VMOG5−v2,v3,v8,v9R61=Rv01,v03VMOG5−u4,v2,v7R62=Rv01,v04VMOG5−u4R63=Rv01,v05VMOG5−u2,v3,v8R64=Rv01,v06VMOG5−u2R65=Rv01,v07VMOG5−u5,v4,v9R66=Rv01,v08VMOG5−u5R67=Rv01,v09VMOG5−u3,v5R68=Rv01,v10VMOG5−u3,v5,v6R69=Rv02,v03VMOG5−u4,v7,v8R70=Rv02,v04VMOG5−u4,v3,v8R71=Rv02,v05VMOG5−u2R72=Rv02,v06VMOG5−u2,v4,v9R73=Rv02,v07VMOG5−u5R74=Rv02,v08VMOG5−u5,v5,v10R75=Rv02,v09VMOG5−u6R76=Rv02,v10VMOG5−u3,v1,v6R77=Rv03,v05VMOG5−u5,v4,v9R78=Rv03,v06VMOG5−u5R79=Rv03,v07VMOG5−u3,v5,v10R80=Rv03,v08VMOG5−u3R81=Rv03,v09VMOG5−u1,v1,v6R82=Rv03,v10VMOG5−u1R83=Rv04,v05VMOG5−u5,v9,v10R84=Rv04,v06VMOG5−u5,v5,v10R85=Rv04,v07VMOG5−u3R86=Rv04,v08VMOG5−u3,v1,v6R87=Rv04,v09VMOG5−u1R88=Rv04,v10VMOG5−u1,v2,v7R89=Rv05,v07VMOG5−u1,v1,v6R90=Rv05,v08VMOG5−u1R91=Rv05,v09VMOG5−u4,v42,v7R92=Rv05,v10VMOG5−u4R93=Rv06,v07VMOG5−u1,v1,v2R94=Rv06,v08VMOG5−u1,v2,v7R95=Rv06,v09VMOG5−u4R96=Rv06,v10VMOG5−u4,v3,v8R97=Rv07,v09VMOG5−u2,v3,v8R98=Rv07,v10VMOG5−u2R99=Rv08,v09VMOG5−u2,v3,v4R100=Rv08,v10VMOG5−u2,v4,v9Proof.
Case 1: whenn=3, then the RNs are as follows.Since, for1≤t≤24, the cardinality of each RN Ret is 6, as given in Table 1, which is less than the cardinalities of all other RNs Rm of MOG3, as given in Table 2, where 1≤m≤12. Moreover, ∪t=124Ret=VMOG3; this implies that ∪t=124Ret=9 and Rm∩∪t=124Ret>Ret|=6.Consequently,dimfMOG3=∑t=191/6≤3/2.Case 2: whenn=5, as shown in Figure 1, the RNs are as follows.Since, for1≤t≤5, the cardinality of each RN Ret is 8, as given in Table 3, which is less than the cardinalities of all other RNs Rm of MOG5, as given in Table 4, where 1≤m≤100. Moreover, ∪t=15Ret=VMOG5; this implies that ∪t=15Ret=10 and Rm∩∪t=15Ret>Ret|=8.Consequently,dimfMOG5≤∑t=1101/8=5/4.Case 3: whenn=7, as shown in Figure 1, then the RNs are as follows.
Since, for1≤t≤7, the cardinality of each RN Ret is 8, as given in Table 5, which is less than the cardinalities of all other RNs Rm of MOG5, as given in Table 6, where 1≤m≤203. Moreover, ∪t=17Ret=VMOG7; this implies that ∪t=17Ret=14 and Rm∩∪t=17Ret>Ret|=8.
Consequently,dimfMOG5≤∑t=1141/8≤7/4.Table 5
FMD of metal organic graphs.
Resolving sets(n=5)ElementsRe1=Rv1,v2v1,v2,v3,v4,v5,v12,v13,v14Re2=Rv3,v4v1,v2,v3,v4,v5,v6,v7,v14Re3=Rv5,v6v2,v3,v4,v5,v6,v7,v8,v9Re4Rv7,v8v4,v5,v6,v7,v8,v9,v10,v11Re5=Rv9,v10v6,v7,v8,v9,v10,v11,v12,v13Re6=Rv11,v12v1,v8,v9,v10,v11,v12,v13,v14Re7=Rv13,v14v1,v2,v3,v10,v11,v12,v13,v14Table 6
FMD of MOG(7).
Resolving sets(n=7)ElementsR1=Ru1,u2VMOG7−u5,v9,v10R2=Ru1,u3VMOG7−u2,v3,v4R3=Ru1,u4VMOG7−u6,v11,v12R4=Ru1,u5VMOG7−u3,v5,v06R5=Ru1,u6VMOG7−u7,v13,v14R6=Ru1,u7VMOG7−u4,v7,v08R7=Ru1,v1VMOG7−v2,v3,v12,v13R8=Ru1,v2VMOG7−v1,v4,v5,v14R9=Ru1,v3VMOG7−u2,u3,u4,v2,v6,v7,v8R10=Ru1,v4VMOG7−u2,u3,u4,v8R11=Ru1,v5VMOG7−u5,v3,v9,v10R12=Ru1,v6VMOG7−u5,v4,v10R13=Ru1,v7VMOG7−u3R14=Ru1,v8VMOG7−u3,v5,v11R15=Ru1,v9VMOG7−u6,v6,v12R16=Ru1,v10VMOG7−u6R17=Ru1,v11VMOG7−u4,v7R18=Ru1,v12VMOG7−u4,v7,v8,v14R19=Ru1,v13VMOG7−u5,u6,u7,v9R20=Ru1,v14VMOG7−u5,u6,u7,v1,v9R21=Ru2,u3VMOG7−u6,v11,v12R22=Ru2,u4VMOG7−u3,v5,v6R23=Ru2,u5VMOG7−u7,v13,v14R24=Ru2,u6VMOG7−u4,v7,v8R25=Ru2,u7VMOG7−u1,v1,v2,R26=Ru2,v1VMOG7−u1,u6,u7,v11,R27=Ru2,v2VMOG7−u1,u6,u7,v3,u11,v12,v13R28=Ru2,v3VMOG7−v1,v4,v5,v14R29=Ru2,v4VMOG7−v2,v3,v6,v7R30=Ru2,v5VMOG7−u3,u4,u5,v4,v8,v9,v10R31=Ru2,v6VMOG7−u3,u4,u5,v10R32=Ru2,v7VMOG7−u6,v5,v11,v12R33=Ru2,v8VMOG7−u6,v6,v12R34=Ru2,v9VMOG7−u4R35=Ru2,v10VMOG7−u4,v7,v13R36=Ru2,v11VMOG7−v8,v14R37=Ru2,v12VMOG7−u7R38=Ru2,v13VMOG7−u5,v1,v9R39=Ru2,v14VMOG7−u5,v2,v9,v10R40=Ru3,u4VMOG7−u7,v13,v14,R41=Ru3,u5VMOG7−u4,v7,v8R42=Ru3,u6VMOG7−u1,v1,v2R43=Ru3,u7VMOG7−u5,v9,v10R44=Ru3,v1VMOG7−u6,v3,v11R45=Ru3,v2VMOG7−u6,v4,v11,v12R46=Ru3,v3VMOG7−u1,u2,u7,v13,v14R47=Ru3,v4VMOG7−u1,u2,u7,v1,v5,v13,v14R48=Ru3,v5VMOG7−v2,v3,v6,v7R49=Ru3,v6VMOG7−v4,v5,v8,v9R50=Ru3,v7VMOG7−u4,u5,u6,v6,v10,v11,v12R51=Ru3,v8VMOG7−u4,u5,u6,v12R52=Ru3,v9VMOG7−u7,v7,v13R53=Ru3,v10VMOG7−u7,v8,v14R54=Ru3,v11VMOG7−u5R55=Ru3,v12VMOG7−u5,v1R56=Ru3,v13VMOG7−u1,u2,v10R57=Ru3,v14VMOG7−u1,v10R58=Ru4,u5VMOG7−u1,v1,v2R59=Ru4,u6VMOG7−u5,v9,v10R60=Ru4,u7VMOG7−u2,v3,v4R61=Ru4,v1VMOG7−u2,u7,v4,v12R62=Ru4,v2VMOG7−u2,u7R63=Ru4,v3VMOG7−u7,v5,v13R64=Ru4,v4VMOG7−u7,u6,v13,v14R65=Ru4,v5VMOG7−u2,u3,v1R66=Ru4,v6VMOG7−u2,u3,u7,v1,v2,v3,v7R67=Ru4,v7VMOG7−v4,v5,v8,v9R68=Ru4,v8VMOG7−v6,v7,v10,v11R69=Ru4,v9VMOG7−u5,u6,v7,v8,v12,v13R70=Ru4,v10VMOG7−u5,u6,u7,v14R71=Ru4,v11VMOG7−u1,v1,v2,v9R72=Ru4,v12VMOG7−u1,v1,v2,v10R73=Ru4,v13VMOG7−u6,v11R74=Ru4,v14VMOG7−u6,v3,v11R75=Ru5,u6VMOG7−u2,v3,v4R76=Ru5,u7VMOG7−u6,v11,v12R77=Ru5,v1VMOG7−u7R78=Ru5,v2VMOG7−u7,u5,v13R79=Ru5,v3VMOG7−u3,v6,v14R80=Ru5,v4VMOG7−u3R81=Ru5,v5VMOG7−u1,v1,v7R82=Ru5,v6VMOG7−u1,v1,v2,v8R83=Ru5,v7VMOG7−u2,u3,u4,v3R84=Ru5,v8VMOG7−u2,u3,u4,v3,v4,v5,v9R85=Ru5,v9VMOG7−v6,v7,v10,v11R86=Ru5,v10VMOG7−v8,v9,v12,v13R87=Ru5,v11VMOG7−u1,u6,u7,v1v2,v10,v14R88=Ru5,v12VMOG7−u1,u6,u7,v2R89=Ru5,v13VMOG7−u2,v3,v4,v11R90=Ru5,v14VMOG7−u2,v4,v12R91=Ru6,u7VMOG7−u3,v5,v6R92=Ru6,v1VMOG7−u3,v5,v6,v13R93=Ru6,v2VMOG7−u1,v6R94=Ru6,v3VMOG7−u1R95=Ru6,v4VMOG7−u1,v1R96=Ru6,v5VMOG7−u1,u4,v2,v8R97=Ru6,v6VMOG7−u1,u4R98=Ru6,v7VMOG7−u2,v3,v9R99=Ru6,v8VMOG7−u2,v3,v4,v10R100=Ru6,v9VMOG7−u3,u4,u5,v5R101=Ru6,v10VMOG7−u3,u4,u5,v5,v6,v7R102=Ru6,v11VMOG7−v8,v9,v12,v13R103=Ru6,v12VMOG7−v1,v10,v11,v14R104=Ru6,v13VMOG7−u1,u2,u7,v2,v3,v4R105=Ru6,v14VMOG7−u1,u2,u7,v4R106=Ru7,v1VMOG7−u1,u2,u3,v4,v5,v6,v14R107=Ru7,v2VMOG7−u1,u2,u3,v6R108=Ru7,v3VMOG7−u4,v1,v7,v8R109=Ru7,v4VMOG7−u4,v2,v8R110=Ru7,v5VMOG7−u2R111=Ru7,v6VMOG7−u2,v3,v9R112=Ru7,v7VMOG7−u5,v4,v10R113=Ru7,v8VMOG7−u5,v4R114=Ru7,v9VMOG7−u3,v5,v11R115=Ru7,v10VMOG7−u3,v5,v6,v12R116=Ru7,v11VMOG7−u4,u5,u6,v7R117=Ru7,v12VMOG7−u4,u5,u6,v7,v8,v9,v13R118=Ru7,v13VMOG7−v1,v10,v11,v14R119=Ru7,v14VMOG7−v2,v3,v12,v13R120=Rv1,v3VMOG7−u5,v2,v9,v10R121=Rv1,v4VMOG7−u5,v9,v10R122=Rv1,v5VMOG7−u2,v3R123=Rv1,v6VMOG7−u2R124=Rv1,v7VMOG7−u6,v4,v11R125=Rv1,v8VMOG7−u6R126=Rv1,v9VMOG7−u3,v5,v12R127=Rv1,v10VMOG7−u3,v5,v6R128=Rv1,v11VMOG7−u7,v13R129=Rv1,v12VMOG7−u7R130=Rv1,v13VMOG7−u4,v7,v8,v14R131=Rv1,v14VMOG7−u4,v7,v8R132=Rv2,v3VMOG7−u5,v9,v10R133=Rv2,v4VMOG7−u5,v3,v9,v10R134=Rv2,v5VMOG7−u2R135=Rv2,v6VMOG7−u2,v4R136=Rv2,v7VMOG7−u6,u11,v12R137=Rv2,v8VMOG7−u6,v5,v12R138=Rv2,v9VMOG7−u3R139=Rv2,v10VMOG7−u3,v6,v13R140=Rv2,v11VMOG7−u7R141=Rv2,v12VMOG7−u7,v14R142=Rv2,v13VMOG7−u4,v7,v8R143=Rv2,v14VMOG7−u4,v1,v7,v8R144=Rv3,v5VMOG7−u6,v4,v11,v12R145=Rv3,v6VMOG7−u6,v11,v12R146=Rv3,v7VMOG7−u3,v5R147=Rv3,v8VMOG7−u3R148=Rv3,v9VMOG7−u7,v6,v13R149=Rv3,v10VMOG7−u7R150=Rv3,v11VMOG7−u4,v7,v14R151=Rv3,v12VMOG7−u4,v7,v8R152=Rv3,v13VMOG7−u1,v1R153=Rv3,v14VMOG7−u1R154=Rv4,v5VMOG7−u6,v11,v12R155=Rv4,v6VMOG7−u6,v5,v11,v12R156=Rv4,v7VMOG7−u3R157=Rv4,v8VMOG7−u3,v6R158=Rv4,v9VMOG7−u7,v13,v14R159=Rv4,v10VMOG7−u7,v7,v14R160=Rv4,v11VMOG7−u4R161=Rv4,v12VMOG7−u4,v1,v8R162=Rv4,v13VMOG7−u1R163=Rv4,v14VMOG7−u1,v2R164=Rv5,v7VMOG7−u7,v6R165=Rv5,v8VMOG7−u7,u12,v14R166=Rv5,v9VMOG7−u4,v7R167=Rv5,v10VMOG7−u4R168=Rv5,v11VMOG7−u1,v1,v8R169=Rv5,v12VMOG7−u1R170=Rv5,v13VMOG7−u5,v2,v9R171=Rv5,v14VMOG7−u5,v9,v10R172=Rv6,v7VMOG7−u7,v13,v14R173=Rv6,v8VMOG7−u7,v7,v13,v14R174=Rv6,v9VMOG7−u4R175=Rv6,v10VMOG7−u4,v8,R176=Rv6,v11VMOG7−u1,v1,v2R177=Rv6,v12VMOG7−u1,v2,v9R178=Rv6,v13VMOG7−u5R179=Rv6,v14VMOG7−u5,v3,v10R180=Rv7,v9VMOG7−u1,v1,v2,v8R181=Rv7,v10VMOG7−u1,v1,v2R182=Rv7,v11VMOG7−u5,v9R183=Rv7,v12VMOG7−u5R184=Rv7,v13VMOG7−u3,v3,v10R185=Rv7,v14VMOG7−u2R186=Rv8,v9VMOG7−u1,v1,v2R187=Rv8,v10VMOG7−u1,v1,v2,v9R188=Rv8,v11VMOG7−u5R189=Rv8,v12VMOG7−u5,v10R190=Rv8,v13VMOG7−u2,v3,v4R191=Rv8,v14VMOG7−u2,v4,v11R192=Rv9,v11VMOG7−u2,v3,v4,v10R193=Rv9,v12VMOG7−u2,v3,v4R194=Rv9,v13VMOG7−u6,v11R195=Rv9,v14VMOG7−u6R196=Rv10,v11VMOG7−u2,v3,v4R197=Rv10,v12VMOG7−u2,v3,v4,v11R198=Rv10,v13VMOG7−u6R199=Rv10,v14VMOG7−u6,v12R200=Rv11,v13VMOG7−u3,v5,v6,v12R201=Rv11,v14VMOG7−u3,v5,v6R202=Rv12,v13VMOG7−u3,v5,v6R203=Rv12,v14VMOG7−u3,v5,v6,v13Theorem 3.
LetMOGn for n≥9 and n≅1mod2 be the metal organic graph. Then, dimlfMOGn≤n/4.Proof.
In view of Lemmas3–8 for 1≤k≤n, j=i+1, i∈2k−1, 1≤t≤n, Ret=Rvivj=8 and X=∪t=1nRet=2n. Also, we have Rxy≤Ret for all xy∈EMOGn. Moreover, the local resolving neighbourhood of minimum cardinality is not disjoint. Therefore, the fractional metric of MOGn is given as follows:(40)dimfMOGn≤∑t=1X1Ret.
ForX=2n and Ret=8, we have(41)dimfMOGn≤∑t=12n18.
Hence,dimfMOGn≤n/4.


## 5. Conclusion

In this section, we conclude the obtained results as follows:(i)
The FMD ofMOGn for n≡1mod2 is obtained as given in Table 7.(ii)
We note that as we increasen in MOGn for n≡1mod2, 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.Table 7
FMD of metal organic graphs.
MOGn and n≡1mod2Upper bounds of FMDMOG33/2MOG55/4MOG77/4MOGn if n≥9n/4



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*Source: 5539569-2021-03-12.xml*