Characterization of (Molecular) Graphs with Fractional Metric Dimension as Unity

Distance-based dimensions provide the foreground for the identification of chemical compounds that are chemically and structurally different but show similarity in different reactions. &e reason behind this similarity is the occurrence of a set S of atoms and their same relative distances to some ordered set T of atoms in both compounds. In this article, the aforementioned problem is considered as a test case for characterising the (molecular) graphs bearing the fractional metric dimension (FMD) as 1. For the illustration of the theoretical development, it is shown that the FMD of path graph is unity. Moreover, we evaluated the extremal values of fractional metric dimension of a tetrahedral diamond lattice.


Introduction
Day by day, the nexus of chemistry is progressing by the advancements in drug discovery, formation of chemical compounds, and development of testing kits for the diagnosis of different diseases and medical anomalies. Besides different concepts that arose as a result of the emergence of cheminformatics, distance-based dimensions also have their stake in this concern. Assume that, in a graph C, the shortest path between the 2 vertices s, t is given by d(s, t). Let S � s 1 , s 2 , s 3 , . . . , s k ⊆V(C) and u ∈ V(C); then, the k-tuple metric form of S in terms of u is given by r(u|S) � (d(u, s 1 ), d(u, s 2 ), d(u, s 3 ), . . . , d(u, s k )). e set S becomes a resolving set having k elements for a graph C if each pairs of vertices in C bears distinct k-tuple metric forms. e resolving set with minimum cardinality in C forms its metric basis, and its cardinality represents its metric dimension. e terminology of resolving sets was introduced by Slatter [1,2] by naming them as locating sets. Harary and Melter [3] personally discovered these terminologies and called them as the metric dimension of C. Afterward, many researchers have studied different graph structures for the calculation of metric dimensions. e results for the metric dimensions of path, cycle, Peterson, and generalized Peterson graphs can be found in [4][5][6]. For various results on metric dimensions of graphs, we refer to [7][8][9] and [10]. Chartrand et al. [11] employed metric dimension to find the solution of an integer programming problem (IPP). Subsequently, Currie and Oellermann introduced the concept of fractional metric dimension (FMD) and obtained the solution of IPP with higher accuracy [12]. Arumugam and Mathew [13] after discovering the hidden properties of FMD formally defined it. Since then, many researchers have tried their luck in this area by attacking different graph structures.
e results for the FMD of graph structures as obtained from Cartesian, hierarchial, corona, lexicographic, and comb product of connected graph structures can be seen in [14][15][16] and [17,18]. Recently, Liu et al. [19] calculated the fractional metric dimension of the generalized Jahangir graph J 5,k and Raza et al. calculated the FMD of a metal organic network [20,21]. Alisyah et al. presented the concept of local fractional metric dimension (LFMD) and found the LFMD of the corona product of two connected networks [22]. Liu et al. calculated the LFMD of rotationally symmetric and planar networks [23]. Recently, Javaid et al. calculated the bounds for the LFMD of connected and cycle-related networks in [24,25].
Johnson [26,27] employed the concept of metric dimension for creating proficiency of large datasets of chemical graph structures. e mathematical study of chemical structures concerns the development of mathematical classification of chemical compounds. e graphtheoretic version of chemical compounds naturally exists. Despite having different chemical and structural aspects, two chemical compounds show similar behaviour during the reactions. e reason behind this peculiarity is the existence of certain common substructures within these compounds. If in two compounds, the elements of the set S of atoms and the elements of the ordered set T are relatively equidistant, then we call these compounds to be similar or equivalent [28]. Finding a T with minimum cardinality such that the ordered lists associated with every two distinct vertices of S are distinct has applications to classification problems in chemistry, as described in [11].
In this article, we are going to characterise the (molecular) graphs with FMDs as unity. As a test case, we have considered the allotropic form of carbon called by tetrahedral diamond developed by Ali et al. is article propels in the following manner: Section 1 is for introduction, Section 2 is devoted for the applications of FMD in chemistry, Section 3 is for preliminaries, Section 4 concerns with the development of a tool for the characterization of graphs with FMD as 1, and Section 5 deals with the resolving neighbourhood sets of TD(n). In Section 6, we have calculated the FMD of TD(n). Section 7 gives the conclusion.

Applications in Chemistry
In a molecular graph, atoms are denoted by nodes and bond between them by edges. e fraternity of chemists and pharmacists is always in search of finding out chemical compounds in some collection bearing physiochemical properties in common at some particular places. is objective is achieved by the identification of the substructure having the smallest number of atoms. In graph theory, this problem is the same as finding the FMD of the graph under consideration. In this way, druggists and chemists will be able to capture the aforementioned features of these compounds and comprehend whether they are responsible for some pharmacological activity for a newly developed drug. For more on the applications like these, see [11].

Preliminaries
For c ∈ V(C) and a, b Suppose a connected network C(V(C), E(C)) having order p. A function τ: . An RF η of C is known as a minimal resolving function (MRF) if any function ϕ: V(C) ⟶ [0, 1] such that ϕ ≤ η and ϕ(z) ≠ η(z) for at least one z ∈ V(C) that is not an RF of C. en, the FMD of the network C is given by dim f (C) � min |η|:η is the MRF of C}, where |η| � z∈V(C) η(z) [13].

Construction of Tetrahedral Diamond.
e tetrahedral diamond graph is an n-dimensional lattice, comprising n i layers where 1 ≤ i ≤ n. Figures 1 and 2 show TD(n) for 3 ≤ n ≤ 5.
Each n i layer is having n 2 i vertices, ((n i − 2)(n i − 1)/2) hexagons, and three pendent edges. e vertices of each layer are denoted by v e first layer is isomorphic to K 1 , and layer two is isomorphic to K 1.3 , whereas for 1 ≤ i ≤ n, each n i−1 layer is the subgraph of the n i -th layer. Hence, the graph formed by each layer is denoted by S  Figure 3 shows all the subsets of TD(n).
It can be seen from the figure that, in each layer, v Apart from them, every vertex with an odd label in the n i − 1 layer is adjacent to the vertex with an even label in the n i layer and vice versa.

Characterization of Graphs with FMD as Unity
In this section of the article, we are giving generic criteria for identifying graphs with FMD as 1. ese criteria have been shaped up as a theorem given below.

Theorem 1. Let C be a connected graph and R a, b { } be a resolving neighbourhood set of the pair of vertices
where |V(C)| ≥ 3.
where c is a real number that approaches to 1 and X � V(C). For a, b ∈ V(C) and c ⟶ 1, where w � (|V(C)||V(C) − 1|/2). It implies that ψ is a resolving function. To check that ψ is a minimal resolving function, assume that there is another minimal resolving function τ such that τ ≤ ψ. By definition, τ(x) < ψ(x) for some x ∈ X. Now, for some resolving neighbourhood set R, we have Consequently, τ(R) < 1 which implies that τ is not a resolving function. us, ψ is a minimal resolving function. Let ψ be another minimal resolving function of C. Now, we have the following possibilities: for all x ∈ X, then for each resolving neighbourhood set R, ψ(R) < 1⇒ψ is not a resolving function; therefore, this case does not hold.   Journal of Chemistry 3 for all x ∈ X, then we have the following subcases: As for some x ∈ X, this case is a consequent of the abovementioned two cases (Case I and II); therefore, we have dim frac (C) � 1.
Consequently, from Case 1-3, we arrive at the following conclusion: Using the result presented above, we are now going to prove the following fact: Proposition 1. Suppose that, for any n ≥ 3, G � P n ; then, dim frac (G) � 1.

Proof
Case 1: for n � 3: the resolving neighbourhood sets for the current case are R 1 � R a 1 , a 2 � a 1 , a 2 , a 3 }, It can be seen that ∩ 3 t�1 R t � a 1 , a 3 ≠ Φ. erefore, from eorem 1, we arrive at the conclusion thatdim frac (P 3 ) � 1. Case 2: for n ≥ 4: the resolving neighbourhood sets of P n are It can be seen that ∩ erefore, from eorem 1, it implies that dim frac P n � 1.

Resolving Neighbourhood Sets of TD(n)
In this section, we present some the results regarding the resolving neighbourhood sets of TD(n). Lemma 1 deals with the resolving neighbourhoods of TD(n) having minimum cardinality followed by Lemma 2 and Lemma 3 that are concerned with resolving neighbourhood sets of maximum cardinalities. Lemma 1. Suppose that C � TD(n) is an n-dimensional tetrahedral diamond lattice. en, the minimum resolving neighbourhood sets are as follows: (b) First of all, we introduce a notation for simplification.
Also, by the symmetry of the network, . It can be seen that □ Lemma 3. Suppose that C � TD(n) is an n-dimensional tetrahedral diamond lattice with n ≥ 4 and n ≡ 0(modn). en,

Fractional Metric Dimension of TD(n)
In this section, the FMD of TD(n) is calculated and the criterion of their evaluation is devised by the following result.
Case 2: when n ≥ 5. e required minimum resolving neighbourhood sets are . Also, the resolving neighbourhood sets with maximum cardinality of |V(C)|, as clarified by Lemma  Moreover, Journal of Chemistry To find the minimum value for the dim frac (C), we define a mapping κ: where c � η � (n(2n 2 + 3n + 1)/6). Assigning the labels to the elements of ⋃ ∪ 2 L�1 R � L and summing them up, we get |κ| � Similarly, for the maximum value of dim frac (C), we define another mapping τ: It can be seen that τ is a resolving function for C with n ≥ 3 because On the contrary, assume that there is another resolving function ρ such that ρ(u) ≤ τ(u), for at least one u ∈ V(C)ρ(u) ≠ τ(u). As a consequence, is a resolving neighbourhood of C with minimum cardinality λ. It shows that ρ is not a resolving function which is a contradiction. erefore, τ is a minimal resolving function that attains minimum |τ| for C. Since all the R t have nonempty intersection, there is another minimal resolving function of τ of C such that |τ| ≤ |τ|. Hence, assigning (1/λ) to the vertices of C in ∪ 3 t�1 R t and calculating the summation of all the weights, we get In the end, we arrive at the following finding: □ Theorem 3. If C � TD(n) is an n-dimensional tetrahedral diamond lattice with n ≥ 4 and n ≡ 0(mod2), then Proof Case 1: when n � 4. e resolving neighbourhood sets are as shown in Tables 4-6.

Conclusions
We conclude our discussion by the following remarks: (i) In this article, we have made a characterization of graphs having the FMD as unity (ii) It is computed that the FMD of the path is 1 that strengthens the result proved in [13] (iii) We have calculated the extremal values of FMD of TD(n) as (i) for n ≡ 0(mod2), 1 < dim frac (C) ≤ 2 and (ii) for n ≡ 1(mod2), 1 < dim frac (C) ≤ 2(n 3 + 6n 2 + 11n − 30/n 3 + 3n 2 + 5n + 3) (iv) Now, we close our discussion with the open problem that investigates the families of graphs other than P n having FMD as unity

Data Availability
All the data are included within this article. However, the reader may contact the corresponding author for more details of the data.

Conflicts of Interest
e authors have no conflicts of interest.