On Acyclic Structures with Greatest First Gourava Invariant

Hindawi Journal of Chemistry Volume 2022, Article ID 6602899, 9 pages https://doi.org/10.1155/2022/6602899 Research Article On Acyclic Structures with Greatest First Gourava Invariant Mariam Imtiaz,1 Maria Naseem,2 Misbah Arshad,3 Salma Kanwal ,4 and Maria Liaqat4 1 University of Engineering and Technology, KSK Campus, Lahore, Pakistan Department of Mathematics, Faculty of Science, University of Central Punjab, Lahore, Pakistan 3 COMSATS University Islamabad, Sahiwal Campus, Pakistan 4 Lahore College for Women University, Lahore, Pakistan 2 Correspondence should be addressed to Salma Kanwal; [email protected] Received 27 December 2021; Accepted 1 April 2022; Published 25 April 2022 Academic Editor: Haidar Ali Copyright © 2022 Mariam Imtiaz et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Let ξ be a simple connected graph. The first Gourava index of graph ξ is defined as GO1 ðξÞ = ∑μη∈EðξÞ ½dðμÞ + dðηÞ + dðμÞdðηފ, where dðμÞ indicates the degree of vertex μ. In this paper, we will find the upper bound of GO1 ðξÞ for trees of given diameter, order, size, and pendent nodes, by using some graph transformations. We will find the extremal trees and also present an ordering of these trees having this index in decreasing order. 1. 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 W ðξÞ = 〠 dξ ðμ, ηÞ, ð1Þ fμ,ηg M 2 ðξÞ = 〠 μη∈EðξÞ μη∈EðξÞ
dξ ðμÞdξ ðηÞ : EM1 ðξÞ = 〠 f1 ∈EðξÞ  d f1 2 , ð2Þ ð3Þ EM 2 ðξÞ = 〠 d f1 dg1 : f1 ~g1 The 1st and 2nd Gourava indices were presented by V. R. Kulli in 2017 [6]. These indices are defined as GO1 ðξÞ = 〠 ½d ðμÞ + d ðηÞ + dðμÞdðηފ, μη∈EðξÞ where dξ 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
2 M 1 ðξÞ = 〠 dξ ðμÞ = 〠 dξ ðμÞ + dξ ðηÞ, μ,η∈V ðξÞ Some properties about these indices are depicted in [3, 4]. The 1st and 2nd reformulated Zagreb indices were regenerated by Milic̆evic ′ et al. [5] in terms of edge degree, defined as GO2 ðξÞ = 〠 ½d ðμÞ + d ðηފðd ðμÞd ðηÞÞ: ð4Þ μη∈EðξÞ 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 2 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 1st and 2nd 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 N½μŠ = 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 1st 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 Journal of Chemistry η1 μ1,2 μ1,1 η1,1 μ1 η1 η1,2 μ1,2 η1,h μ1, f μ1,1 η1,1 η1,2 μ1 μ1, f η1,h Figure 1: B1 -tranform 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. 2. 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 GO1 for trees. These transformations are narrated below. In B1 ‐transformation, let ξ be a nontrivial connected graph having vertices η, μ ∈ ξ, such that Nðη1 Þ = μ1 , η1,1 , η1,2 , ⋯, η1,h and Nðμ1 Þ = η1 , μ1,1 , μ1,2 , ⋯, μ1, f , where η1 and μ1 have no common neighbors in ξ, f ≥ 0 and h ≥ 1. Let ξ ′ be the graph obtained after applying B1 − transformation, such that ξ ′ = ξ − η1 η1,1 , η1 η1,2 , ⋯, η1 η1,h + μ1 η1,1 , μ1 η1,2 , ⋯, μ1 η1,h as explained in Figure 1. Lemma 1. Let ξ be a connected graph with no cycle and ξ ′ be a graph obtained after applying B1 -transformation (as shown in Figure 1), and then, GO1 ðB1 ðξÞÞ = GO1 ðξ ′ Þ > GO1 ðξÞ for any f , h ≥ 1. Proof. From Figure 1, d ξ′ = f + h + 1 and d ξ ′ ðη1 Þ = 1. We can easily guess that degree of μ1 increases, while degree of η1 decreases after applying transformation, and all other vertices preserve their degrees.   GO1 ξ ′ − GO1 ðξÞ f n o

= 〠 d ξ ′ μ1,i + d ξ ′ ðμ1 Þ + dξ ′ μ1,i d ξ ′ ðμ1 Þ i=1 h n     o + 〠 d ξ ′ η1, j + d ξ ′ μ1 + dξ , η1, j dξ ′ ðμ1 Þ j=1 n + d ξ , ðμ1 Þ + d ξ ′ ðη1 Þ + dξ ′ ðμ1 Þd ξ ′ ðη1 Þ o f 

− 〠 dξ μ1,i + d ξ μ1 + dξ μ1,i dξ ðμ1 Þ i=1 h n     o − 〠 d ξ η1, j + dξ ðη1 + d ξ η1, j d ξ ðη1 Þ j=1  − d ξ ðμ 1 Þ + d ξ ðη 1 Þ + d ξ ðμ 1 Þd ξ ð η 1 Þ Journal of Chemistry 3 η1,1 η1,h γ1,1 μ1,1 η1,1 μ1 η1 γ1 μ1,1 b1 μ1, f γ1,1 η1,h μ1 η1 γ1 b1 μ1, f Figure 2: B2 -transform. Proof. Here d ξ ′ ðμ1,i Þ = d ξ ðμ1,i Þ, and d ξ ′ ðη1,i Þ = dξ ðη1,i Þ, and after transformation, dξ ′ ðμ1 Þ > d ξ ðμ1 Þ and d ξ ′ ðη1 Þ < dξ ðη1 Þ. f hn o

= 〠 dξ , μ1,i + d ξ ′ ðμ1 Þ + dξ ′ μ1,i d ξ , ðμ1 Þ i=1 i 

− dξ μ1,i + d ξ ðμ1 Þ + dξ μ1,i dξ ðμ1 Þ   GO1 ξ ′ − GO1 ðξÞ h hn     o + 〠 d ξ , η1, j + d ξ , ðμ1 + d ξ , η1, j dξ , ðμ1 Þ f n o

= 〠 d ξ ′ μ1,i + d ξ ′ ðμ1 Þ + dξ ′ μ1,i d ξ ′ ðμ1 Þ j=1 i=1   n   oi − d ξ η1, j + dξ ðη1 + d ξ η1, j d ξ ðη1 Þ n o + d ξ , ðμ1 Þ + d ξ , ðη1 Þ + dξ ′ ðμ1 Þdξ ′ ðη1 Þ  − dξ ðμ1 Þ + d ξ ðη1 Þ + d ξ ðμ1 Þd ξ ðη1 Þ h n     o + 〠 dξ ′ η1, j + dξ ′ ðμ1 Þ + d ξ ′ η1, j dξ ′ ðμ1 Þ j=1 n o + dξ ′ ðμ1 Þ + d ξ ′ ðη1 Þ + dξ ′ ðμ1 Þd ξ ′ ðη1 Þ n o + dξ ′ ðη1 Þ + d ξ ′ ðγ1 Þ + d ξ ′ ðη1 Þd ξ ′ ðγ1 Þ f hn

o = 〠 dξ ′ μ1,i + ð f + h + 1Þ + ð f + h + 1Þd ξ ′ μ1,i i=1 f 
− 〠 d ξ μ1,i dξ ðη1 Þ

i − dξ μ1,i + ð f + 1Þ + ð f + 1Þd ξ μ1,i i=1 h hn    o + 〠 d ξ ′ η1, j + ð f + h + 1Þ + ð f + h + 1Þdξ ′ η1, j j=1 n    oi − d ξ η1, j + ðh + 1Þ + ðh + 1Þdξ η1, j + f ð f + h + 1Þ + 1 + ð f + h + 1Þg − d ξ ðμ1 Þ + dξ ðη1 Þ + dξ ðμ1 Þd ξ ðη1 Þ  − d ξ ðη1 Þ + dξ ðγ1 Þ + d ξ ðη1 Þdξ ðγ1 Þ f hn o

= 〠 d ξ ′ μ1,i + d ξ ′ ðμ1 Þ + dξ ′ μ1,i d ξ ′ ðμ1 Þ i=1 i 

− d ξ μ1,i + d ξ ðμ1 Þ + d ξ μ1,i d ξ ðμ1 Þ − f ð f + 1Þ + ðh + 1Þ + ð f + 1Þðh + 1Þg f h n  o 
− fh = 〠 h + hd ξ μ1,i + 〠 f + f dξ η1, j i=1  h +〠 j=1 j=1 n  o 
− fh = f h + hd ξ μ1,i + h f + f dξ η1, j n   o
= hf d ξ μ1,i + hf d ξ η1, j + hf n   o
= hf d ξ μ1,i + d ξ η1, j + 1 > 0 ⇒ GO1 ðB1 ðξÞÞ   = GO1 ξ ′ > GO1 ðξÞ ð5Þ   hn   o d ξ ′ η1, j + d ξ ′ ðμ1 Þ + dξ ′ η1, j d ξ ′ ðμ1 Þ   n   oi − dξ η1, j + d ξ ðη1 Þ + dξ η1, j dξ ðη1 Þ n o + dξ ′ ðμ1 Þ + d ξ ′ ðη1 Þ + dξ ′ ðμ1 Þd ξ ′ ðη1 Þ  − d ξ ðμ1 Þ + dξ ðη1 Þ + dξ ðμ1 Þd ξ ðη1 Þ n o + dξ ′ ðη1 Þ + d ξ ′ ðγ1 Þ + d ξ ′ ðη1 Þd ξ ′ ðγ1 Þ  − d ξ ðη1 Þ + dξ ðγ1 Þ + d ξ ðη1 Þdξ ðγ1 Þ f hn
o
= 〠 d ξ ′ μ1,i + ð f + h + 1Þ + ð f + h + 1Þd ξ ′ μ1,i i=1 Lemma 2. Let ξ be an acyclic graph, and ξ ′ is obtained after applying B2 -transformation (as shown in Figure 2) where d ξ ðγ1 , b1 Þ ≥ 1; then,   GO1 ðB2 ðξÞÞ = GO1 ξ ′ > GO1 ðξÞ, for any f > 1 and h, ℓ ≥ 1. 
i
− d ξ μ1,i + ð f + 1Þ + ð f + 1Þdξ μ1,i h +〠 j=1 ð6Þ hn    o d ξ ′ η1, j + ð f + h + 1Þ + ð f + h + 1Þd ξ ′ η1, j n    oi − dξ η1, j + ðh + 2Þ + ðh + 2Þd ξ η1, j + f ð f + h + 1Þ + 2 + 2ð f + h + 1Þg 4 Journal of Chemistry γ1,1 η1,h η1,h γ1,1 η1,1 η1,1 µ1 g1 g2 η1 γ1 g3 µ1 µ1,1 g1 g2 η1 γ1 g3 µ1,f Figure 3: B3 -transform. h n     o − 〠 d ξ η1, j + dξ ðη1 Þ + dξ η1, j dξ ðη1 Þ − f ð f + 1Þ + ðh + 2Þ + ð f + 1Þðh + 2Þg + f2 + ðℓ + 1Þ + 2ðℓ + 1Þg j=1 − fðh + 2Þ + ðℓ + 1Þ + ðh + 2Þðℓ + 1Þg f h  i
i h h = 〠 h + hd ξ ′ μ1,i + 〠 ð f − 1Þ + ð f − 1Þdξ ′ η1, j i=1 j=1 + ðh − hf Þ + ð−2h − hℓÞ  i h
i h = f h + hd ξ ′ μ1,i + h ð f − 1Þ + ð f − 1Þdξ ′ η1, j − dξ ðμ1 Þ + d ξ ðg1 Þ + d ξ ðμ1 Þd ξ ðg1 Þ  − dξ ðg2 Þ + d ξ ðη1 Þ + d ξ ðg2 Þd ξ ðη1 Þ  − dξ ðη1 Þ + d ξ ðγ1 Þ + d ξ ðη1 Þd ξ ðγ1 Þ f hn o

= 〠 dξ ′ μ1,i + dξ ′ ðμ1 Þ + d ξ ′ μ1,i dξ ′ ðμ1 Þ i=1 i 

− dξ μ1,i + d ξ ðμ1 Þ + dξ μ1,i dξ ðμ1 Þ + ðh − hf Þ + ð−2h − hℓÞ = hf + hf + hf − h + hf − h + h − hf − 2h − hℓ = 3hf − 3h − hℓ = h½3f − ðℓ + 3ފ > 0 ⇒ GO1 ðB2 ðξÞÞ   = GO1 ξ ′ > GO1 ðξÞ: h hn     o + 〠 d ξ ′ η1, j + d ξ ′ ðμ1 Þ + d ξ ′ η1, j d ξ ′ ðμ1 Þ j=1 ð7Þ Lemma 3. Let ξ be an acyclic graph and B3 ðξÞ = ξ ′ is obtained after applying B3 -transformation (as shown in Figure 3), where d ξ ðμ1 , η1 Þ = dB3 ðξÞ ðμ1 , η1 Þ ≥ 2 and d ξ ðγ1 , g3 Þ = dB3 ðγ1 , g3 Þ ≥ 0. If f > 1, h, ℓ ≥ 1, then,   GO1 ðB3 ðξÞÞ = GO1 ξ ′ > GO1 ðξÞ  ð8Þ   n   oi − d ξ η1, j + dξ ðη1 Þ + dξ η1, j d ξ ðη1 Þ n o + d ξ ′ ðμ1 Þ + dξ ′ ðg1 Þ + d ξ ′ ðμ1 Þd ξ ′ ðg1 Þ  − dξ ðμ1 Þ + d ξ ðg1 Þ + d ξ ðμ1 Þd ξ ðg1 Þ n o + d ξ ′ ðg2 Þ + dξ ′ ðη1 Þ + d ξ ′ ðg2 Þdξ ′ ðη1 Þ  − dξ ðg2 Þ + d ξ ðη1 Þ + d ξ ðg2 Þd ξ ðη1 Þ n o + d ξ ′ ðη1 Þ + dξ ′ ðγ1 Þ + d ξ ′ ðη1 Þdξ ′ ðγ1 Þ  − dξ ðη1 Þ + d ξ ðγ1 Þ + d ξ ðη1 Þd ξ ðγ1 Þ f hn
o
= 〠 dξ ′ μ1,i + ð f + h + 1Þ + ð f + h + 1Þdξ ′ μ1,i i=1

i − dξ μ1,i + ð f + 1Þ + ð f + 1Þd ξ μ1,i Proof. Like the previous lemma, we have h hn  o   + 〠 d ξ ′ η1, j + ð f + h + 1Þ + ð f + h + 1Þdξ ′ η1, j j=1 f GO1 ðξÞ = 〠 i=1 hn oi

dξ ′ μ1,i + dξ ′ ðμ1 Þ + d ξ ′ μ1,i dξ ′ ðμ1 Þ h hn     oi + 〠 dξ ′ η1, j + dξ ′ ðμ1 Þ + d ξ ′ η1, j dξ ′ ðμ1 Þ j=1 n o + d ξ ′ ðμ1 Þ + dξ ′ ðg1 Þ + dξ ′ ðμ1 Þdξ ′ ðg1 Þ n o + d ξ ′ ðg2 Þ + d ξ ′ ðη1 Þ + dξ ′ ðg2 Þdξ ′ ðη1 Þ n o + d ξ ′ ðη1 Þ + dξ ′ ðγ1 Þ + dξ ′ ðη1 Þdξ ′ ðγ1 Þ f 

− 〠 d ξ μ1,i + d ξ ðμ1 Þ + d ξ μ1,i d ξ ðμ1 Þ i=1 n    oi − d ξ η1, j + ðh + 2Þ + ðh + 2Þd ξ η1, j + f ð f + h + 1Þ + 2 + 2ð f + h + 1Þg − f ð f + 1Þ + 2 + 2ð f + 1Þg + f2 + 2 + 4g − fð2 + ðh + 2Þ + 2ðh + 2Þg + fð2 + ðℓ + 1Þ + 2ðℓ + 1Þg − fðh + 2Þ + ðℓ + 1Þ + ðh + 2Þðℓ + 1Þg f h  i
i h h = 〠 h + hdξ ′ μ1,i + 〠 ð f − 1Þ + ð f − 1Þd ξ ′ η1, j i=1 j=1 − hℓ − 2h = f ½h + hŠ + h½ ð f − 1Þ + ð f − 1ފ − hℓ − 2h Journal of Chemistry 5 γ1,1 μ1,1 γ1,1 μ1,1 γ1,2 g1 μ1 μ1, f g2 γ1,2 γ1 μ1 μ1, f g1 g2 γ1 Figure 4: B4 -transform. = f h + f h + f h − h + f h − h − hℓ − 2h = 4hf − 4h − hℓ = h½hf − ðℓ + 4ފ > 0 ⇒ GO1 ðB3 ðξÞÞ   = GO1 ξ ′ > GO1 ðξÞ ð9Þ Lemma 4. Let ξ be acyclic connected graph, and ξ ′ = B4 ðξÞ is obtained after applying B4 -transformation(as shown in Figure 4) for any f > ðℓ − 1Þ; we have GO1 ðB4 ðξÞÞ = GOðξ ′ Þ > GO1 ðξÞ. Proof. Since dξ ðμ1 , γ1 Þ ≥ 1, and if dξ ðμ1 , γ1 Þ ≥ 2, then d ξ′ ðμ1 Þ + d ξ′ ðγ1 Þ = ðf + 1Þ + ðℓ + 1Þ = dξ ðμ1 Þ + dξ ðγ1 Þ; now by using definition of GO1 ðξÞ, we have j=1 + f1 + ð f + 2Þ + ð f + 2Þg − f1 + ðℓ + 1Þ + ðℓ + 1Þg ℓ−1n     o + 〠 d ξ ′ γ1, j + d ξ ′ ðγ1 Þ + d ξ ′ γ1, j dξ ′ ðγ1 Þ j=1 i=1 = f ð2Þ + ð−2Þðℓ − 1Þ + f1 + ð f + 2Þ + ð f + 2Þg − f1 + ðℓ + 1Þ + ðℓ + 1Þg + 3 − 3  − 〠 dξ μ1,i + d ξ ðμ1 Þ + dξ μ1,i dξ ðμ1 Þ i=1 − 〠 d ξ γ1, j + dξ ðγ1 Þ + d ξ γ1, j d ξ ðγ1 Þ j=1 o  − dξ ðγℓ Þ + d ξ ðγ1 Þ + d ξ ðγℓ Þd ξ ðγ1 Þ  − dξ ðμ1 Þ + dξ ðg1 Þ + d ξ ðμ1 Þdξ ðg1 Þ  − dξ ðγ1 Þ + d ξ ðg2 Þ + d ξ ðγ1 Þd ξ ðg2 Þ f hn o
= 〠 dξ ′ μ1,i + dξ ′ ðμ1 Þ + d ξ ′ μ1,i dξ ′ ðμ1 Þ i=1  − dξ μ1,i + d ξ ðμ1 Þ + dξ μ1,i dξ ðμ1 Þ ℓ−1n     o + 〠 d ξ ′ γ1, j + d ξ ′ ðγ1 Þ + d ξ ′ γ1, j dξ ′ ðγ1 Þ j=1 j=1 + f1 + ð f + 2Þ + ð f + 2Þg − f1 + ðℓ + 1Þ + ðℓ + 1Þg + 3 − 3 f  + f ð f + 2 Þ + 2 + 2 ð f + 2 Þg − f ð f + 1 Þ + 2 + 2 ð f + 1 Þg + fℓ + 2 + 2ðℓÞg − fðℓ + 1Þ + 2 + 2ðℓ + 1Þg f h i ℓ−1  = 〠 1 + d ξ ′ ðμ1 , iÞ + 〠 −1 − dξ ðμ1 , jÞ n o + d ξ ′ ðγℓ Þ + dξ ′ ðμ1 Þ + d ξ ′ ðγℓ Þdξ ′ ðμ1 Þ n o + d ξ ′ ðμ1 Þ + dξ ′ ðg1 Þ + d ξ ′ ðμ1 Þd ξ ′ ðg1 Þ n o + d ξ ′ ðγ1 Þ + dξ ′ ðg2 Þ + d ξ ′ ðγ1 Þdξ ′ ðg2 Þ   o   d ξ ′ γ1, j + ðℓÞ + ℓd ξ ′ γ1, j n    oi − d ξ γ1, j + ðℓ + 1Þ + ðℓ + 1Þd ξ γ1, j i=1  n oi − d ξ ′ μ1,i + ð f + 1Þ + ð f + 1Þdξ ′ μ1,i ℓ−1hn f n o = 〠 d ξ ′ μ1,i + d ξ ′ ðμ1 Þ + dξ ′ μ1,i d ξ ′ ðμ1 Þ  i=1 +〠   GO1 ξ ′ − GO1 ðξÞ ℓ−1n   n   o − dξ γ1, j + d ξ ðγ1 Þ + dξ γ1, j d ξ ðγ1 Þ n + d ξ ′ ðγℓ Þ + dξ ′ ðμ1 Þ + d ξ ′ ðγℓ Þdξ ′ ðμ1 Þ  − dξ ðγℓ Þ + d ξ ðγ1 Þ + d ξ ðγℓ Þd ξ ðγ1 Þ n o + d ξ ′ ðμ1 Þ + dξ ′ ðg1 Þ + d ξ ′ ðμ1 Þd ξ ′ ðg1 Þ  − dξ ðμ1 Þ + dξ ðg1 Þ + d ξ ðμ1 Þdξ ðg1 Þ n o + d ξ ′ ðγ1 Þ + dξ ′ ðg2 Þ + d ξ ′ ðγ1 Þdξ ′ ðg2 Þ  − dξ ðγ1 Þ + d ξ ðg2 Þ + d ξ ðγ1 Þd ξ ðg2 Þ f hn o = 〠 d ξ ′ μ1,i + ð f + 2Þ + ð f + 2Þd ξ ′ μ1,i = 4f − 4ℓ + 4 = 4ð f − ðℓ − 1ÞÞ > 0 ⇒ GO1 ðB4 ðξÞÞ   = GO1 ξ ′ > GO1 ðξÞ ð10Þ If dξ ðμ1 , γ1 Þ = 1, then   GO1 ξ ′ − GO1 ðξÞ f n o = 〠 dξ ′ μ1,i + dξ ′ ðμ1 Þ + d ξ ′ μ1,i d ξ ′ ðμ1 Þ i=1 ℓ−1n     o + 〠 dξ ′ γ1, j + dξ ′ ðγ1 Þ + dξ ′ γ1, j d ξ ′ ðγ1 Þ j=1 6 Journal of Chemistry n o + dξ ′ ðγℓ Þ + dξ ′ ðμ1 Þ + d ξ ′ ðγℓ Þd ξ ′ ðμ1 Þ n o + d ξ ′ ðμ1 Þ + d ξ ′ ðg1 Þ + dξ ′ ðμ1 Þdξ ′ ðg1 Þ n o + d ξ ′ ðγ1 Þ + d ξ ′ ðg2 Þ + dξ ′ ðγ1 Þd ξ ′ ðg2 Þ f  − 〠 d ξ μ1,i + dξ ðμ1 Þ + d ξ μ1,i d ξ ðμ1 Þ i=1 ℓ−1n     o − 〠 dξ γ1, j + d ξ ðγ1 Þ + dξ γ1, j d ξ ðγ1 Þ j=1  − d ξ ðγℓ Þ + dξ ðγ1 Þ + dξ ðγℓ Þdξ ðγ1 Þ  − d ξ ðμ1 Þ + d ξ ðg1 Þ + dξ ðμ1 Þd ξ ðg1 Þ  − d ξ ðγ1 Þ + dξ ðg2 Þ + dξ ðγ1 Þdξ ðg2 Þ f hn o
= 〠 d ξ ′ μ1,i + d ξ ′ ðμ1 Þ + dξ ′ μ1,i d ξ ′ ðμ1 Þ i=1  − d ξ μ1,i + dξ ðμ1 Þ + d ξ μ1,i d ξ ðμ1 Þ ℓ−1n     o + 〠 dξ ′ γ1, j + dξ ′ ðγ1 Þ + d ξ ′ γ1, j d ξ ′ ðγ1 Þ j=1   n   o − d ξ γ1, j + d ξ ðγ1 Þ + dξ γ1, j dξ ðγ1 Þ n + d ξ ′ ðγℓ Þ + d ξ ′ ðμ1 Þ + dξ ′ ðγℓ Þd ξ ′ ðμ1 Þ  − d ξ ðγℓ Þ + dξ ðγ1 Þ + dξ ðγℓ Þdξ ðγ1 Þ n o + d ξ ′ ðμ1 Þ + d ξ ′ ðg1 Þ + dξ ′ ðμ1 Þdξ ′ ðg1 Þ  − d ξ ðμ1 Þ + d ξ ðg1 Þ + dξ ðμ1 Þd ξ ðg1 Þ n o + d ξ ′ ðγ1 Þ + d ξ ′ ðg2 Þ + dξ ′ ðγ1 Þd ξ ′ ðg2 Þ  − d ξ ðγ1 Þ + dξ ðg2 Þ + dξ ðγ1 Þdξ ðg2 Þ f hn o = 〠 d ξ ′ μ1,i + ð f + 2Þ + ð f + 2Þdξ ′ μ1,i = 4f − 4ℓ + 4 = 4ð f − ðℓ − 1ÞÞ > 0 ⇒ GO1 ðB4 ðξÞÞ   = GO1 ξ ′ > GO1 ðξÞ ð11Þ 3. Ordering Trees Having Maximum GO1 In this section, we identify the trees having maximum first Gourava index, and also, we give a sequence of these trees having first five largest values of GO1 . Theorem 5. Let T ∗ be a set of trees having order λ and diameter d ∗ , where λ ≥ 3 and 2 ≤ d∗ ≤ λ − 1. Then, GO1 is maximum value for T ∗ = S∗λ,λ−d∗ +1 . Proof. First, we apply B1 -transformation to those vertices of T ∗ which are other than diametral path, and we observe that maximum value of GO1 is obtained for MSðλ1 , λ2 , ⋯, λd∗ −1 Þ. Then, we apply all those transformations which are explained above, and we conclude that the maximum value is acquired only for λ1 = λ − d∗ , λ2 = λ3 = ⋯ = 0, λd∗ −1 = 1 for S∗λ,λ−d∗ +1 . Corollary 6. (a) Let T ∗ denotes the set of trees with order λ. Then, max GO1 ðT ∗ Þ > diamðT ∗ Þ=ℓ max diamðT ∗ Þ=m ′ GO1 ðT ∗ Þ, ð12Þ where 2 ≤ ℓ ≤ m ′ ≤ λ − 1. (b) Let the order of set of trees of T ∗ be λ with diameter 3 ≤ d∗ ≤ λ − 2; then the greatest value of GO1 ðT ∗ Þ for these graphs is in the following order: MSðλ − d∗ , 0, ⋯, 0, 1Þ, MS ðλ − d ∗ − 1, 0, ⋯, 0, 2Þ, ⋯, MSðdλ − d∗ + 1/2e, 0, ⋯, 0, bλ − d∗ + 1/2cÞ: i=1 n oi − d ξ ′ μ1,i + ð f + 1Þ + ð f + 1Þd ξ ′ μ1,i ℓ−1hn +〠 j=1  o   dξ ′ γ1, j + ðℓÞ + ℓdξ ′ γ1, j n    oi − d ξ γ1, j + ðℓ + 1Þ + ðℓ + 1Þdξ γ1, j + f1 + ð f + 2Þ + ð f + 2Þg − f1 + ðℓ + 1Þ + ðℓ + 1Þg + f ð f + 2Þ + 2 + 2ð f + 2Þg − f ð f + 1Þ + 2 + 2ð f + 1Þg + fℓ + 2 + 2ðℓÞg − fðℓ + 1Þ + 2 + 2ðℓ + 1Þg f h i ℓ−1  = 〠 1 + dξ ′ ðμ1 , iÞ + 〠 −1 − d ξ ðμ1 , jÞ i=1 j=1 + f1 + ð f + 2Þ + ð f + 2Þg − f1 + ðℓ + 1Þ + ðℓ + 1Þg + 3 − 3 = f ð2Þ + ð−2Þðℓ − 1Þ + f1 + ð f + 2Þ + ð f + 2Þg − f1 + ðℓ + 1Þ + ðℓ + 1Þg + 3 − 3 Proof. We can achieve MSðλ − ℓ, 0, ⋯, 0, 1Þ from MSðλ − m ′ , 0, ⋯, 0, 1Þ by repeated use of first transformation (as described in Lemma 1). (b) We use first three transformations (as explained in Lemmas 1 to 3, and then, we use the fourth transformation to multistars MSðg1 , 0, ⋯, 0, g2 Þ with order λ where g1 + g2 = λ − d ∗ + 1. Theorem 7. For λ > 10, the trees having the greatest 1st Gourava index can be given in the following order (shown in Figure 5): GO1 ðK 1,λ−1 Þ > GO1 ðBSðλ − 3, 1ÞÞ > GO1 ðBSðλ − 4, 2ÞÞ
> GO1 S∗λ,λ−3 > GO1 ðBSðλ − 5, 3ÞÞ: ð13Þ Proof. In the family of trees having diameter 2, the star K 1,λ−1 is the only tree which by above corollary possesses Journal of Chemistry 7 κ1 , λ – 1 Sλ , λ – 2 = BS (λ – 3, 1) BS (λ – 4, 2) Sλ , λ – 3 BS (λ – 5, 3) Figure 5: Five trees having maximum first Gourava index. the greatest 1st Gourava index. The second maximum value of 1st Gourava index reaches for S∗λ,λ−2 = BSðλ − 3, 1Þ for the trees having diameter 3. The third maximum value of this sequence can be obtained for BSðλ − 4, 2Þ which also belongs to the class of trees having diameter 3. For λ = 5, the BSðλ − 4, 2Þ coincides with BSðλ − 3, 1Þ. The next graph in the class of trees with diameter 3 is BSðλ − 5, 3Þ. The next maximum value is obtained by S∗λ,λ−3 which has diameter 4. We obtain
GO1 ðBSðλ − 4, 2ÞÞ > GO1 S∗λ,λ−3 : ð14Þ Applying τ1 -transform to S∗λ,λ−3 , we obtain BSðλ − 4, 2Þ. It follows that for λ ≥ 6, the order of trees possessing maximum value is GO1 ðK 1,λ−1 Þ > GO1 ðBSðλ − 3, 1ÞÞ > GO1 ðBSðλ − 4, 2ÞÞ. For the fourth term of this series, we have GO1 ðBSðλ − 5, 3Þ − GO1 S∗λ,λ−3
ð16Þ = 2ðλ − 4Þ > 0: For λ > 4, MSðλ − 5, 0, 2Þ gets the 2nd maximum value of GO1 in the set of trees of diameter 4. Applying 1st transformation (as described earlier) to MSðλ − 5, 0, 2Þ, we get MSðλ − 5, 0, 0, 1Þ which attains maximum value of 1st Gourava index in the set of trees of diameter 5 and MS ðλ − 5, 0, 0, 1Þ < MSðλ − 5, 0, 2Þ which terminates the proof. For example, for λ = 12, we see that GO1 ðK 1,λ−1 Þ > GO1 ðBSðλ − 3, 1ÞÞ > GO1 ðBSðλ − 4, 2ÞÞ
> GO1 S∗λ,λ−3 > GO1 ðBSðλ − 5, 3ÞÞ, ð17Þ which as a result verifies our main result of this work. = ½ðλ − 5Þð1 + λ − 4 + λ − 4Þ + ðλ − 4 + 4 + 4ðλ − 4ÞÞ + 3ð4 + 1 + 4ފ − ½ðλ − 4Þð1 + λ − 3 + λ − 3Þ + ðλ − 3 + 2 + 2ðλ − 3ÞÞ + ð2 + 2 + 4Þ + ð1 + 2 + 2Þ = ½ðλ − 5Þð2λ − 7Þ + ð5λ − 16Þ + 3ð9ފ − ½ðλ − 4Þð2λ − 5Þ + ð3λ − 7Þ + 8 + 5Š  2    = 2λ − 12λ + 46 − 2λ2 − 10λ + 26 GO1 ðK 1,λ−1 Þ = 〠 ½d ðμÞ + dðηÞ + dðμÞd ðηފ μη∈ðEðK 1,λ−1 ÞÞ = 11½11 + 1 + 11Š = 25GO1 ðBSðλ − 3, 1ÞÞ 〠 = ½d ðμÞ + d ðηÞ + d ðμÞd ðηފ μη∈ðEBSðλ−3,1ÞÞ = 9½1 + 10 + 10Š + ½10 + 2 + 20Š + ½2 + 1 + 2Š = 2½10 − λŠ < 0 = 226, ð15Þ This implies that GO1 ðBSðλ − 5, 3Þ < GO1 ðS∗λ,λ−3 Þ for λ > 10. So for λ > 10, the fourth member in the above constructed sequence is S∗λ,λ−3 . For next member, we calculate GO1 ðBSðλ − 5, 3Þ − GO1 ðMSðλ − 5, 0, 2ÞÞ = ½ðλ − 5Þð1 + λ − 4 + λ − 4Þ + ðλ − 4 + 4 + 4ðλ − 4ÞÞ + 3ð4 + 1 + 4ފ − ½ðλ − 5Þð1 + λ − 4 + λ − 4Þ + ðλ − 4Þ + 2 + 2ðλ − 4Þ + ð2 + 3 + 6Þ + 2ð1 + 3 + 3ފ = ½ðλ − 5Þð2λ − 7Þ + ð5λ − 16Þ + 3ð9ފ − ½ðλ − 5Þð2λ − 7Þ + ð3λ − 6Þ + 11 + 14Š  2    = 2λ − 12λ + 46 − 2λ2 − 14λ + 54 GO1 ðBSðλ − 4, 2ÞÞ = 〠 ½d ðμÞ + dðηÞ + dðμÞd ðηފ μη∈ðEBSðλ−4,2ÞÞ = 8½1 + 9 + 9Š + ½9 + 3 + 27Š + 2½1 + 3 + 3Š = 205, GO1 ðSλ,λ−3 Þ = 〠 ½d ðμÞ + dðηÞ + dðμÞd ðηފ μη∈ðESλ,λ−3 Þ = 8½1 + 9 + 9Š + ½9 + 2 + 18Š + ½8 + 4 + 32Š + ½ 2 + 2 + 4Š + ½ 2 + 1 + 2Š = 194, 8 Journal of Chemistry Table 1: Comparison of different values of GO1 ðT ∗ðλ−1Þ for λ = 14:   GO T ∗ðλ−1Þ T ∗ðλ−1Þ κ1,13 351 BSð11, 1Þ 318 BSð10, 2Þ 291 S14,11 278 BSð9, 3Þ 270 〠 GO1 ðBSðλ − 5, 3ÞÞ = ½dðμÞ + d ðηÞ + d ðμÞd ðηފ Since dðℓ0 ≥ 2, we have h n    o     d y ′ − 1 + d ðℓ0 Þ + d y ′ − 1 d ðℓ0 Þ n     oi − d y ′ + d ðℓ 0 Þ + d y ′ d ð ℓ 0 Þ n   o n   o ≤ 3d y ′ − 1 − 3d y ′ + 2 : ð23Þ For the remaining dðy ′ Þ − 2 nodes ℓ ∈ Nðy ′ Þ \ fx ′ , ℓ0 g, we conclude that μη∈ðEBSðλ−5,3ÞÞ = 7½1 + 8 + 8Š + ½8 + 4 + 32Š + 3½1 + 4 + 4Š = 190: ð18Þ Hence proved GO1 ðK 1,λ−1 Þ > GO1 ðBSðλ − 3, 1ÞÞ > GO1 ðBSðλ − 4, 2ÞÞ
> GO1 S∗λ,λ−3 > GO1 ðBSðλ − 5, 3ÞÞ: ð19Þ Table 1 provides certain trees T ∗ðλÞ of order λ, along with their value of GO1 . Theorem 8. Let T ∗ be a tree having order λ ≥ 5 with α-leaf nodes, where 3 ≤ α ≤ λ − 2. Then,  GO1 ðT ∗ Þ ≤ 2 α2 − 3α + 4λ − 5 : ð20Þ Equality holds if T ∗ = S∗λ,α . ð24Þ This implies that GO1 ðT ∗ Þ − GO1 ðT ∗ Þ h   i hn     = 2d y ′ + 1 − d y ′ − 1 + d ðℓ0 Þ       oi o n   + d y ′ − 1 dðℓ0 Þ − d y ′ + d ðℓ0 Þ + d y ′ dðℓ0 Þ    hn        o − d y′ − 2 d y ′ − 1 + dðℓÞ + d y ′ − 1 d ðℓÞ   n   oi − d y ′ + dðℓÞ + d y ′ dðℓÞ h   i hn   o n   oi ≤ 2d y ′ + 1 − 3d y ′ − 1 − 3d y ′ + 2 hn   o n   oi    − 2d y ′ − 1 − 2d y ′ + 1 4 d y ′ : ð25Þ Proof. First under the supposition of theorem, we prove that if x ′ is a pendant node attached to a vertex y ′ , then GO1 ðT ∗ Þ − GO1 ðT ∗ − xÞ ≤ 4α: hn    o     d y ′ − 1 + d ðℓÞ + d y ′ − 1 d ðℓÞ n   oi   − d y ′ + d ðℓ Þ + d y ′ d ðℓ Þ n   o n   o ≤ 2d y ′ − 1 − 2d y ′ + 1 : ð21Þ Since dðy ′ Þ ≤ α, ⇒GO1 ðT ∗ Þ − GO1 ðT ∗ Þ ≤ 4α: ð26Þ S∗λ,α Equality holds for and for dðy ′ Þ = α. Here, we notice that there exist a vertex ℓ0 ∈ Nðy ′ Þ \ fx ′ g such that dðℓ0 Þ ≥ 2 since otherwise T ∗ would be a star, which is against the supposition of theorem. Now, GO1 ðT ∗ Þ − GO1 ðT ∗ Þ h    i = d y′ + 1 + d y′ h n    o     − 〠 d y ′ − 1 + d ð ℓÞ + d y ′ − 1 d ð ℓÞ z∈N ðy ′ Þfx ′ g n   oi   − d y ′ + d ðℓÞ + d y ′ d ðℓÞ : ð22Þ Equality holds if dðy ′ Þ = α, one neighbor of y ′ has degree two, while all the neighbor are leaf nodes, i.e., T ∗ = S∗λ,α . Now, the proof follows by induction on λ. For λ = 5, we obtain α = 3, and S∗5,3 = BS∗ ð1, 2Þ is a single tree of order 5 having three pendant nodes. Let α ≥ 6 and suppose that the theorem is true for all the trees of order λ − 1 having α-leaf nodes where 3 ≤ α ≤ λ − 3. Let x ′ be the pendant node that is attached to node y ′ . Here, we examine two subcases: (a) When dðy ′ Þ = 2 (b) When dðy ′ Þ ≥ 3 Journal of Chemistry 9 (a) For dðy ′ Þ = 2, we have   GO1 ðT ∗ Þ − GO1 T ∗ − x ′ = ðd ðℓÞ + 2 + 2dðℓÞÞ + ð2 + 1 + 2Þ   − ðd ðℓÞ + 1 + dðℓÞÞGO1 ðT ∗ Þ − GO1 T ∗ − x ′ = 6 + d ðℓÞGO1 ðT ∗ Þ   = 6 + d ðℓÞ + GO1 T ∗ − x ′ GO1 ðT ∗ Þ  ≤ 6 + d ðℓÞ + 2α2 − 3α + 4ðλ − 1Þ − 5  = 2 α2 − 3α + 4ðλÞ − 5 − 8 + 6 + dðℓÞ ⟹ GO1 ðT ∗ Þ ≤ S∗λ,α + dðℓÞ − 2 ð27Þ Equality holds for dðℓÞ = 2. If α = λ − 2, then T − x ′ has λ − 1 nodes and λ − 2 pendent nodes, i.e., T ∗ − x ′ = κ1,λ−2 and T ∗ = S∗λ,λ−2 = Sλ,α . (b) For dðy ′ Þ ≥ 3, then T ∗ − x ′ has λ − 1 nodes and α − 1 leaf nodes. 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