Fractional metric dimension of metal-organic frameworks

Mohsin Raza; Muhammad Javaid; Naeem Saleem

doi:10.1515/mgmc-2021-0012

Open Access Published by De Gruyter May 9, 2021

Fractional metric dimension of metal-organic frameworks

Mohsin Raza , Muhammad Javaid and Naeem Saleem

From the journal Main Group Metal Chemistry

https://doi.org/10.1515/mgmc-2021-0012

Abstract

Metal-organic frameworks (MOF(n)) are organic-inorganic hybrid crystalline porous materials that consist of a regular array of positively charged metal ions surrounded by organic ‘linker’ molecules. The metal ions form nodes that bind the arms of the linkers together to form a repeating, cage-like structure. Moreover, in a chemical structure or molecular graph, edges and vertices are known as bonds and atoms, respectively. Metric dimension being a subsets of atoms with minimum cardinality is used in the substrcturing of the chemical compounds in the molecular structures. Fractional metric dimension is weighted version of metric dimension that associate a numeric value to the identified subset of atoms. In this paper, we have computed the fractional metric dimension of metal organic framework (MOF(n)) for n ≡ 0(mod)2.

Keywords: metal organic framework; fractional metric dimension; resolving neighborhoods

1 Introduction and preliminaries

Metal organic framework (MOF) is a graph that consists of metal atoms. These atoms are linked with the help of organic ligands which acts like a linker, having large pore volume which is known as pours coordination polymer. Therefore, these frameworks led to a new world of remarkable applications. MOF also have large surface area that allow these chemicals compounds to absorb huge quantity of several gases such as carbon dioxide hydrogen and methane acting as a gas storage chemical compounds. These frameworks are also used for environmental protection and cleaning energy with the help of capturing carbon dioxide. Being small density, high surface, structure edibility and tune able 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 industries and pharmaceutical. MOF is also used in heterogeneous catalyst, drugs delivery and sensing conductivity. Graphical nomenclature of MOF that is represented by a graph G, metal atoms and organic ligands are represented by vertices and edges, respectively (Hasan and Jhung, 2015; Jiao et al., 2019; Liu et al., 2014; Pettinari et al., 2017).

Let G be the connected graph and distance u, v ∈ V(G) is represented by d_G(u, v), which is length of shortest path from u to v in G. Define a resolving set such that, for u, v ∈ V(G), R_G(u, v) = {y ∈ V(G): d(u, y) ≠ d(v, y)}. A vertex set W ⊆ V(G) is called resolving set of G if W ∩ R_G(u, v) ≠ ∅ for any two distinct vertices of G. The minimum cardinality among all the resolving set of G is called metric dimension of G. A resloving function g is called minimal if any function f: V(G) → [0,1] such that f ≤ g and f(v) ≠ g(v) for at least one v ∈ V is not a resolving function of G. The fractional metric dimesnison of G is denoted by dim_f(G) and defined as dim_f(G) = min{|g| : g is minimal resolving function of G}, where |g| = Σ_v∈V(G) g(v).

Metric dimension has several applications in chemistry, e.g., the substructures of a chemical compound which can be denoted by a set of functional groups. Moreover, in a chemical structure or molecular graph edges and vertices are known as bonds and atoms, respectively. Furthermore, the sub graphs are simply deliberated as substructures and functional groups. Now after altering the position of functional groups, the formed collections of compounds are distinguished as substructures being similar to each other. Later on using the method of traditional view, we can investigate if any two compounds hold the same functional group at the same point, while in drugs discovery comparative statement contributes a critical part to determine pharmacology activities related to the feature of compounds.

Metric dimension initially came under investigation through the works of Harary and Melter (1976). Chartrand et al. (2000) used the concept of metric dimension of graphs for the solution integer programming problem (IPPS). Later Currie et al. (2001) used the concept the fractional metric dimension for the better solution of IPPS. Yu et al. (2020) computed the upper bounds of conves polytopes. Mufti et al. (2020) computed the edge metric dimension of barycentric subdivision of Cayley graphs.

Arumugam et al. (2012) introduced many characteristics of the fractional metric dimension for a graph regarding its order. Various results regarding fractional metric dimension of the Cartesian, hierarchical, corona, lexicographic, and comb products of connected graphs, are studied by Feng and Kong (2018), Feng and Wang (2013), and Yi (2015). Feng et al. (2014) find the metric dimension and fractional metric dimension of graphs. Liu et al. (2019) computed the fractional metric dimension of generalized Jahangir graph. Javaid et al. (2020) and Liu et al. (2020) calculated local fractional metric dimensions of different connected graphs.

Let MOF(n) for n ≥ 2 be a metal organic frame works graph with the V(MOF(n)) = {u_k: 1 ≤ k ≤ n} U{v_l: 1 ≤ l ≤ 2n} and E(MOF(n)) = {u_ku_k+1: 1 ≤ k ≤ n − 1} U{v_lv_l+1: 1 ≤ l ≤ 2n − 1} ∪ {u_nu₁, v_2nv₁} ∪ {u_kv_l,u_kv_l+1 1 ≤ k ≤ n, 1 ≤ l ≤ 2n} are vertices and edges set, respectively, as shown in Figure 1 and 2, where |V (MOF (n))| = 3n and |E (MOF (n))| = 5n.

Figure 1

Metal-organic framework (4).

Figure 2

Metal-organic framework (6).

In this paper, we have followed the sequence that first section includes introduction and preliminaries, second section having resolving neighborhood of each possible pair of vertices of (MOF (n)) for n = 0(mod)2. Section 3 consists of main results. The final section of the paper is conclsion.

2 Resolving neighborhoods of metal organic frame works (MOF)

Resolving neighborhood of the metal organic frame work (MOF (n)) for n ≥ 8 and n ≡ 0(mod 2).

Lemma 1

Let MOF (n) be the metal organic frame work for n ≥ 8, n ≡ 0(mod 2). Then, for 1 ≤ k ≤ n, j = i + 1, i ∈ [2k − 1], and 1 ≤ t ≤ n:

|Rt|=|R{vi,vj}|=8.

Moreover, ∪t=1nRt={vm:1≤m≤2n} and β=|∪t=1nRt|=2n .

Proof

The resolving neighborhoods of metal organic framework for 1 ≤ k ≤ n, 1 ≤ t ≤ n, j = i + 1, i ∈ [2k − 1] is:

R(vivj)={vl 2k-1≤l≤k-32k-4≤l≤2k-2

with,

|Rt|=8∪t=1nRt={vp:1≤p≤2n} and |∪t=1nRt|=2n.

Lemma 2

Let MOF (n) be the metal organic frame work for n ≥ 8, (n ≡ 0(mod 2). Then, for 1 ≤ i ≤ n, 1 ≤ k ≤ n − 1, and 1 ≤ t ≤ n:

(a) |R_t| < |R(u_i, u_j)| and |R(ui,uj)∩∪t=1nRt|≥|Rt| , for j − i ≡ 1(mod 2), j ∈ {i + k},

(b) |R_t| < |R(u_i, u_j)| and |R(ui,uj)∩∪t=1nRt|≥|Rt , for j − i ≡ 0(mod 2), j ∈ {i + k}.

Proof

(a) The resolving neighborhood for j − i ≡ 1(mod 2):

R(ui,uj)={up:1≤p≤nvp:1≤p≤2n

with |R(u_i, u_j)| = 3n > 8 = |R_t| and R(ui,uj)∩∪t=1nRt=,{v1,v2,v3,,,,,,,,,,,,,,,v2n,} , therefore, |R(ui,uj)∩∪t=1nRt|=2n>|Rt|

(b) The resolving neighborhood for j − i ≡ 0(mod 2) for 1 ≤ i ≤ n, 1 ≤ t ≤ n, j ∈ {i + k}, 1 ≤ k ≤ n − 1:

R(ui,uj)={up:1≤p≤n:p≠i+j2=m and p≠n+i+j2=r,vq:1≤q≤2n:q≠2m,2m-1 and q≠2r,2r-1

with |R(u_i, u_j)| = 3n − 6 > 8 = |R_t| and R(ui,uj)∩∪t=1nRt=,{vq, &1≤q≤2n:q≠2m,2m-1 and q≠2r,2r-1} , therefore, |R(ui,uj)∩∪t=1nRt|=2n-4>|Rt| .

Lemma 3

Let MOF (n) be the metal organic frame work for and n ≥ 8 and n ≡ 0(mod 2). Then, for 1 ≤ i ≤ n, 1 ≤ t ≤ n, |R_t| < |R(u_i,v_j)| and |R(ui,uj)∩∪t=1nRt|≥|Rt| :

(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 where k = 4y − 3, 4y − 2 for 1≤y≤n2-5 }

(l) j ∈ {2i + 10 + k: 1 ≤ k ≤ 2n − 22 where k = 4y − 1, 4y for 1≤y≤n2-6 }.

Proof

(a) The resolving neighborhood for n ≥ 8, 1 ≤ i ≤ n, 1 ≤ t ≤ n, and j ∈ {2i − 1, 2i}.

R(ui,vj)={up:1≤p≤n,vp:1≤q≤2n:q≠j+1,j+2,j-2,j-3 when j∈{2i-2}q≠j+2,j+3,j-1,j-2 when j∈{2i}

with |R(u_i, v_j)| = 3n − 4 > 8 = |R_t| and R(ui,uj)∩∪t=1nRt={vq:1≤q≤2n and q ≠ j + 1, j + 2, j − 2, j − 3 when j ∈ {2i − 2} q ≠ j + 2, j + 3, j − 1, j − 2 when j ∈ {2i}, therefore, |R(ui,uj)∩∪t=1nRt|=2n-4>|Rt| .

(b) The resolving neighborhood for n ≥ 8, 1 ≤ i ≤ n, 1 ≤ t ≤ n, and j ∈ {2i + 1}.

R(ui,vj)={up:1≤p≤n:p≠m, j-i≤m≤n2,vq:1≤q≤2n:q≠j-1,q≠t and j+3≤t≤n2+3

with |R(u_i, v_j)| = 3n − 16 > 8 = |R_t| and R(ui,vj)∩∪t=1nRt={vq, 1≤q≤2n:q≠j-1,q≠t and j+3≤t≤n2+3} , therefore, |R(ui,uj)∩∪t=1nRt|=2n-10>|Rt| .

(c) The resolving neighborhood for n ≥ 8, 1 ≤ i ≤ n, 1 ≤ t ≤ n, and j ∈ {2i + 2}.

R(ui,vj)={up:1≤p≤n:p≠m, j-i-1≤m≤n2,vq:1≤q≤2n:q≠s and 2(j-i+1)≤s≤n2+1

with |R(u_i, v_j)| = 3n − 13 > 8 = |R_t| and R(ui,vj)∩∪t=1nRt={vq, 1≤q≤2n:q≠s and 2(j-i+1)≤s≤n2+1} , therefore, |R(ui,uj)∩∪t=1nRt|=2n-7>|Rt| .

(d) The resolving neighborhood for n ≥ 8, 1 ≤ i ≤ n, 1 ≤ t ≤ n, and j ∈ {2i + 3, 2i + 4}.

R(ui,vj)={up:1≤p≤nvq:1≤q≤2n:q≠j-2

with |R(u_i, v_j)| = 3n − 1 > 8 = |R_t| and R(ui,vj)∩∪t=1nRt={vq:1≤q≤2n:q≠j-2} , therefore, |R(ui,uj)∩∪t=1nRt|=2n-1>|Rt| .

(e) The resolving neighborhood for n ≥ 8, 1 ≤ i ≤ n, 1 ≤ t ≤ n, and j ∈ {2i + 5}.

R(ui,vj)={up:1≤p≤n:p≠i+2,p≠n2+2=svq:1≤q≤2n:q≠2s,2s-1

with |R(u_i, v_j)| = 3n − 4 > 8 = |R_t| and R(ui,vj)∩∪t=1nRt={vq, 1≤q≤2n:q≠2s,2s-1} , therefore, |R(ui,vj)∩∪t=1nRt|=2n-2>|Rt| .

(f) The resolving neighborhood for n ≥ 8, 1 ≤ i ≤ n, 1 ≤ t ≤ n, and j ∈ {2i + 6}.

R(ui,vj)={up:1≤p≤n:p≠i+2,p≠n2+2=svq:1≤q≤2n:q≠2s,2s-1,j2+1

with |R(u_i, v_j)| = 3n − 5 > 8 = |R_t| and R(ui,vj)∩∪t=1nRt={vq, 1≤q≤2n:q≠2s,2s-1,j2+1} , therefore, |R(ui,uj)∩∪t=1nRt|=2n-3>|Rt| .

(g) The resolving neighborhood for n ≥ 8, 1 ≤ i ≤ n, 1 ≤ t ≤ n, and j ∈ {2i + 7}.

R(ui,vj)={up:1≤p≤n:vq:1≤q≤2n:q≠j-3

with |R(u_i, v_j)| = 3n − 1 > 8 = |R_t| and R(ui,vj)∩∪t=1nRt={vq, 1≤q≤2n:q≠j-3} , therefore, |R(ui,vj)∩∪t=1nRt|=2n-3>|Rt| .

(h) The resolving neighborhood for n ≥ 8, 1 ≤ i ≤ n, 1 ≤ t ≤ n, and j ∈ {2i + 8}.

R(ui,vj)={up:1≤p≤nvq:1≤q≤2n

with |R(u_i, v_j)| = 3n > 8 = |R_t| and R(ui,vj)∩∪t=1nRt={vq, 1≤q≤2n} , therefore, |R(ui,vj)∩∪t=1nRt|=2n>|Rt| .

(i) The resolving neighborhood for n ≥ 8, 1 ≤ i ≤ n, 1 ≤ t ≤ n, and j ∈ {2i + 9}.

R(ui,vj)={up:1≤p≤n:p≠i+3,p≠n+j-52=svq:1≤q≤2n:q≠2s,2s-1,j-4

with |R(u_i, v_j)| = 3n − 5 > 8 = |R_t| and R(ui,vj)∩∪t=1nRt={vq, 1≤q≤2n} , therefore, |R(ui,vj)∩∪t=1nRt|=2n-3>|Rt| .

(j) The resolving neighborhood for n ≥ 8, 1 ≤ i ≤ n, 1 ≤ t ≤ n, and j ∈ {2i + 10}.

R(ui,vj)={up:1≤p≤n:p≠i+3=s,p≠n+j-52=mvq:1≤q≤2n:q≠2s,2s-1,2m,2m-1

with |R(u_i, v_j)| = 3n − 6 > 8 = |R_t| and R(ui,vj)∩∪t=1nRt={vq, 1≤q≤2n} , therefore, |R(ui,vj)∩∪t=1nRt|=2n-4>|Rt| .

(k) The resolving neighborhood for n ≥ 8, 1 ≤ i ≤ n, 1 ≤ t ≤ n, j ∈ {2i + 10 + k: 1 ≤ k ≤ 2n − 22} where k = 4t − 3, 4t − 2 for 1≤t≤n2-5 }.

R(ui,vj)={up:1≤p≤nvq:1≤q≤2n

with |R(u_i, v_j)| = 3n > 8 = |R_t| and R(ui,vj)∩∪t=1nRt={vq, 1≤q≤2n} , therefore, |R(ui,vj)∩∪t=1nRt|=2n>|Rt| .

(l) The resolving neighborhood for n ≥ 8, 1 ≤ i ≤ n, 1 ≤ t ≤ n and j ∈ {2i + 10 + k: 1 ≤ k ≤ 2n − 22} where k = 4t − 1, 4t for 1≤t≤n2-6 },

R(ui,vj)={up:1≤p≤n:p≠i+92=s,p≠n2+i+3=mvq:1≤q≤2n:q≠2s,2s-1,2m,2m-1

with |R(u_i, v_j)| = 3n − 6 > 8 = |R_t| and R(ui,vj)∩∪t=1nRt={vq,1≤q≤2n:q≠2s,2s-1,2m,2m-1} , therefore, |R(ui,vj)∩∪t=1nRt|=2n-4>|Rt| .

Corollary

Let MOF (n) be the metal organic frame work for n ≥ 8 for n ≡ 0 (mod 2). Following is the symmetry of cardinalities of resloving neighbourhoods of metal organic framework.

R(ui,vj))|=|R(ui,vj)|,for 1≤i≤n j∈{2i-1,2i,… … ..2i+10} and j∈{2i-2,2i-3,… … ..2i-13}.

Lemma 4

Let MOF (n) be the metal organic frame work for n ≥ 8 for n ≡ 0(mod 2). Then for 1 ≤ k ≤ n, 1 ≤ t ≤ n, i ∈ {2k − 1}. |R_t| < |R(v_i, v_j)| and |R(vi,vj)∩∪t=1nRt|≥|Rt| ,

(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 for n ≥ 8, 1 ≤ k ≤ n, 1 ≤ t ≤ n i ∈ {2k − 1}, j ∈ {i +2. i + 6}.

R(vi,vj)={up:1≤p≤nvq:1≤q≤2n,q≠i+j2

with |R(v_i, v_j)| = 3n − 1 > 8 = |R_t| and R(vi,vj)∩∪t=1nRt={vq,1≤q≤2n,q≠i+j2} , therefore, |R(vi,vj)∩∪t=1nRt|=2n-1>|Rt| .

(b) The resolving neighborhood for n ≥ 8, 1 ≤ k ≤ n, 1 ≤ t ≤ n, i ∈ {2k − 1}, and j ∈ {i +3, i + 7}.

R(vi,vj)={up:1≤p≤nvq:1≤q≤2n

with |R(v_i, v_j)| = 3n > 8 = |R_t| and R(vi,vj)∩∪t=1nRt={vq, 1≤q≤2n} , therefore, |R(vi,vj)∩∪t=1nRt|=2n>|Rt| .

(c) The resolving neighborhood for n ≥ 8, 1 ≤ k ≤ n, 1 ≤ t ≤ n, i ∈ {2k − 1}, and j ∈ {i + 4, i + 8}.

When j ∈ {i + 4}:

R(vi,vj)={up:1≤p≤n, p≠i+j+24,n+j-12=mvq:1≤q≤2n,q≠i+j3,2m,2m-1

When j ∈ {i + 8}:

R(vi,vj)={up:1≤p≤n, p≠i+j+24,n+j-12=svq:1≤q≤2n,q≠i+j2,2s,2s-1

with |R(v_i, v_j)| = 3n − 5 > 8 = |R_t| and R(vi,vj)∩∪t=1nRt={1≤q≤2n,q≠i+j3,2m,2m-1 when j∈{i+4} q≠i+j2,2s,2s-1 when j ∈ {i + 8}, therefore, |R(vi,vj)∩∪t=1nRt|=2n-3>|Rt| .

(d) The resolving neighborhood for n ≥ 8, 1 ≤ k ≤ n, 1 ≤ t ≤ n, i ∈ {2k − 1}, and j ∈ {i + 5}.

R(vi,vj)={up:1≤p≤n:p≠i+j+14=s,p≠n+j-42=mvq:1≤q≤2n:q≠2s,2s-1,2t,2t-1

with |R(v_i, v_j)| = 3n − 6 > 8 = |R_t| and R(vi,vj)∩∪t=1nRt={vq, &1≤q≤2n:q≠2m,2m-1,2t,2t-1} , therefore, |R(vi,vj)∩∪t=1nRt|=2n-4>|Rt| .

Lemma 5

Let MOF (n) be the metal organic frame work for n ≥ 8 and n ≡ 0(mod 2). Then for 1 ≤ k ≤ n, 1 ≤ t ≤ n, i ∈ {2k}. |R_t| < |R(v_i, v_j)|, and |R(vi,vj)∩∪t=1nRt|≥|Rt| , (a) j ∈ {i + 1} (b) j ∈ {i + 2, i + 6} (c) j ∈ {i + 4, i + 8} (d) j ∈ {i + 3, i + 7} (e) j ∈ {i + 4, i + 8}.

Proof

(a) The resolving neighborhood for n ≥ 8, 1 ≤ k ≤ n, 1 ≤ t ≤ n i ∈ {2k}, and j ∈ {i + 1}:

R(vi,vj)={up:1≤p≤nvq:1≤q≤2n

with |R(v_i, v_j)| = 3n > 8 = |R_t| and R(vi,vj)∩∪t=1nRt={vq,1≤q≤2n} , therefore, |R(vi,vj)∩∪t=1nRt|=2n>|Rt| .

(b) The resolving neighborhood for n ≥ 8, 1 ≤ k ≤ n, 1 ≤ t ≤ n, i ∈ {2k}, and j ∈ {i + 2, i + 6}:

R(vi,vj)={up:1≤p≤nvq:1≤q≤2n,q≠i+j2

with |R(v_i, v_j)| = 3n − 1 > 8 = |R_t| and R(vi,vj)∩∪t=1nRt={vq,1≤q≤2n,q≠i+j2} , therefore, |R(vi,vj)∩∪t=1nRt|=2n-1>|Rt| .

(c) The resolving neighborhood for n ≥ 8, 1 ≤ k ≤ n, 1 ≤ t ≤ n, i ∈ {2k}, j ∈ {i + 3. i + 7}:

R(vi,vj)={up:1≤p≤n:p≠i+j+14,p≠n+j-42=tvq:1≤q≤2n:q≠2t,2t-1

with |R(v_i, v_j)| = 3n − 4 > 8 = |R_t| and R(vi,vj)∩∪t=1nRt={vq, 1≤q≤2n:q≠2t,2t-1} , therefore, |R(vi,vj)∩∪t=1nRt|=2n-2>|Rt| .

(d) The resolving neighborhood for n ≥ 8, 1 ≤ k ≤ n, 1 ≤ t ≤ n, i ∈ {2k}, and j ∈ {i + 4. i + 8}:

R(vi,vj)={up:1≤p≤n:p≠i+j4,p≠n+j-42=mvq:1≤q≤2n:q≠2m,2m-1,i+j2

with |R(v_i, v_j)| = 3n − 5 > 8 = |R_t| and R(ui,vj)∩∪t=1nRt={vq, 1≤q≤2n:q≠2m,2m-1,i+j2} , therefore, |R(vi,vj)∩∪t=1nRt|=2n-3>|Rt| .

(e) The resolving neighborhood for n ≥ 8, 1 ≤ k ≤ n, 1 ≤ t ≤ n, i ∈ {2k}, and j ∈ {i + 5}:

R(vi,vj)={up:1≤p≤nvq:1≤q≤2n

with |R(v_i, v_j)| = 3n > 8 = |R_t| and R(vi,vj)∩∪t=1nRt={vq,1≤q≤2n} , therefore, |R(vi,vj)∩∪t=1nRt|=2n>|Rt| .

Lemma 6

Let MOF (n) be the metal organic frame work for n ≥ 8 and (n ≡ 0(mod 2). Then for 1 ≤ i ≤ 2n, 1 ≤ t ≤ n:

(a) |R_t| < |R(v_i, v_j)| and |R(ui,vj)∩∪t=1nRt|≥|Rt| , j ∈ {i + 9 + k: 1 ≤ k ≤ 2n − 18 where k = 4t − 3, 4t − 2 for 1≤t≤n2-4 },

(b) |R_t| < |R(v_i, v_j)| and |R(ui,vj)∩∪t=1nRt|≥|Rt| , j ∈ {i + 9 + k: 1 ≤ k ≤ 2n − 18 where k = 4t − 1, 4t for 1≤t≤n2-5 }.

Proof

(a) The resolving neighborhood for n ≥ 8, 1 ≤ i ≤ 2n, 1 ≤ t ≤ n, and j ∈ {i + 9 + k: 1 ≤ k ≤ 2n − 18 where k = 4t − 3, 4t − 2 for 1≤t≤n2-4 }:

R(ui,vj)={up:1≤p≤nvq:1≤q≤2n

with |R(v_i, v_j)| = 3n > 8 = |R_t| and R(vi,vj)∩∪t=1nRt={vq, 1≤q≤2n} , therefore, |R(vi,vj)∩∪t=1nRt|=2n>|Rt| .

(b) The resolving neighborhood for n ≥ 8, 1 ≤ i ≤ 2n, 1 ≤ t ≤ n, j ∈ {i + 9 + k: 1 ≤ k ≤ 2n − 18 where k = 4t − 1, 4t for 1≤t≤n2-5 }:

R(ui,vj)={up:1≤p≤n:p≠i+72=s,p≠n2+i+3=mvq:1≤q≤2n:q≠2m,2m-1,2t,2t-1

with |R(v_i, v_j)| = 3n − 6 ≥ 8 = |R_t| and R(ui,vj)∩∪t=1nRt={vq,1≤q≤2n:q≠2s,2s-1,2m,2m-1} , therefore, |R(ui,vj)∩∪t=1nRt|=2n-4>|Rt| .

Corollary

Let MOF (n) be the metal organic frame work for n ≥ 8 for n ≡ 0(mod 2). Following is the symmetry of cardinalities of resloving neighbourhoods of metal organic framework.

(i) |for 1 ≤ k ≤ n, i ∈ {2k − 1} R(v_i, v_j))| = |R(v_i, v_j)|, where j ∈ {i + 2, … … … i + 8} and j ∈ {i − 2, … … … i − 8}.

(ii) |R(v_i, v_j))| = |R(v_i, v_j)|, for 1 ≤ k ≤ n, i ∈ {2k} j ∈ {i + 2, … … … i + 7} and j ∈ {i − 2, … … … i − 7}.

3 Fractional metric dimension of metal organic frame work

In this section fractional metric dimension metal organic frame work MOF (n) for n ≡ 0(mod 2) is calculated.

Theorem 1

The fractional metric dimension of metal organic frame work MOF(n) for 2 ≤ n ≤ 6 is L

dimf(MOF(n))={32 if n=22 if n=432 if n = 6

Proof

Case 1: When n = 2, then resolving neighbourhoods are as presented in Tables 1 and 2.

For 1 ≤ t ≤ 12, each resloving neighbourhood having cardinality 4 which is less than all other cardinalities of resolving neighbourhood R^m of MOF(2), where, 1 ≤ m ≤ 3. Moreover, ∪t=112Rt=V(MOF(2) which implies that |∪t=112Rt|=10 and |Rm∩∪t=112Rt|≥|Rt|=8 . Consequently, dimf(MOF(2))<∑1614=32 .

Case 2: When n = 4, then resolving neighbourhoods are as presented in Tables 3 and 4.

For 1 ≤ t ≤ 2, each resloving neighbourhood (R_t) having cardinality 6 which is less than all other cardinalities of resolving neighbourhood (R^m) of MOF(2), where, 1 ≤ m ≤ 64. Moreover, ∪t=112Rt=V(MOF(2) which implies that |∪t=12Rt|=10 and |Rm∩∪t=12Rt|=2n-4≥|Rt|=8 . Consequently, dimf(MOF(2))<∑t=11216=2 .

Case 3: When n = 6, then resolving neighbourhoods are are presented in Tables 5 and 6.

For 1 ≤ t ≤ 6, each resloving neighbourhood (R_t) having cardinality 8 which is less than all other cardinalities of resolving neighbourhood (R^m) of MOF(6), where, 1 ≤ m ≤ 147. Moreover, ∪t=16Rt=V(MOF(6) which implies that |∪t=16Rt|=10 and |Rm∩∪t=16Rt|≥|Rt|=8 . Consequently, dimf(MOF(6))<∑t=11218=32 .

Table 1

Resolving neighbourhoods for n = 2

Resolving sets n = 2	Elements
R¹ = R(u₁, u₂)	V(MOF(2))
R² = R(v₁, v₄)	V(MOF(2))
R³ = R(v₂, v₃)	V(MOF(2))

Table 2

Resolving neighbourhoods with minimum cardnality for n = 2

Resolving sets n = 2	Elements
R₁ = R(u₁, v₁)	V(MOF(2))–{v₂,v₃}
R₂ = R(u₁, v₂)	V(MOF(2))–{v₁,v₄}
R₃ = R(u₁, v₃)	V(MOF(2))–{u₂,v₂}
R₄ = R(u₁, v₄)	V(MOF(2))–{u₂,v₁}
R₅ = R(u₂, v₁)	V(MOF(2))–{u₁,v₄}
R₆ = R(u₂, v₂)	V(MOF(2))–{u₁,v₃}
R₇ = R(u₂, v₃)	V(MOF(2))–{v₁,v₄}
R₈ = R(u₂, v₄)	V(MOF(2))–{v₂,v₃}
R₉ = R(v₁, v₂)	V(MOF(2))–{u₁,u₂}
R₁₀ = R(v₁, v₃)	V(MOF(2))–{v₂,v₄}
R₁₁ = R(v₂, v₄)	V(MOF(2))–{u₁,u₃}
R₁₂ = R(v₃, v₄)	V(MOF(2))–{u₁,u₂}

Table 3

Resolving neighbourhoods for n = 4

Resolving sets n = 4	Elements
R¹ = R(u₁, u₂)	V(MOF(4))
R² = R(u₁, u₄)	V(MOF(4))
R³ = R(u₁, v₁)	V(MOF(4))–{v₂,v₃,v₆,v₇}
R⁴ = R(u₁, v₂)	V(MOF(4))–{v₁,v₄,v₅,v₈}
R⁵ = R(u₁, v₃)	V(MOF(4))–{u₂,u₃,v₂,v₆}
R⁶ = R(u₁, v₄)	V(MOF(4))–{u₂,u₃}
R⁷ = R(u₁, v₅)	V(MOF(4))–{u₃,u₄,v₃,v₇}
R⁸ = R(u₁, v₆)	V(MOF(4))–{v₄,v₈}
R⁹ = R(u₁, v₇)	V(MOF(4))–{u₃,u₄}
R¹⁰ = R(u₁, v₈)	V(MOF(4))–{u₃,u₄,v₁,v₅}
R¹¹ = R(u₂, u₃)	V(MOF(4))
R¹² = R(u₂, v₁)	V(MOF(4))–{u₁,u₄}
R¹³ = R(u₂, v₂)	V(MOF(4))–{u₁,u₄,v₇,v₈}
R¹⁴ = R(u₂, v₃)	V(MOF(4))–{v₁,v₄,v₅,v₈}
R¹⁵ = R(u₂, v₄)	V(MOF(4))–{v₂,v₃,v₆,v₇}
R¹⁶ = R(u₂, v₅)	V(MOF(4))–{u₃,u₄,v₄,v₈}
R¹⁷ = R(u₂, v₆)	V(MOF(4))–{u₃,u₄}
R¹⁸ = R(u₂, v₇)	V(MOF(4))–{v₁v₅}
R¹⁹ = R(u₂, v₈)	V(MOF(4))–{v₂,v₆}
R²⁰ = R(u₃, u₄)	V(MOF(4))
R²¹ = R(u₃, v₁)	V(MOF(4))–{v₃,v₇}
R²² = R(u₃, v₂)	V(MOF(4))–{v₃,v₈}
R²³ = R(u₃, v₃)	V(MOF(4))–{u₁,u₂}
R²⁴ = R(u₃, v₄)	V(MOF(4))–{u₁,u₂,v₁,v₅}
R²⁵ = R(u₃, v₅)	V(MOF(4))–{v₂,v₃,v₆,v₇}
R²⁶ = R(u₃, v₆)	V(MOF(4))–{v₁,v₄,v₅,v₈}
R²⁷ = R(u₃, v₇)	V(MOF(4))–{u₁,u₄,v₂,v₆}
R²⁸ = R(u₃, v₈)	V(MOF(4))–{u₁,u₄}
R²⁹ = R(u₄, v₁)	V(MOF(4))–{u₁,u₂,v₄,v₈}
R³⁰ = R(u₄, v₂)	V(MOF(4))–{u₁,u₂,}
R³¹ = R(u₄, v₃)	V(MOF(4))–{v₁,v₅}
R³² = R(u₄, v₄)	V(MOF(4))–{v₂,v₆}
R³³ = R(u₄, v₅)	V(MOF(4))–{u₂,u₃}
R³⁴ = R(u₄, v₆)	V(MOF(4))–{u₂,u₃,v₃,v₇}
R³⁵ = R(u₄, v₇)	V(MOF(4))–{v₁,v₄,v₅,v₈}
R³⁶ = R(u₄, v₈)	V(MOF(4))–{v₂,v₃,v₆,v₇}
R³⁷ = R(v₁, v₂)	V(MOF(4))–{u₁,u₂,u₃,u₄}
R³⁸ = R(v₁, v₃)	V(MOF(4))–{v₂,v₆}
R³⁹ = R(v₁, v₄)	V(MOF(4))
R⁴⁰ = R(v₁, v₅)	V(MOF(4))–{u₂,u₄,v₆,v₇}
R⁴¹ = R(v₁, v₆)	V(MOF(4))–{u₂,u₄}
R⁴² = R(v₁, v₇)	V(MOF(4))–{v₄,v₈}
R⁴³ = R(v₁, v₈)	V(MOF(4))
R⁴⁴ = R(v₂, v₃)	V(MOF(4))
R⁴⁵ = R(v₂, v₄)	V(MOF(4))–{v₃,v₇}
R⁴⁶ = R(v₂, v₅)	V(MOF(4))–{u₂,u₄}
R⁴⁷ = R(v₂, v₆)	V(MOF(4))–{u₂,u₄,v₄,v₈}
R⁴⁸ = R(v₂, v₇)	V(MOF(4))
R⁴⁹ = R(v₂, v₈)	V(MOF(4))–{v₁,v₅}
R⁵⁰ = R(v₃, v₄)	V(MOF(4))–{u₁,u₂,u₃,u₄}
R⁵¹ = R(v₃, v₅)	V(MOF(4))–{v₄,v₈}
R⁵² = R(v₃, v₆)	V(MOF(4))
R⁵³ = R(v₃, v₇)	V(MOF(4))–{u₁,u₃,v₁,v₅}
R⁵⁴ = R(v₃, v₈)	V(MOF(4))–{u₁,u₃}
R⁵⁵ = R(v₄, v₅)	V(MOF(4))
R⁵⁶ = R(v₄, v₆)	V(MOF(4))–{v₁,v₅}
R⁵⁷ = R(v₄, v₇)	V(MOF(4))–{u₁,u₃}
R⁵⁸ = R(v₄, v₈)	V(MOF(4))–{u₁,u₃,v₂,v₆}
R⁵⁹ = R(v₅, v₆)	V(MOF(4))–{u₁,u₂,u₃,u₄}
R⁶⁰ = R(v₅, v₇)	V(MOF(4))–{v₂,v₆}
R⁶¹ = R(v₅, v₈)	V(MOF(4))
R⁶² = R(v₆, v₇)	V(MOF(4))
R⁶³ = R(v₆, v₈)	V(MOF(4))–{v₃,v₇}
R⁶⁴ = R(v₇, v₈)	V(MOF(4))–{u₁,u₂,u₃,u₄}

Table 4

Resolving neighbourhoods with minimum cardnality for n = 4

Resolving sets n = 4	Elements
R₁ = R(u₁, u₃)	V(MOF(4))–{u₂,u₄,v₃,v₄,v₇,v₈}
R₂ = R(u₂, u₄)	V(MOF(4))–{u₁,u₃,v₁,v₂,v₆,v₇}

Table 5

Resolving neighbourhoods for n = 6

Resolving sets n = 6	Elements
R¹ = R(u₁, u₂)	V(MOF(6))
R² = R(u₁, u₃)	V(MOF(6))–{u₂,u₅,v₃,v₄,v₉,v₁₀}
R³ = R(u₁, u₄)	V(MOF(6))
R⁴ = R(u₁, u₅)	V(MOF(6))–{u₃,u₆,v₅,v₆,v₁₁,v₁₂}
R⁵ = R(u₁, u₆)	V(MOF(6))
R⁶ = R(u₁, v₁)	V(MOF(6))–{v₂,v₃,v₁₀,v₁₁}
R⁷ = R(u₁, v₂)	V(MOF(6))–{v₁,v₄,v₅,v₁₂}
R⁸ = R(u₁, v₃)	V(MOF(6))–{u₂,u₃,u₄,v₂,v₆,v₇,v₈}
R⁹ = R(u₁, v₄)	V(MOF(6))–{u₂,u₃,u₄,v₈}
R¹⁰ = R(u₁, v₅)	V(MOF(6))–{u₃,}
R¹¹ = R(u₁, v₆)	V(MOF(6))–{v₄,v₉}
R¹² = R(u₁, v₇)	V(MOF(6))–{u₃,u₅,v₁₀}
R¹³ = R(u₁, v₈)	V(MOF(6))–{u₃,u₅,v₅}
R¹⁴ = R(u₁, v₉)	V(MOF(6))–{v₆,v₁₁}
R¹⁵ = R(u₁, v₁₀)	V(MOF(6))–{v₁₂}
R¹⁶ = R(u₁, v₁₁)	V(MOF(6))–{u₄,u₅,u₆,v₇}
R¹⁷ = R(u₁, v₁₂)	V(MOF(6))–{u₄,u₅,u₆,v₁,v₇,v₈,v₉}
R¹⁸ = R(u₂, u₃)	V(MOF(6))
R¹⁹ = R(u₂, u₄)	V(MOF(6))–{u₃,u₆,v₅,v₆,v₁₁,v₁₂}
R²⁰ = R(u₂, u₅)	V(MOF(6))
R²¹ = R(u₂, u₆)	V(MOF(6))–{u₁,u₄,v₁,v₂,v₇,v₈}
R²² = R(u₂, v₁)	V(MOF(6))–{u₁,u₅,u₆,v₉}
R²³ = R(u₂, v₂)	V(MOF(6))–{u₁,u₅,u₆,v₃,v₉,v₁₀,v₁₁}
R²⁴ = R(u₂, v₃)	V(MOF(6))–{v₁,v₄,v₁₂}
R²⁵ = R(u₂, v₄)	V(MOF(6))–{v₂,v₃,v₆,v₇}
R²⁶ = R(u₂, v₅)	V(MOF(6))–{u₃,u₅,u₆,v₅,v₈,v₉,v₁₀}
R²⁷ = R(u₂, v₆)	V(MOF(6))–{u₃,u₄,u₅,v₁₀}
R²⁸ = R(u₂, v₇)	V(MOF(6))–{v₅,}
R²⁹ = R(u₂, v₈)	V(MOF(6))–{v₆,v₁₁}
R³⁰ = R(u₂, v₉)	V(MOF(6))–{u₄,u₆,v₁₂}
R³¹ = R(u₂, u₁₀)	V(MOF(6))–{u₄,u₆,u₇}
R³² = R(u₂, v₁₁)	V(MOF(6))–{v₁,v₈}
R³³ = R(u₂, v₁₂)	V(MOF(6))–{v₂,}
R³⁴ = R(u₃, u₄)	V(MOF(6))
R³⁵ = R(u₃, u₅)	V(MOF(6))–{u₁,u₄,v₁,v₂,v₇,v₈}
R³⁶ = R(u₃, u₆)	V(MOF(6))
R³⁷ = R(u₃, v₁)	V(MOF(6))–{v₃,v₁₀}
R³⁸ = R(u₃, v₂)	V(MOF(6))–{v₄,}
R³⁹ = R(u₃, v₃)	V(MOF(6))–{u₁,u₃,u₆,v₁₁}
R⁴⁰ = R(u₃, v₄)	V(MOF(6))–{u₁,u₂,u₆,v₁,v₅,v₁₂}
R⁴¹ = R(u₃, v₅)	V(MOF(6))–{v₂,v₃,v₆,v₇}
R⁴² = R(u₃, v₆)	V(MOF(6))–{v₄,v₅,v₈,v₉}
R⁴³ = R(u₃, v₇)	V(MOF(6))–{u₄,u₅,u₆,v₆,v₁₀,v₁₁,v₁₂}
R⁴⁴ = R(u₃, v₈)	V(MOF(6))–{u₄,u₅,u₆,v₁₂,}
R⁴⁵ = R(u₃, v₉)	V(MOF(6))–{v₇}
R⁴⁶ = R(u₃, v₁₀)	V(MOF(6))–{v₁,v₈}
R⁴⁷ = R(u₃, v₁₁)	V(MOF(6))–{u₁,u₅,v₂}
R⁴⁸ = R(u₃, v₁₂)	V(MOF(6))–{u₁,u₅,v₉}
R⁴⁹ = R(u₄, u₅)	V(MOF(6))
R⁵⁰ = R(u₄, u₆)	V(MOF(6))–{u₂,u₅,v₃,v₄,v₉,v₁₀}
R⁵¹ = R(u₄, v₁)	V(MOF(6))–{u₂,u₆,v₄}
R⁵² = R(u₄, v₂)	V(MOF(6))–{u₂,u₆,v₁₁}
R⁵³ = R(u₃, v₃)	V(MOF(6))–{v₅,v₁₂}
R⁵⁴ = R(u₄, v₄)	V(MOF(6))–{v₆}
R⁵⁵ = R(u₄, v₅)	V(MOF(6))–{u₁,u₂,u₃,v₁}
R⁵⁶ = R(u₄, v₆)	V(MOF(6))–{u₁,u₂,u₃,v₁,v₂,v₃}
R⁵⁷ = R(u₄, v₇)	V(MOF(6))–{v₄,v₅,v₈,v₉}
R⁵⁸ = R(u₄, v₈)	V(MOF(6))–{v₆,v₇,v₁₀,v₁₁}
R⁵⁹ = R(u₄, v₉)	V(MOF(6))–{u₁,u₅,u₆,v₁,v₂,v₈,v₁₂}
R⁶⁰ = R(u₄, v₁₀)	V(MOF(6))–{u₁,u₅,u₆,v₂}
R⁶¹ = R(u₄, v₁₁)	V(MOF(6))–{v₉}
R⁶² = R(u₄, v₁₂)	V(MOF(6))–{v₃,v₁₀}
R⁶³ = R(u₅, u₆)	V(MOF(6))
R⁶⁴ = R(u₅, v₁)	V(MOF(6))–{v₃,v₁₀}
R⁶⁵ = R(u₅, v₂)	V(MOF(6))–{u₅,u₁₂}
R⁶⁶ = R(u₅, v₃)	V(MOF(6))–{u₁,v₆}
R⁶⁷ = R(u₅, v₄)	V(MOF(6))–{u₁,v₁}
R⁶⁸ = R(u₅, v₅)	V(MOF(6))–{v₂,v₇}
R⁶⁹ = R(u₅, v₆)	V(MOF(6))–{v₈}
R⁷⁰ = R(u₅, v₇)	V(MOF(6))–{v₃,u₂,u₃,u₄,v₃,}
R⁷¹ = R(u₅, v₈)	V(MOF(6))–{u₂,u₃,u₄,v₃,v₄,v₅,v₉}
R⁷² = R(u₅, v₉)	V(MOF(6))–{v₆,v₇,v₁₀,v₁₁}
R⁷³ = R(u₅, v₁₀)	V(MOF(6))–{v₁,v₈,v₉,v₁₂}
R⁷⁴ = R(u₅, v₁₁)	V(MOF(6))–{u₁,u₂,u₆,v₂,v₃,v₄,v₁₀}
R⁷⁵ = R(u₅, v₁₂)	V(MOF(6))–{u₁,u₂,u₆,v₁,v₃,v₄}
R⁷⁶ = R(u₆, v₁)	V(MOF(6))–{u₁,u₂,u₃,v₄,v₅,v₆,v₁₂}
R⁷⁷ = R(u₆, v₂)	V(MOF(6))–{u₁,u₂,u₃,v₆}
R⁷⁸ = R(u₆, v₃)	V(MOF(6))–{v₁}
R⁷⁹ = R(u₆, v₄)	V(MOF(6))–{v₂}
R⁸⁰ = R(u₆, v₅)	V(MOF(6))–{u₂,u₄,v₈}
R⁸¹ = R(u₆, v₆)	V(MOF(6))–{u₂,u₄,v₃}
R⁸² = R(u₆, v₇)	V(MOF(6))–{u₄,v₉}
R⁸³ = R(u₆, v₈)	V(MOF(6))–{v₉}
R⁸⁴ = R(u₆, v₉)	V(MOF(6))–{u₃,u₄,u₅,v₅}
R⁸⁵ = R(u₆, v₁₀)	V(MOF(6))–{u₃,u₄,u₅,v₅,v₆,v₇,v₁₁}
R⁸⁶ = R(u₆, v₁₁)	V(MOF(6))–{v₁,v₉,v₁₂}
R⁸⁷ = R(u₆, v₁₂)	V(MOF(6))–{v₂,v₃,v₁₀,v₁₁}
R⁸⁸ = R(v₁, v₃)	V(MOF(6))–{v₂}
R⁸⁹ = R(v₁, v₄)	V(MOF(6))
R⁹⁰ = R(v₁, v₅)	V(MOF(6))–{u₂,u₅,v₃,v₉}
R⁹¹ = R(v₁, v₆)	V(MOF(6))–{u₁,u₅}
R⁹² = R(v₁, v₇)	V(MOF(6))–{v₅}
R⁹³ = R(v₁, v₈)	V(MOF(6))
R⁹⁴ = R(v₁, v₉)	V(MOF(6))–{u₃,u₆,u₅,v₆,v₁₁}
R⁹⁵ = R(v₁, v₁₀)	V(MOF(6))–{u₃,u₆,v₅,v₆}
R⁹⁶ = R(v₁, v₁₁)	V(MOF(6))–{v₁₂}
R⁹⁷ = R(v₁, v₁₂)	V(MOF(6))
R⁹⁸ = R(v₂, v₃)	V(MOF(6))
R⁹⁹ = R(v₂, v₄)	V(MOF(6))–{v₃}
R¹⁰⁰ = R(v₂, v₅)	V(MOF(6))–{u₂,u₅,v₉,v₁₀}
R¹⁰¹ = R(v₂, v₆)	V(MOF(6))–{u₂,u₅,v₄,v₁₀}
R¹⁰² = R(v₂, v₇)	V(MOF(6))–{v₄}
R¹⁰³ = R(v₂, v₈)	V(MOF(6))–{v₅,v₁₁}
R¹⁰⁴ = R(v₂, v₉)	V(MOF(6))–{u₃,u₆,}
R¹⁰⁵ = R(v₂, v₁₀)	V(MOF(6))–{u₃,u₆,v₆,v₁₂}
R¹⁰⁶ = R(v₂, v₁₁)	V(MOF(6))
R¹⁰⁷ = R(v₂, v₁₂)	V(MOF(6))–{v₁}
R¹⁰⁸ = R(v₃, v₅)	V(MOF(6))–{v₄}
R¹⁰⁹ = R(v₃, v₆)	V(MOF(6))
R¹¹⁰ = R(v₃, v₇)	V(MOF(6))–{u₃,u₆,v₅,v₁₁}
R¹¹¹ = R(v₃, v₈)	V(MOF(6))–{u₃,u₆,v₁₁,v₁₂}
R¹¹² = R(v₃, v₉)	V(MOF(6))–{v₆,v₁₂}
R¹¹³ = R(v₃, v₁₀)	V(MOF(6))
R¹¹⁴ = R(v₃, v₁₁)	V(MOF(6))–{u₁,u₄,v₁,v₇}
R¹¹⁵ = R(v₃, v₁₂)	V(MOF(6))–{u₁,u₄,v₇,v₈}
R¹¹⁶ = R(v₄, v₅)	V(MOF(6))
R¹¹⁷ = R(v₄, v₆)	V(MOF(6))–{v₅}
R¹¹⁸ = R(v₄, v₇)	V(MOF(6))–{u₃,u₆,v₁₁,v₁₂}
R¹¹⁹ = R(v₄, v₈)	V(MOF(6))–{u₃,u₆,v₆,v₁₂}
R¹²⁰ = R(v₄, v₉)	V(MOF(6))
R¹²¹ = R(v₄, v₁₀)	V(MOF(6))–{v₁,v₇}
R¹²² = R(v₄, v₁₁)	V(MOF(6))–{u₄,u₆,}
R¹²³ = R(v₄, v₁₂)	V(MOF(6))–{u₁,u₄,v₂,v₈}
R¹²⁴ = R(v₅, v₇)	V(MOF(6))–{v₆}
R¹²⁵ = R(v₅, v₈)	V(MOF(6))
R¹²⁶ = R(v₅, v₉)	V(MOF(6))–{u₁,u₄,v₁,v₇}
R¹²⁷ = R(v₅, v₁₀)	V(MOF(6))–{u₁,u₄,}
R¹²⁸ = R(v₅, v₁₁)	V(MOF(6))–{v₂,v₈}
R¹²⁹ = R(v₅, v₁₂)	V(MOF(6))
R¹³⁰ = R(v₆, v₇)	V(MOF(6))
R¹³¹ = R(v₆, v₈)	V(MOF(6))–{v₇}
R¹³² = R(v₆, v₉)	V(MOF(6))–{u₁,u₄,v₁,v₂}
R¹³³ = R(v₆, v₁₀)	V(MOF(6))–{u₁,u₄,v₂,v₈}
R¹³⁴ = R(v₆, v₁₁)	V(MOF(6))
R¹³⁵ = R(v₆, v₁₂)	V(MOF(6))–{u₃,u₉}
R¹³⁶ = R(v₇, v₉)	V(MOF(6))–{v₈}
R¹³⁷ = R(v₇, v₁₀)	V(MOF(6))
R¹³⁸ = R(v₇, v₁₁)	V(MOF(6))–{u₂,u₅,v₃,v₉}
R¹³⁹ = R(v₇, v₁₂)	V(MOF(6))–{u₁,u₅,}
R¹⁴⁰ = R(v₈, v₉)	V(MOF(6))
R¹⁴¹ = R(v₈, v₁₀)	V(MOF(6))–{v₉}
R¹⁴² = R(v₈, v₁₁)	V(MOF(6))–{u₂,u₅,v₃,v₄}
R¹⁴³ = R(v₈, v₁₂)	V(MOF(6))–{u₂,u₅,v₄,v₁₀}
R¹⁴⁴ = R(v₉, v₁₁)	V(MOF(6))–{v₁₀}
R¹⁴⁵ = R(v₉, v₁₂)	V(MOF(6))
R¹⁴⁶ = R(v₁₀, v₁₁)	V(MOF(6))
R¹⁴⁷ = R(v₁₀, v₁₂)	V(MOF(6))–{v₁₁}

Table 6

Resolving neighbourhoods with minimum cardnality for n = 6

Resolving sets n = 6	Elements
R¹ = R(v₁, v₂)	V(MOF(6))–{u₁,u₂,u₃,u₄,u₅,u₆,v₆,v₇,v₈,v₉}
R² = R(v₃, v₄)	V(MOF(6))–{u₁,u₂,u₃,u₄,u₅,u₆,v₈,v₉,v_10,v₁₁}
R³ = R(v₅, v₆)	V(MOF(6))–{u₁,u₂,u₃,u₄,u₅,u₆,v₁,v₁₀,v_11,v₁₂}
R⁴ = R(v₇, v₈)	V(MOF(6))–{u₁,u₂,u₃,u₄,u₅,u₆,v₁,v₂,v_3,v₁₂}
R⁵ = R(v₉, v₁₀)	V(MOF(6))–{u₁,u₂,u₃,u₄,u₅,u₆,v₂,v₃,v_4,v₅}
R⁶ = R(v₁₁, v₁₂)	V(MOF(6))–{u₁,u₂,u₃,u₄,u₅,u₆,v₄,v₅,v_6,v₇}

Theorem 2

Let MOF (n) for n ≥ 8 and n ≡ 0(mod 2) be metal organic frame work. Then, dimf(MOF(n))<n4 .

Proof

From Lemma 1–6 for n ≥ 8 and n ≡ 0(mod 2), 1 ≤ k ≤ n, 1 ≤ t ≤ n, j = i +1, i ∈ [2k − 1], |R_t| = |R{v_i, v_j}| = 8, β=|∪t=1nRt|=2n . |R{v_i, v_j}| ≤ |R{x, y}| for all x, y ∈ V(MOF(n)) for n ≥ 8 such that x ≠ v_i and x ≠ v_j, all resloving sets having minimum cardinalities are disjoint. Therefore,

dimf(MOF(n))<∑1β1|Rt|

where β = 2n, for 1 ≤ k ≤ n, 1 ≤ t ≤ n, j = i + 1, i ∈ [2k − 1], we have hence:

dimf(MOF(n))<n4.

4 Conclusions

In this paper, fractional metric dimension of metal organic frame work MOF(n) for n ≡ 0(mod2) is computed. The problem is still open to compute the FMD of MOF (n) n ≡ 1(mod2). The details of the obtained results of FMD is given in Table 7.

Table 7

Fractional metric dimension of MOF(n)

*MOF (n), n ≡ 0(mod* 2)**	Upper bounds of fractional metric dimension
n = 2	3/2
n = 4	2
n = 6	3/2
n ≥ 8	n/4

Research funding: Authors state no funding involved.
Author contributions: Mohsin Raza: writing – original draft, methodology; Muhammad Javaid: resources, formal analysis, visualization, methodology; Naeem Saleem: writing – review and editing, eesources.
Conflict of interest: One of the authors (Muhammad Javaid) is a Guest Editor of the Main Group Metal Chemistry’s Special Issue “Topological descriptors of chemical networks: Theoretical studies” in which this article is published.
Data availability statement: 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.

References

Arumugam S., Mathew V., The Fractional Metric Dimension of Graphs. Discrete Math. 2012, 312, 184–190.Search in Google Scholar

Chartrand C., Eroh L., Oellermann O.R., Resolvability in Graphs and the Metric Dimension of a Graph. Discrete Appl. Math., 2000, 105, 99–113.10.1016/S0166-218X(00)00198-0Search in Google Scholar

Chu Y.M., Nadeem M.F, Azeem M., Siddiqui M.K, On sharp bounds on partition dimension of convex polytopes. IEEE Access, 2020, 8, 224781–224790.10.1109/ACCESS.2020.3044498Search in Google Scholar

Currie J., Oellermann O.R, The Metric Dimension and Metric Independence of a Graph. J. Combin. Math. Combin. Comput., 2001, 39, 157–167.Search in Google Scholar

Feng M., Kong Q., On the Fractional Metric Dimension of Corona Product Graphs and Lexicographic Product Graphs. Ars Combinatoria, 2018, 138, 249–260.Search in Google Scholar

Feng M., Lv B., Wang K., On the Metric Dimension and Fractional Metric Dimension of Graphs. Appl. Anal. Discr. Math., 2014, 170, 55–63.10.1016/j.dam.2014.01.006Search in Google Scholar

Feng M., Wang K., On the Metric Dimension and Fractional Metric Dimension of the Hierarchical Product of Graphs. Appl. Anal. Discr. Math., 2013, 7, 302–313.10.2298/AADM130521009FSearch in Google Scholar

Harary F., Melter R.A., The Metric Dimension of a Graph. Ars Combinatoria, 1976, 2, 191–195.Search in Google Scholar

Hasan Z., Jhung S.H., Removal of Hazardous Organics from Water Using Metal Organic Frameworks (MOFs): Plausible Mechanisms for Selective Adsorptions. Hazard. Mat., 2015, 283, 329–339.10.1016/j.jhazmat.2014.09.046Search in Google Scholar PubMed

Javaid M., Raza M., Kumam P., Liu J.-B., Sharp Bounds of Local Fractional Metric Dimensions of Connected Networks. IEEE, 2020, 8, 172329–172342.10.1109/ACCESS.2020.3025018Search in Google Scholar

Jiao L., Ruseow J.Y., Scotskinner W., Wang Z.U., Metal Organic Frameworks: Structures and Functional Applications. Mater. Today, 2019, 27, 43–68.10.1016/j.mattod.2018.10.038Search in Google Scholar

Liu J., Chen L., Cui H., Zhang J., Zhang L., Su C.Y., Applications of Metal Organic Frameworks in Heterogeneous Supramolecular Catalysis. Chem. Soc. Rev., 2014, 43, 6011–6061.10.1039/C4CS00094CSearch in Google Scholar

Liu J.-B., Kamaran M., Javaid M., Local Fractional Metric Dimensions of Rotationally Symmetric and Planar Networks. IEEE, 2020, 8, 82404–82420.10.1109/ACCESS.2020.2991685Search in Google Scholar

Liu J.-B., Kashif A., Rashid T., Javaid M., Fractional Metric Dimension of Generalized Jahangir Graph. Mathematics, 2019, 7, 1–10.10.3390/math7010100Search in Google Scholar

Mufti Z.S., Nadeem M.F., Ahmad A., Ahmad Z., Computation of Edge Metric Dimension of Barycentric Subdivision of Cayley Graphs. Ital. J. Pure Appl. Math., 2020, 44, 714–722.Search in Google Scholar

Pettinari C., Marchetti F., Drozdovc A., Applications of Metal Organic Frameworks. Polym. Int., 2017, 66, 731–744.10.1002/pi.5315Search in Google Scholar

Yi E., The Fractional Metric Dimension of Permutation Graphs. Acta Math. Sin., 2015, 31, 367–382.10.1007/s10114-015-4160-5Search in Google Scholar

Received: 2020-11-25

Accepted: 2021-03-01

Published Online: 2021-05-09

This work is licensed under the Creative Commons Attribution 4.0 International License.

Fractional metric dimension of metal-organic frameworks

Abstract

1 Introduction and preliminaries

2 Resolving neighborhoods of metal organic frame works (MOF)

Lemma 1

Proof

Lemma 2

Proof

Lemma 3

Proof

Corollary

Lemma 4

Proof

Lemma 5

Proof

Lemma 6

Proof

Corollary

3 Fractional metric dimension of metal organic frame work

Theorem 1

Proof

Theorem 2

Proof

4 Conclusions

References

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