Research On Acyclic Structures with Greatest First Gourava Invariant

Let ξ be a simple connected graph. The ﬁ rst Gourava index of graph ξ is de ﬁ ned as GO , where d ð μ Þ indicates the degree of vertex μ . In this paper, we will ﬁ nd the upper bound of GO 1 ð ξ Þ for trees of given diameter, order, size, and pendent nodes, by using some graph transformations. We will ﬁ nd the extremal trees and also present an ordering of these trees having this index in decreasing order.


Introduction
Topological index is very basic tool in chemical modeling. In molecular graph, atoms are considered as vertices and chemical bonds as edges. In short, the graph is a combination of vertices and edges. First chemical index was Wiener index introduced by Wiener [1] in 1947 to compare the boiling points of few alkanes isomers; he revealed that this index is highly agreed with the boiling point of molecules of alkanes. Later study on QSAR manifested that this index is also helpful to correlate with other quantities like density, critical point, and surface tension. The mathematical formula of this index is given as where d ξ ðμ, ηÞ indicates the distance between the vertices μ and η in ξ. The most studied degree-based indices, i.e., Zagreb indices introduced by Gutman and Das [2], are defined as follows Some properties about these indices are depicted in [3,4]. The 1 st and 2 nd reformulated Zagreb indices were regenerated by Milicevic′ et al. [5] in terms of edge degree, defined as The 1 st and 2 nd Gourava indices were presented by V. R. Kulli in 2017 [6]. These indices are defined as A topological index is a mathematical formula, which has significant applications in chemical graph theory, because it is used as a molecular descriptor to investigate physical as well as chemical properties of chemical structure. Therefore, it is a powerful technique in avoiding high-cost and longterm laboratory experiments. There are 3,000 topological invariants registered till now. All these indices have their applications in chemical graph theory. In these molecular descriptors, Gourava and hyper-Gourava invariants are used to find out the physical and chemical properties (such as entropy, acentric factor, and DHAVP) of octane isomers. The 1 st and 2 nd Gourava invariants highly correlate with entropy and acentric factor, respectively.
In [7], the graph operations for Gourava index are presented. In our present study, we considered that all graphs are simple and connected. For any graph, the degree of a vertex is defined as the number of edges attached to it. The smallest degree of graph ξ is represented by δðξÞ. The vertex in a graph whose degree is one is known as pendent vertex. The neighborhood of a vertex μ is the set of all nodes attached with μ, represented by NðμÞ. There are two types of neighborhood, open neighborhood and closed neighborhood. If NðμÞ includes all the other nodes except μ, then it is called open neighborhood, but if it includes the node μ, then it is called closed neighborhood. Closed neighborhood is defined as N½μ = NðμÞ ∪ fμg (for further notations in graph theory, we refer [8]).
Some bounds of reformulated Zagreb indices are given in [9]. In 2012, Xu and Das [10] established some graph transformations that maximize or minimize the multiplicative sum Zagreb index of graphs and used these graph transformations to determine the extremal graphs among trees, unicyclic, and bicyclic graphs for multiplicative sum Zagreb index. Two years later, in 2014, Ji et al. [11] extended the work of Xu and Das [10] for the 1 st reformulated Zagreb index. In 2017, Gao et al. [12] used the same graph transformations as given in [11] to compute the similar results as computed in [11] but for the hyper-Zagreb index.
Tomescu and Kanwal [13] in 2013 introduced some graph transformations to compute the general sumconnectivity index for acyclic connected graphs of given diameter, order, and pendant vertices and determined the corresponding extremal trees and gave the ordering of trees with minimum general sum-connectivity index. Ilic ′ et al. in 2011 [14] used some graph transformations to find the bounds for unicyclic and bicyclic graphs with respect to degree distance index. Liu et al. [15] analyzed the newly introduced chemical invariant termed as Mostar invariant for tree-like phenylenes and provided a detailed discussion for the obtained results. Liu et al. [16], provided an ordering of acyclic, bicyclic, and tricyclic structures with respect to recently introduced invariants Sombor and reduced Sombor invariants. In [17], Liu et al. determined some degree-based chemical invariants for octahedron networks. Qi et al. [18] put forward computations of several degree-based chemical invariants for rhombus-type silicate and oxide structures. In [19], the authors investigated several degree-based invariants for planar octahedron networks and made comparison of obtained numerical results. Hu et al. [20] analyzed certain distance-based invariants for chemical interconnection networks and analyzed their behavior.
In this work, we are aimed to determine the acyclic structures having maximum values of first Gourava invariant and put forward acyclic structures attaining first five greatest values of first Gourava invariant. Plan of work and methodology behind attaining main results of this work is to apply certain edge swapping transformations to acyclic graphs and observe the behavior of first Gourava invariant. We will see that it increased for the resultant graph and eventually leads us to acyclic structures with the first five bigger values of above-mentioned invariant.

Gourava Index and Graph Transformations
In this section, we use certain graph transformations presented by Ji et al. [11]. Further, we will notice that these transformations increase the GO 1 for trees. These transformations are narrated below.
In B 1 -transformation, let ξ be a nontrivial connected graph having vertices η, μ ∈ ξ, such that Let ξ′ be the graph obtained after applying Figure 1.

Lemma 1.
Let ξ be a connected graph with no cycle and ξ′ be a graph obtained after applying B 1 -transformation (as shown in Figure 1), and then, GO 1 ðB 1 ðξÞÞ = GO 1 ðξ′Þ > GO 1 ðξÞ for any f , h ≥ 1.
Proof. From Figure 1, d ξ′ = f + h + 1 and d ξ ′ ðη 1 Þ = 1. We can easily guess that degree of μ 1 increases, while degree of η 1 decreases after applying transformation, and all other vertices preserve their degrees.
Proof. First, we apply B 1 -transformation to those vertices of T * which are other than diametral path, and we observe that maximum value of GO 1 is obtained for MSðλ 1 , λ 2 , ⋯, λ d * −1 Þ. Then, we apply all those transformations which are explained above, and we conclude that the maximum value is acquired only for λ 1 Corollary 6. (a) Let T * denotes the set of trees with order λ. Then, where 2 ≤ ℓ ≤ m′ ≤ λ − 1.
For example, for λ = 12, we see that