Computing Bounds of Fractional Metric Dimension of Metal Organic Graphs

<jats:p>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 <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1">
                        <mi>x</mi>
                     </math>
                  </jats:inline-formula> resolves the vertices <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2">
                        <mi>u</mi>
                     </math>
                  </jats:inline-formula> and <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3">
                        <mi>v</mi>
                     </math>
                  </jats:inline-formula> of a graph <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M4">
                        <mi>G</mi>
                     </math>
                  </jats:inline-formula> if <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M5">
                        <mi>d</mi>
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mi>u</mi>
                              <mo>,</mo>
                              <mi>x</mi>
                           </mrow>
                        </mfenced>
                        <mo>≠</mo>
                        <mi>d</mi>
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mi>v</mi>
                              <mo>,</mo>
                              <mi>x</mi>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula>. For a pair <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M6">
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mi>u</mi>
                              <mo>,</mo>
                              <mi>v</mi>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula> of vertices of <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M7">
                        <mi>G</mi>
                     </math>
                  </jats:inline-formula>, <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M8">
                        <mi>R</mi>
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mi>u</mi>
                              <mo>,</mo>
                              <mi>v</mi>
                           </mrow>
                        </mfenced>
                        <mo>=</mo>
                        <mfenced open="{" close="}" separators="|">
                           <mrow>
                              <mi>x</mi>
                              <mo>∈</mo>
                              <mi>V</mi>
                              <mfenced open="(" close=")" separators="|">
                                 <mrow>
                                    <mi>G</mi>
                                 </mrow>
                              </mfenced>
                              <mo>:</mo>
                              <mi>d</mi>
                              <mfenced open="(" close=")" separators="|">
                                 <mrow>
                                    <mi>x</mi>
                                    <mo>,</mo>
                                    <mi>u</mi>
                                 </mrow>
                              </mfenced>
                              <mo>≠</mo>
                              <mi>d</mi>
                              <mfenced open="(" close=")" separators="|">
                                 <mrow>
                                    <mi>x</mi>
                                    <mo>,</mo>
                                    <mi>v</mi>
                                 </mrow>
                              </mfenced>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula> is called its resolving neighbourhood set. For each pair of vertices <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M9">
                        <mi>u</mi>
                     </math>
                  </jats:inline-formula> and <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M10">
                        <mi>v</mi>
                     </math>
                  </jats:inline-formula> in <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M11">
                        <mi>V</mi>
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mi>G</mi>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula>, if <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M12">
                        <mi>f</mi>
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mi>R</mi>
                              <mfenced open="(" close=")" separators="|">
                                 <mrow>
                                    <mi>u</mi>
                                    <mo>,</mo>
                                    <mi>v</mi>
                                 </mrow>
                              </mfenced>
                           </mrow>
                        </mfenced>
                        <mo>≥</mo>
                        <mn>1</mn>
                     </math>
                  </jats:inline-formula>, then <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M13">
                        <mi>f</mi>
                     </math>
                  </jats:inline-formula> from <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M14">
                        <mi>V</mi>
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mi>G</mi>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula> to the interval <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M15">
                        <mfenced open="[" close="]" separators="|">
                           <mrow>
                              <mn>0,1</mn>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula> is called resolving function. Moreover, for two functions <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M16">
                        <mi>f</mi>
                     </math>
                  </jats:inline-formula> and <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M17">
                        <mi>g</mi>
                     </math>
                  </jats:inline-formula>, <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M18">
                        <mi>f</mi>
                     </math>
                  </jats:inline-formula> is called minimal if <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M19">
                        <mi>f</mi>
                        <mo>≤</mo>
                        <mi>g</mi>
                     </math>
                  </jats:inline-formula> and <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M20">
                        <mi>f</mi>
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mi>v</mi>
                           </mrow>
                        </mfenced>
                        <mo>≠</mo>
                        <mi>g</mi>
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mi>v</mi>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula> for at least one <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M21">
                        <mi>v</mi>
                        <mo>∈</mo>
                        <mi>V</mi>
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mi>G</mi>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula>. The fractional metric dimension (FMD) of <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M22">
                        <mi>G</mi>
                     </math>
                  </jats:inline-formula> is denoted by <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M23">
                        <msub>
                           <mrow>
                              <mtext>dim</mtext>
                           </mrow>
                           <mrow>
                              <mi>f</mi>
                           </mrow>
                        </msub>
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mi>G</mi>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula> and defined as <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M24">
                        <msub>
                           <mrow>
                              <mtext>dim</mtext>
                           </mrow>
                           <mrow>
                              <mi>f</mi>
                           </mrow>
                        </msub>
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mi>G</mi>
                           </mrow>
                        </mfenced>
                        <mo>=</mo>
                        <mi mathvariant="normal">min</mi>
                        <mfenced open="{" close="}" separators="|">
                           <mrow>
                              <mfenced open="|" close="|" separators="|">
                                 <mrow>
                                    <mi>g</mi>
                                 </mrow>
                              </mfenced>
                              <mo>:</mo>
                              <mi>g</mi>
                              <mtext> </mtext>
                              <mtext>is a minimal resolving function of </mtext>
                              <mi>G</mi>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula>, where <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M25">
                        <mfenced open="|" close="|" separators="|">
                           <mrow>
                              <mi>g</mi>
                           </mrow>
                        </mfenced>
                        <mo>=</mo>
                        <mstyle displaystyle="true">
                           <msub>
                              <mrow>
                                 <mo stretchy="false">∑</mo>
                              </mrow>
                              <mrow>
                                 <mi>v</mi>
                                 <mo>∈</mo>
                                 <mi>V</mi>
                                 <mfenced open="(" close=")" separators="|">
                                    <mrow>
                                       <mi>G</mi>
                                    </mrow>
                                 </mfenced>
                              </mrow>
                           </msub>
                           <mrow>
                              <mi>g</mi>
                              <mfenced open="(" close=")" separators="|">
                                 <mrow>
                                    <mi>v</mi>
                                 </mrow>
                              </mfenced>
                           </mrow>
                        </mstyle>
                     </math>
                  </jats:inline-formula>. If we take a pair of vertices <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M26">
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mi>u</mi>
                              <mo>,</mo>
                              <mi>v</mi>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula> of <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M27">
                        <mi>G</mi>
                     </math>
                  </jats:inline-formula> as an edge <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M28">
                        <mi>e</mi>
                        <mo>=</mo>
                        <mi>u</mi>
                        <mi>v</mi>
                     </math>
                  </jats:inline-formula> of <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M29">
                        <mi>G</mi>
                     </math>
                  </jats:inline-formula>, then it becomes local fractional metric dimension (LFMD) <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M30">
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <msub>
                                 <mrow>
                                    <mtext>dim</mtext>
                                 </mrow>
                                 <mrow>
                                    <mi>l</mi>
                                    <mi>f</mi>
                                 </mrow>
                              </msub>
                              <mrow>
                                 <mfenced open="(" close=")" separators="|">
                                    <mrow>
                                       <mi>G</mi>
                                    </mrow>
                                 </mfenced>
                              </mrow>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula>. In this paper, local fractional and fractional metric dimensions of <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M31">
                        <mtext>MOG</mtext>
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mi>n</mi>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula> are computed for <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M32">
                        <mi>n</mi>
                        <mo>≅</mo>
                        <mn>1</mn>
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mi mathvariant="normal">mod</mi>
                              <mn>2</mn>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula> 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.</jats:p>


Introduction
For a connected graph G, a vertex x ∈ V(G) is said to resolve a pair (u, v) of vertices of G if d(x, u) ≠ d (x, v). A set S ⊆ V(G) is called a resolving set of G if each pair of vertices of G is resolved by some vertex in S. e metric dimension of G is denoted by dim(G) and is defined as dim(G) � min |S|: S is a resolving set of G . (1) For a pair (u, v) of vertices of G, the resolving neighborhood A resolving function is a real-valued function g: V(G) ⟶ [0, 1] such that g(R(u, v)) ≥ 1 for each distinct pair of vertices of G, where g(R(u, v)) � x∈R (u,v) g (x). A resolving 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. e fractional metric dimension (FMD) of G is denoted by dim f (G) and defined as dim f (G) � min |g|: g is a minimal resolving function of G , (2) where |g| � ≤ v∈V(G) g (v). 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 R(u, v), minimal resolving function g, and FMD dim f (G) become local resolving neighborhood (LR(uv)), local minimal resolving function, and local fractional metric dimension (dim lf (G)), 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][7][8][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. e 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][14][15][16][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 (MOG) consists of metal atoms, where atoms are linked with thes help of organic ligands which act like a linker. erefore, 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. ese frame works are also used in heterogeneous catalyst, drugs delivery, and sensing conductivity [22][23][24][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 MOG is calculated in Section 4. Conclusion is presented in Section 5.

Construction of Metal Organic Graphs
In this section, we describe the construction of metal organic graphs. Let MOG(n) for n ≥ 3 be a metal organic graph with vertex set V(MOG(n)) � u i : Figure 1 shows MOG(n) for n ∈ 5, 7, 9 { }.

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 eorem 1.

Lemma 1.
Let MOG(n) for n ≡ 1 (mod2) and n ≥ 5 be metal organic graph, then Proof. e local resolving neighborhood of metal organic graphs, Let MOG(n) for n ≡ 1(mod2) and n ≥ 9 be a metal organic graph with 1 ≤ t ≤ n. en, the following holds: Proof. (a) e local resolving neighborhood for 1 ≤ k ≤ n, with □ Theorem 1. Let MOG(n) for n ≡ 1(mod2) and n ≥ 5 be the metal organic graphs, then dim lf (MOG(n)) ≤ n/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 eorems 2 and 3.

Lemma 3.
Let MOG(n) for n ≡ 1(mod2) and n ≥ 9 be metal organic graph, then Proof. e resolving neighborhood sets of metal organic □ Lemma 4. Let MOG(n) for n ≡ 1(mod2) and n ≥ 9 be metal organic graphs, then for Proof. (a) e resolving neighborhood for When □ Lemma 6. Let MOG(n) for n ≡ 1(mod2) and n ≥ 9 be metal organic graph. en, the following holds.
Proof. e resolving neighborhood for 1 ≤ i ≤ n, Journal of Chemistry 5 Lemma 7. Let MOG(n) for n ≡ 1(mod2) and n ≥ 9 be metal organic graph.

e FMD of metal organic graph MOG(n) for
Proof. Case 1: when n � 3, then the RNs are as follows.
Since, for 1 ≤ t ≤ 24, the cardinality of each RN R(e t ) is 6, as given in Table 1, which is less than the cardinalities of all other RNs R m of MOG(3), as given in Table 2, this implies that ∪ 24 t�1 R(e t ) � 9 and |R m ∩ ∪ 24 t�1 R(e t )| > R(e t )| � 6.

Proof. In view of Lemmas 3-8 for
. Also, we have |R(xy)| ≤ |R(e t )| for all xy ∈ E(MOG(n)). Moreover, the local resolving neighbourhood of minimum cardinality is not disjoint. erefore, the fractional metric of MOG(n) is given as follows: For |X| � 2n and |R(e t )| � 8, we have Hence, dim f MOG(n) ≤ n/4. Resolving sets (n � 5) Elements Table 5: FMD of metal organic graphs.

Conclusion
In this section, we conclude the obtained results as follows: (i) e FMD of MOG(n) for n ≡ 1(mod2) is obtained as given in Table 7. (ii) We note that as we increase n in MOG(n) for n ≡ 1(mod2), the FMD also increases. (iii) is is one of the important graphs that has same FMD and LFMD having unique resolving and local resolving neighbourhood sets. (iv) e problem is still open to characterize the graphs with same FMD and LFMD.

Data Availability
e 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
ere is no funding source. Elements  Resolving sets (n � 7) Elements