An Approach to the Geometric-Arithmetic Index for Graphs under Transformations’ Fact over Pendent Paths

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Introduction
e advancement in technology mainly networking, computer, biological, and electrical networks made practicable the accurate data transfer within very small duration. e Internet, social media, biological, ecological, and neural networks are few examples of such networks. Telecommunication is based on interconnection networks which used to share data files. Similarly, data exchange using computing devices is also based on computer network through data linkage, optical fiber cable (OFC), and wireless media such as Wi-Fi. Different algorithms are used for directing, arranging/determining numerical calculations, and image processing. Multiprocessor interconnection networks (MINs) are used to design powerful microprocessors and memory chips [1,2].
Graph theory provides a fundamental tool for designing and analyzing such networks. Naturally, the interconnection system is modeled by the graph with processor nodes as vertices and links between these nodes as edges of such graph. Graph theory and interconnection networks provide a thorough understanding of these interrelated topics through their topology. e topology of a graph provides information about the manner in which vertices joined in a graph. e topological indices are graph invariants used to study the topology of graphs. Other than computer networks, graph theory is considered as a powerful tool in different areas of research, such as in coding theory, database management system, circuit design, secret sharing schemes, and theoretical chemistry [3]. e topological descriptors of several interconnection networks are already been computed in [4][5][6]. Along with interconnection networks, these invariants are equally important in chemical graph theory which deals with problems in chemistry using associated graph of chemical compounds [7]. e study of underlying substance using their graph with the help of graph invariants plays an important role in cheminformatics, pharmaceutical sciences, materials science, engineering, and so forth [8,9]. Among theoretical molecular descriptors, topological indices have an impact in chemistry due to the prediction of physio-chemical properties of the underlying substance. Its role in the QSPR/ QSAR analysis to model physical and chemical properties of molecules is also remarkable [10][11][12]. Actually, topological indices are designed on the ground of transformation which associates a numeric value with the graph which characterizes its topology [13]. e first topological index, named Winner index, was proposed in 1947 by Winner [14]. It provides best correlation with the boiling points of alkanes. e discovery of the Winner index provides emerging research platform to the research community. e interest in accurate prediction of physio-chemical properties encouraged the researchers to propose a large class of topological indices. For the first time, an index is defined on the base of end vertices' degrees of the edges by Milan Randić named Randić connectivity index [15]: Due to this reason, it has attained a great attraction of the researchers till now. In 2009, Vukičević and Furtula [16] introduced the geometric-arithmetic index: GA has correlation coefficient of 0.972 with heat of formation of benzene hydrocarbons. Also, in case of "standard enthalpy of vaporization," its accuracy is 9% more than the Randić index. Due to this reason, GA was studied more than all other indices in the last decade. e bonds and extremal characterization of graphs regarding the GA index were studied at some extent in [17][18][19][20][21][22][23][24]. It encouraged us to study the GA index for Γ k,l n and transformed graphs A α (Γ k,l n ) and A β α (Γ k,l n ) under the fact of transformations A α and A β α , 0 ≤ α ≤ l 0 ≤ β ≤ k − 1, respectively. We characterize extremal graphs for all of these families of graphs.

Results and Discussion
roughout this work, let graph Γ k,l n comprise with n-vertex simple connected graph Γ along with k pendent paths of length l ≥ 2 attached with v ∈ Γ, having degree d v ≥ 2. e order of Γ k,l n is n + kl, size is m + kl, and Let graph Γ � Γ(V, E) be with the degree of vertex u ∈ Γ and δ Γ ≤ deg u ≤ Δ Γ and δ Γ ≤ deg v ≤ Δ Γ + 1 be the degrees of v ∈ Γ k,l n . For validity of our proved results, we defined the following list of useful graphs.
Type I: let δ Γ ≤ deg u ≤ Δ Γ , where u ∈ V(Γ). Γ k,l n of type I is obtained by attaching pendent paths of length l with vertices of degree deg u ≥ 2 in such a way that the vertices with pendent path are adjacent to the vertices without pendent paths. e graph of type I is shown in Figure 1(a). Type II: Γ k,l n of type II is the graph of type I with deg u � Δ Γ , ∀u ∈ V(Γ). e graph of type II is shown in Figure 1(b). Before attempting the major problem, we prove the following preposition.
Proof. Let x ≥ 2: 2 Complexity e above calculations implies □ Theorem 1. Let graph Γ k,l n comprise of n-vertex simple connected graph Γ along with k pendent paths of length l ≥ 2 attached with v ∈ Γ of degree d v ≥ 2, maximum degree Δ Γ + 1, and minimum δ Γ . en, Equality holds for graphs of type II. And, Equality holds for graph of type II.
Proof. Let a simple graph Γ be of order n, size m, maximum degree Δ Γ , and minimum δ Γ . Γ k,l n be the graph formed by k number of paths having length l pendent at distinct vertices e edge set of Γ k,l n partitioned as and From equation (10), we have After simplification, we obtain Now, again set which implies from Proposition 1 and the characteristics of f(x) � 2 �� � xy √ /x + y in equation (10). We get the following inequality:

Complexity
After simplification, we obtain Inequalities (13) and (16) complete the proof. Corollary 1 shows generalization of the above defined inequalities. One can get more inequalities of their desire by replacing GA(Γ) with already defined bonds of the GA index. □ Corollary 1. Let graph Γ k,l n comprise of n-vertex simple connected graph Γ along with k pendent paths of length l ≥ 2 attached with v ∈ Γ of degree d v ≥ 2, maximum degree Δ Γ + 1, and minimum δ Γ . en, Equality holds for regular graph of the type II.
Proof. Using results of eorem 1 and inequality regarding the geometric index proved in [25,26], as we get desired results. We use the following transformations as used in [27]. ese transformations have solid effect over GA of Γ k,l n .
Transformation A: let w j ∈ V(Γ), deg w j ≥ 2, for 1 ≤ j ≤ k ≤ n, and paths pendent at w j of the form e transformation A is shown in Figure 2. In eorem 2, we discuss the effect of transformation A over the GA index. Theorem 2. Let graph Γ k,l n comprise of n-vertex simple connected graph Γ along with k pendent paths of length l ≥ 2 attached with v ∈ Γ of degree d v ≥ 2, maximum degree of v ∈ Γ k,l n is Δ Γ + 1, and minimum δ Γ . en, Equality holds for all graphs of type II:

Complexity 5
Equality holds for all graphs of the type II and α � 0.
Proof. Let a simple graph Γ be of order n, size m, minimum degree δ Γ , and maximum Δ Γ . Let Γ k,l n be the graph formed by k number of paths of length l pendent at distinct fully connected vertices of Γ. e geometric-arithmetic index of any graph Γ is e construction of Γ k,l n , l ≥ 2, implies |E(Γ k,l n )| � m + kl. 6 Complexity e cardinality of A 3 is k, i.e.,

After successive applications of transformation A as
/a + x is decreasing, where a ≤ x is a constant. So, for δ Γ minimum degree of Γ and Δ Γ maximum, for any graph, Substituting these changes in equation (24), we have the following inequality: After simplification, we get the required result Now, again, from equation (24) and inequalities, After simplification, we obtain (30) Inequalities (27) and (30) complete the proof. Transformation B: let w j ∈ V(Γ), deg w j ≥ 2, for 1 ≤ j ≤ k ≤ n, and paths pendent at w j of the form w j u 1 j , u 1 j u 2 j , u 2 j u 3 j , . . . , u l− 1 j u l j which comprises Γ k,l n . en, for fixed vertex w 1 , e transformation B is shown in Figure 3 and A β α shown in Figure 4.
Transformation A β α : let 0 ≤ α ≤ l − 1 and 0 ≤ β ≤ k − 1. e transformation A β α is the composition of successive applications of transformation A and B as A α and B β , respectively [27].
In eorem 3, we discuss the effect of transformation A β α over the GA index.
□ Theorem 3. Let graph Γ k,l n comprise of n-vertex simple connected graph Γ along with k pendent paths of length l ≥ 2 attached with v ∈ Γ of degree d v ≥ 2, maximum degree of v ∈ Γ k,l n is Δ Γ + 1, and minimum δ Γ . en, 8 Complexity Equality holds for graph of the type II with α � 0 and β � 0.