On the Inverse Problem for Some Topological Indices

)e study of the inverse problem (IP) based on the topological indices (TIs) deals with the numerical relations to TIs. Mathematically, the IP can be expressed as follows: given a graph parameter/TI that assigns a non-negative integer value (g) to every graph within a given family (G) of graphs, find some G ∈ G for which TI(G) � g. It was initiated by the Zefirov group in Moscow and later Gutman et al. proposed it. In this paper, we have established the IP only for the Y-index, Gourava indices, second hyper-Zagreb index, reformulated first Zagreb index, and reformulated F-index since they are closely related to each other. We have also studied the same which is true for the molecular, tree, unicyclic, and bicyclic graphs.


Introduction
roughout the paper, we consider Ł � (V(Ł), E(Ł)) as a simple (without loops and multiple edge loops) finite graph that contains |V(Ł)| � n vertices and |E(Ł)| � m edges, respectively. e notation d(u/Ł) denotes the degree of a vertex u ∈ V(Ł). All other notations and terminologies used but not clearly stated in this paper may be followed from [1].
In chemical graph theory, a TI, usually known as a molecular descriptor, can be expressed by a real number calculated from a chemical/molecular graph which is the representation of a chemical compound by replacing atoms with vertices and bonds with edges. e TI is calculated for evaluating the information about the atomic constitution and bond characteristics of a molecule/chemical compound. e TI of a molecular graph is a numerical number that enables us to collect information about the concerned chemical structure. It helps us to know its hidden properties without performing experiments [2][3][4]. e TIs also correlate and predict several physical, chemical, biological, pharmaceutical, pharmacological activities/properties from molecular structures of graphs corresponding to real-life situations. e IP is defined as the feasibility of finding/modeling the chemical structure represented by a graph whose index value is equal to a given nonnegative integer for the integer-valued problem. In the QSAR and QSPR studies [5], a method by which it is possible to predict the properties of a given molecular structure is called a forward problem. e inverse problem is concerned that, one can design the exact molecular structure that satisfies the given target properties by applying the forward problem solution.
e most popular as well as the oldest degree-based graph indices are the first and second Zagreb indices. Gutman et al. introduced the first Zagreb index M 1 (Ł) in [6] and second Zagreb index M 2 (Ł) in [7]. ey are defined, respectively, as In 2016, Farahani et al. [8] defined the second hyper-Zagreb index as follows: (2) Kulli [9] introduced the first and second Gourava indices, defined, respectively, as Milicevic proposed the reformulated first Zagreb index EM 1 (Ł) in [10], defined as Liu et al. [11] put forward the reformulated F-index, and it is defined as follows: Alameri et al. [12] introduced Y-index, and it is defined by Graph theory, a branch of mathematics, provides the tools for solving problems of information theory, computer sciences, physics, and chemistry [13][14][15]. e study of the IP is encountered in various fields of science, especially in mathematics and chemistry. e IP for a TI is defined as follows: for a given TI and a non-negative integer value (x), find (chemical) graph (Ł) for which TI(Ł) � x. Also, the inverse existence problem [16] for the pair (G, Γ) can be asked as follows: given (G) a class of graph and function Γ: G ⟶ H, for which g ∈ H there is G ∈ G with Γ(G) � g? e idea of the IP based on TIs was initiated by the Zefirov group in Moscow [5,17,18] and was first proposed by Gutman et al. in [19]. e IP for Wiener index was solved in [20]. In [21], the IP was studied for sigma index as well as for acyclic, unicyclic, and bicyclic graphs. e same problem for the Steiner Wiener index was also solved in [22]. is type of problem for the Zagreb indices, forgotten Zagreb index, and the hyper-Zagreb index was studied in [23]. Tavakoli et al. [24] addressed the IP for first Zagreb index. Also, the IP for some graph indices was investigated in [25]. In [26], Czabarka et al. solved the IP for certain tree parameters. To study more about the inverse problems and the topological indices of graph operations, one can see these references [27][28][29][30][31][32].
ere are so many benefits in the solution of IP. It helps to design of combinatorial libraries for drug discovery in combinatorial chemistry [33]. It may be helpful in speeding up the discovery of lead compounds with desired properties [34]. Also, the IP plays its significance in the application of trees [35] such as the field of algorithms, chemical graph theory, signal processing, and electrical circuits.

Preliminaries
To study the IP for Y-index, we will use the following crucial observation.
Let Ł be a graph having u and v as vertices which are adjacent to each other. We subdivide each edge (uv) by introducing a new vertex (w) (of degree 2) to construct a new graph Ł * (see Figure 1).
Here, the Y-index of the new graph Ł * will be sixteen which is more than the graph Ł. Lemma 1. By applying the transformation Ł ⟶ Ł * , the Y-index value will be increased by 16. at is, Proof. Let us consider the graph Ł with vertices u and v of degrees x and y, respectively. Since in the new constructed graph Ł * , a new vertex w is inserted between u and v,

Main Results and Discussion
In graph theory, the IP based on the TIs is an interesting one among the problems associated with the estimation of different graph invariants/TIs such as chromatic number, connectivity, girth, and number of independent sets. e IP plays a crucial role in many areas of science, especially in mathematics. In this section, we have investigated the IP for Y-index, Gourava indices, second hyper-Zagreb index, reformulated first Zagreb index, and reformulated F-index. Additionally, we have developed the same problem for molecular, tree, acyclic, unicyclic, and bicyclic graphs.

e IP for Y-Index.
Here we will discuss the IP for Y-index.
Proof. Proof. To prove the theorem, we establish a set of Consider the cyclic graphs C n for n ≥ 3. Clearly, in Figure 2 and Y(C n ) � 16n for n ≥ 3. Now we apply Lemma 1 for each graph in Figure 2. us, Y(Y 0 ) takes all those even positive integer values which are divisible by 16, except 16 and 32.
In Figure 2(b), Y(P 2 ) � 2. By applying the transformation in Lemma 1, we arrive at graphs whose Y-index values are 18, 34, 50, 66, and so on. ere are the path graphs. ere exist no connected graphs with the Y-indices mentioned in Table 1

Corollary 2. Let Ł be a connected unicyclic or bicyclic graph.
en, there exists the Y-index of the form 16h + 2k for all nonnegative integers where h ≠ 0 and 1 ≤ k ≤ 7.
Proof. Proof. Let Ł 1 be a unicyclic graph which is obtained by adding a path of length two to any vertex u of the cyclic graph C n for n � 5.

Theorem 5.
e first reformulated Zagreb index of a connected graph can take all positive even integer values except for 4 and 8.

Conclusion
e inverse problem is one of the recent problems of graph theory related to the applicative area. Here, we have studied the IP based on some topological graph indices such as Y-index, Gourava indices, second hyper-Zagreb index, reformulated first Zagreb index, and reformulated F-index. We have studied the inverse problems for the aforesaid indices since they are closely related to each other. We have also investigated the results for tree, molecular, unicyclic, and bicyclic graphs. e inverse problem is still open for other graph indices and other molecular structures.

Data Availability
No data were used to support this study.