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

Authors: Muhammad Asif; Hamad Almohamedh; Muhammad Hussain; Khalid M Alhamed; Abdulrazaq A. Almutairi; Sultan Almotairi
Journal: Complexity
 (2021)

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## Abstract

Graph theory is a dynamic tool for designing and modeling of an interconnection system by a graph. The vertices of such graph are processor nodes and edges are the connections between these processors nodes. The topology of a system decides its best use. Geometric-arithmetic index is one of the most studied graph invariant to characterize the topological aspects of underlying interconnection networks or graphs. Transformation over graph is also an important tool to define new network of their own choice in computer science. In this work, we discuss transformed family of graphs. LetΓnk,l be the connected graph comprises by k number of pendent path attached with fully connected vertices of the n-vertex connected graph Γ. Let AαΓnk,l and AαβΓnk,l be the transformed graphs under the fact of transformations Aα and Aαβ, 0≤α≤l, 0≤β≤k−1, respectively. In this work, we obtained new inequalities for the graph family Γnk,l and transformed graphs AαΓnk,l and AαβΓnk,l which involve GAΓ. The presence of GAΓ makes the inequalities more general than all those which were previously defined for the GA index. Furthermore, we characterize extremal graphs which make the inequalities tight.


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## Body

## 1. Introduction

The advancement in technology mainly networking, computer, biological, and electrical networks made practicable the accurate data transfer within very small duration. The 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. The topology of a graph provides information about the manner in which vertices joined in a graph. The 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]. The topological descriptors of several interconnection networks are already been computed in [4–6]. 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].The study of underlying substance using their graph with the help of graph invariants plays an important role in chem-informatics, 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 [10–12]. Actually, topological indices are designed on the ground of transformation which associates a numeric value with the graph which characterizes its topology [13]. The first topological index, named Winner index, was proposed in 1947 by Winner [14]. It provides best correlation with the boiling points of alkanes. The discovery of the Winner index provides emerging research platform to the research community. The 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]:(1)RAΓ=∑uv∈EΓ1degudegv.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:(2)GAΓ=∑st∈EG2degsdegtdegs+degt.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. The bonds and extremal characterization of graphs regarding the GA index were studied at some extent in [17–24]. It encouraged us to study the GA index for Γnk,l and transformed graphs AαΓnk,l and AαβΓnk,l under the fact of transformations Aα and Aαβ, 0≤α≤l0≤β≤k−1, respectively. We characterize extremal graphs for all of these families of graphs.


## 2. Results and Discussion

Throughout this work, let graphΓnk,l comprise with n-vertex simple connected graph Γ along with k pendent paths of length l≥2 attached with v∈Γ, having degree dv≥2. The order of Γnk,l is n+kl, size is m+kl, and deg1=δΓ≤deg2≤deg3≤⋯≤ΔΓ+1 is its degree sequence.Let graphΓ=ΓV,E be with the degree of vertex u∈Γ and δΓ≤degu≤ΔΓ and δΓ≤degv≤ΔΓ+1 be the degrees of v∈Γnk,l. For validity of our proved results, we defined the following list of useful graphs.Type I: letδΓ≤degu≤ΔΓ, where u∈VΓ. Γnk,l of type I is obtained by attaching pendent paths of length l with vertices of degree degu≥2 in such a way that the vertices with pendent path are adjacent to the vertices without pendent paths.The graph of type I is shown in Figure1(a).Figure 1
GraphΓnk,l. (a) Graph Γnk,l of type I. (b) Graph Γnk,l of type II.
(a)(b)Type II:Γnk,l of type II is the graph of type I with degu=ΔΓ,∀u∈VΓ.The graph of type II is shown in Figure1(b).Before attempting the major problem, we prove the following preposition.Proposition 1.
Letx≥2; then,(3)2xx−12x−1≤2xx+12x+1.Proof.
Letx≥2:(4)2xx+12x+1−2xx−12x−1=2xx+12x−1−2xx−12x+12x+12x−1=2xxx+1+x−1x+1−xx−1−x+1x−12x+12x−1=2xx+1x−x2−1−x−1x−x2−12x+12x−1=2xx−x2−1x+1−x−12x+12x−1≥0.
The above calculations implies(5)2xx−12x−1≤2xx+12x+1.Theorem 1.
Let graphΓnk,l comprise of n-vertex simple connected graph Γ along with k pendent paths of length l≥2 attached with v∈Γ of degree dv≥2, maximum degree ΔΓ+1, and minimum δΓ. Then,(6)GAΓnk,l≥k3l−ΔΓ−2+223+2ΔΓ+12ΔΓ+3+ΔΓδΓΔΓ+δΓ+1+GAΓ.Equality holds for graphs of type II. And,(7)GAΓnk,l≤k22+3l−63+22δΓ+1δΓ+3+2kΔΓ3/2ΔΓ+12ΔΓ+1−δΓδΓ+ΔΓ+GAΓ.Equality holds for graph of type II.Proof.
Let a simple graphΓ be of order n, size m, maximum degree ΔΓ, and minimum δΓ. Γnk,l be the graph formed by k number of paths having length l pendent at distinct vertices u∈Γ such that 2≤degu≤δΓ. The geometric-arithmetic index of any graph Γ is(8)GAΓ=∑st∈EΓ2degsdegtdegs+degt.
The construction ofΓnk,l,l≥2 implies EΓnk,l=m+kl and for st∈EΓnk,ldegs+degt∈3,4,degu+2,degu+degv,degu+degv+1.
The edge set ofΓnk,l partitioned as(9)A3=st∈Γnk,l:degs=1,degt=2,A4=st∈Γnk,l:degs=degt=2,Adegu+2=st∈Γnk,l:δΓ≤degs=degu≤ΔΓ,degt=2,Adegu+degv=st∈Γnk,l:δΓ≤degs=degu,degt=degv≤ΔΓ,and Adegu+degv+1=st∈Γnk,l:δΓ≤degs=degu,degt=degv+1≤ΔΓ,(10)GAΓnk,l=∑st are edges ofpendent paths2degsdegtdegs+degt+∑st are edges ofΓ2degsdegtdegs+degt,GAΓnk,l=∑st∈A32degsdegtdegs+degt+∑st∈A42degsdegtdegs+degt+∑st∈Adegu+32degsdegtdegs+degt+∑st∈Adegu+degv+12degsdegtdegs+degt+∑st∈Adegu+degv2degsdegtdegs+degt.
The construction ofGAΓnk,l implies that the cardinality of A3 is k,i.e.,A3=k,A4=kl−2, Adegu+3=k, Adegu+degu+1≤kΔΓ, and Adegu+degu≤kΔΓ. The function fx=2ax/a+x is decreasing, where a≤x is a constant. So, for δΓ minimum degree of vertices of Γ and maximum degree ΔΓ, we have(11)22∗degs+12+degs+1≥22∗ΔΓ+12+ΔΓ+1,2ΔΓ+1δΓΔΓ+1+δΓ≤2degs+1degtdegs+1+degt,2degs+degtdegs+degt≥GAΓ−ΔΓk.
From equation (10), we have(12)GAΓnk,l≥2k23+kl−2+2k2∗ΔΓ+12+ΔΓ+1+2ΔΓkδΓΔΓ+1δΓ+ΔΓ+1−kΔΓ+GAΓ.
After simplification, we obtain(13)GAΓnk,l≥k3l−ΔΓ−2+223+2ΔΓ+12ΔΓ+3+ΔΓδΓΔΓ+δΓ+1+GAΓ.
Now, again set(14)22∗degs+12+degs+1≤22∗δΓ+12+δΓ+1,∑st∈Adegu+degv+12degsdegtdegs+degt+∑st∈Adegu+degv2degsdegtdegs+degt≤2ΔΓ+1ΔΓΔΓ+1+ΔΓ+GAΓ−2kΔΓδΓΔΓδΓ+ΔΓ,which implies from Proposition 1 and the characteristics of fx=2xy/x+y in equation (10). We get the following inequality:(15)GAΓnk,l≤2k23+kl−2+2k2∗δΓ+12+δΓ+1+2ΔΓkΔΓΔΓ+1ΔΓ+ΔΓ+1+GAΓ−2ΔΓkδΓΔΓδΓ+ΔΓ.
After simplification, we obtain(16)GAΓnk,l≤k22+3l−63+22δΓ+1δΓ+3+2kΔΓ3/2ΔΓ+12ΔΓ+1−δΓδΓ+ΔΓ+GAΓ.
Inequalities (13) and (16) complete the proof.
Corollary1 shows generalization of the above defined inequalities. One can get more inequalities of their desire by replacing GAΓ with already defined bonds of the GA index.Corollary 1.
Let graphΓnk,l comprise of n-vertex simple connected graph Γ along with k pendent paths of length l≥2 attached with v∈Γ of degree dv≥2, maximum degree ΔΓ+1, and minimum δΓ. Then,(17)GAΓnk,l≥k3l−ΔΓ−2+223+2ΔΓ+12ΔΓ+3+ΔΓδΓΔΓ+δΓ+1+2mΔΓδΓδΓ+ΔΓ,GAΓnk,l≤k22+3l−63+22δΓ+1δΓ+3+2kΔΓ3/2ΔΓ+12ΔΓ+1−δΓδΓ+ΔΓ+m.Equality holds for regular graph of the type II.Proof.
Using results of Theorem1 and inequality regarding the geometric index proved in [25, 26], as(18)2mΔΓδΓΔΓ+δΓ≤GAΓ≤m,we get desired results.


### 2.1. Graph Transformations

LetHΓ⊂EΓ, Γ′=Γ−H be the new graph generated by removing set edges of HΓ, and Γ″=Γ−V1Γ be the new graph generated by deleting set of vertices V1Γ⊂VΓ. We use the following transformations as used in [27]. These transformations have solid effect over GA of Γnk,l.TransformationA: let wj∈VΓ,degwj≥2, for 1≤j≤k≤n, and paths pendent at wj of the form wjuj1,uj1uj2,uj2uj3,…,ujl−1ujl comprise Γnk,l. Then,(19)AΓnk,l=Γnk,l−∑j=1kuj2uj3,uj3uj4,…,ujl−1ujl+∑j=1kwjuj2,uj2uj3,…,ujl−1ujl.The transformationA is shown in Figure 2.Figure 2
TransformationA.In Theorem2, we discuss the effect of transformation A over the GA index.Theorem 2.
Let graphΓnk,l comprise of n-vertex simple connected graph Γ along with k pendent paths of length l≥2 attached with v∈Γ of degree dv≥2, maximum degree of v∈Γnk,l is ΔΓ+1, and minimum δΓ. Then,(20)GAAαΓnk,l≥2kΔΓ+α+1αΔΓ+α+2+2ΔΓ+α+3+ΔΓδΓΔΓ+1+α+δΓ+2k23+GAΓ+kl−k2+α+ΔΓ.Equality holds for all graphs of type II:(21)GAAαΓnk,l≤2kδΓ+α+1αδΓ+α+2+2δΓ+α+3+2k23+kl−2−α+GAΓ+2kΔΓ3/3ΔΓ+12ΔΓ+1−δΓΔΓ+δΓ.Equality holds for all graphs of the type II andα=0.Proof.
Let a simple graphΓ be of order n, size m, minimum degree δΓ, and maximum ΔΓ. Let Γnk,l be the graph formed by k number of paths of length l pendent at distinct fully connected vertices of Γ. The geometric-arithmetic index of any graph Γ is(22)GAΓ=∑st∈EG2degsdegtdegs+degt.
The construction ofΓnk,l,l≥2, implies EΓnk,l=m+kl. After successive applications of transformation A as Aα, α≤l−1, the edge set of AαΓnk,l is partitioned as Edegs+degtAαΓnk,l, degs+degt∈3,4,degu+α+2,degu+α+3,degu+degv,degu+α+1+degv:(23)E3AαΓnk,l=st∈Γnk,l:degs=1,degt=2,E4AαΓnk,l=st∈Γnk,l:degs=degt=2,Edegu+α+2AαΓnk,l=st∈Γnk,l:δΓ≤degs=degu+α+1≤ΔΓ+α+1,degt=1,Edegu+α+3AαΓnk,l=st∈Γnk,l:δΓ≤degs=degu+α+1≤ΔΓ+α+1,degt=2,Edegu+α+1AαΓnk,l=st∈Γnk,l:δΓ≤degs=degu+α≤ΔΓ,degt=1,Edegu+degvAαΓnk,l=st∈Γnk,l:δΓ≤degs=degu,degt=degv≤ΔΓ,Edegu+α+1+degvAαΓnk,l=st∈Γnk,l:δΓ≤degv,degu≤ΔΓ,degs=degu+α+1,degt=degv,(24)GAAαΓnk,l=∑Edegs+degtAαΓnk,l⊆EAαΓnk,l∑st∈Edegs+degtAαΓnk,l∑2degsdegtdegs+degt.
The cardinality ofA3 is k,i.e.,E3AαΓnk,l=k,|E4AαΓnk,l=kl−α−2, Edegu+α+2AαΓnk,l=kα, and Edegu+α+3AαΓnk,l=k. The function fx=2ax/a+x is decreasing, where a≤x is a constant. So, for δΓ minimum degree of Γ and ΔΓ maximum, for any graph,(25)22∗degs+1+α2+degs+1+α≥22∗ΔΓ+1+α2+ΔΓ+1+α,2ΔΓ+α+1δΓΔΓ+α+1+δΓ≤2degs+α+1degtdegs+α+1+degt∑st∈Adegu+degv2degs+degtdegs+degt≥GAΓ−ΔΓk.
Substituting these changes in equation (24), we have the following inequality:(26)GAAαΓnk,l≥2k23+kl−2−α+2αk1∗ΔΓ+α+11+ΔΓ+α+1+2k2ΔΓ+α+12+ΔΓ+α+1+2kΔΓΔΓ+α+1δΓΔΓ+α+1+δΓ+GAΓ−ΔΓK.
After simplification, we get the required result(27)GAAαΓnk,l≥2kΔΓ+α+1αΔΓ+α+2+2ΔΓ+α+3+ΔΓδΓΔΓ+1+α+δΓ+2k23+GAΓ+kl−k2+α+ΔΓ.
Now, again, from equation (24) and inequalities,(28)∑st∈Adegu+degv2degs+degtdegs+degt≥GAΓ−2ΔΓΔΓδΓΔΓ+δΓ,2degs+α+11+degs+α+1≤2δΓ+α+11+δΓ+α+1,2ΔΓkΔΓ+1ΔΓΔΓ+1+ΔΓ≥∑st∈Adegu+degv+α+12degs+α+1degtdegs+α+1+degt,22∗degs+α+12+degs+α+1≤22∗δΓ+α+12+δΓ+α+1,(29)GAAαΓnk,l≤2k23+2αk1∗δΓ+α+11+δΓ+α+1+2k2∗δΓ+α+12+δΓ+α+1+2ΔΓkΔΓ+1ΔΓΔΓ+1+ΔΓ+kl−2−α+GAΓ−2kΔΓΔΓδΓΔΓ+δΓ.
After simplification, we obtain(30)GAAαΓnk,l≤2kδΓ+α+1αδΓ+α+2+2δΓ+α+3+2k23+kl−2−α+GAΓ+2kΔΓ3/3ΔΓ+12ΔΓ+1−δΓΔΓ+δΓ.
Inequalities (27) and (30) complete the proof.
TransformationB: let wj∈VΓ,degwj≥2, for 1≤j≤k≤n, and paths pendent at wj of the form wjuj1,uj1uj2,uj2uj3,…,ujl−1ujl which comprises Γnk,l. Then, for fixed vertex w1,(31)BΓnk,l=Γnk,l−uj1uj2,uj2uj3,…,ujl−1ujl+w1uj1,uj1uj2,uj2uj3,…,ujl−1ujl.
The transformationB is shown in Figure 3 and Aαβ shown in Figure 4.
TransformationAαβ: let 0≤α≤l−1 and 0≤β≤k−1. The transformation Aαβ is the composition of successive applications of transformation A and B as Aα and Bβ, respectively [27].
In Theorem3, we discuss the effect of transformation Aαβ over the GA index.Figure 3
TransformationB for fixed vertex w1.Figure 4
TransformationAαβ.Theorem 3.
Let graphΓnk,l comprise of n-vertex simple connected graph Γ along with k pendent paths of length l≥2 attached with v∈Γ of degree dv≥2, maximum degree of v∈Γnk,l is ΔΓ+1, and minimum δΓ. Then,(32)GAAαβΓnk,l≥2k−β−1ΔΓ+α+1αΔΓ+α+2+2ΔΓ+α+3+ΔΓδΓΔΓ+α+1+δΓ+2ΔΓ+β+11+ααβ+11+ΔΓ+β+11+α+2β+12+ΔΓ+β+11+α+2ΔΓΔΓ+β+11+αδΓδΓ+ΔΓ+β+11+α+kl−2−α−ΔΓk−β+2k23+GAΓ,GAAαβΓnk,l≤2k−β−1δΓ+α+1αδΓ+α+2+2δΓ+α+3+kl−2−α+2k23+2δΓ+β+11+ααβ+11+δΓ+β+11+α+2β+12+δΓ+β+11+α+2ΔΓΔΓk−βΔΓ+12ΔΓ+1−δΓΔΓ+δΓ+GAΓ.Equality holds for graph of the type II withα=0 and β=0.Proof.
Let a simple graphΓ of order n, size m, minimum degree δΓ, and maximum ΔΓ. Let Γnk,l be the graph formed by k number of paths of length l pendent at distinct fully connected vertices of Γ. The geometric-arithmetic index of any graph Γ is(33)GAΓ=∑st∈EG2degsdegtdegs+degt.
The construction ofΓnk,l,l≥2, implies EΓnk,l=m+kl. Let u∗ be the fixed vertex. Applications of transformation Aαβ has an effect over the edge set partition as Edegs+degtAαβΓnk,l, degs+degt∈3,4,degu+α+2,degu+α+3,degu+degv,degu+α+1+degv,degu∗+β+1α+1+degv,degu∗+β+1α+1+1,degu∗+β+1α+1+2:(34)E3AαβΓnk,l=st∈Γnk,l:degs=1,degt=2,E4AαβΓnk,l=st∈Γnk,l:degs=degt=2,Edegu+α+2AαβΓnk,l=st∈Γnk,l:δΓ≤degs=degu+α+1≤ΔΓ+α+1,degt=1,Edegu+α+3AαβΓnk,l=st∈Γnk,l:δΓ≤degs=degu+α+1≤ΔΓ+α+1,degt=2,Edegu+degvAαβΓnk,l=st∈Γnk,l:δΓ≤degs=degu,degt=degv≤ΔΓ,Edegu+α+1+degvAαβΓnk,l=st∈Γnk,l:δΓ≤degv=degt,degu≤ΔΓ,degs=degu+α+1,Edegu∗+β+1α+1+degvAαβΓnk,l=st∈Γnk,l:δΓ≤degv=degt,degu≤ΔΓ,degs=degu+β+1α+1,Edegu∗+β+1α+1+1AαβΓnk,l=st∈Γnk,l:δΓ≤degv,degu≤ΔΓ,degs=degu+β+1α+1,degt=1,Edegu∗+β+1α+1+2AαβΓnk,l=st∈Γnk,l:δΓ≤degv,degu≤ΔΓ,degs=degu+β+1α+1,degt=2,(35)GAAαΓnk,l=∑Edegs+degtAαΓnk,l⊆EAαβΓnk,l∑st∈Edegs+degtAαΓnk,l∑2degsdegtdegs+degt.
The cardinality ofE3 is k,i.e.,E3AαβΓnk,l=k,E4AαβΓnk,l=kl−α−2, Edegu+α+2AαβΓnk,l=αk−β−1, Edegu+α+3AαβΓnk,l=k−β−1, Edegu∗+β+1α+1+1AαβΓnk,l=αβ+1, and(36)Edegu∗+β+1α+1+2AαβΓnk,l=β+1.
The functionfx=2ax/a+x is decreasing, where a≤x is a constant. So, for δΓ minimum degree of Γ and ΔΓ maximum, we have(37)22degs+1+α2+degs+1+α≥22ΔΓ+1+α2+ΔΓ+1+α,2ΔΓ+α+1∗1ΔΓ+α+1+1≤2degs+α+1degtdegs+α+1+degt,22∗degs+β+11+α2+degs+β+11+α≥22∗ΔΓ+β+11+α2+ΔΓ+β+11+α,2ΔΓ+β+11+αδΓΔΓ+β+11+α+δΓ≤2degs+β+11+αdegtdegs+β+11+α+degt,(38)∑st∈Adegu+degv2degs+degtdegs+degt≥GAΓ+2ΔΓk−β−1ΔΓ+α+1δΓΔΓ+α+1+δΓ+2ΔΓΔΓ+β+11+αδΓΔΓ+β+11+α+δΓ−ΔΓk−β.
Substituting these changes in equation (35), we obtained the following inequality:(39)GAAαβΓnk,l≥2k23+2αk−β−11∗ΔΓ+α+11+ΔΓ+α+1+2k−β−12ΔΓ+α+12+ΔΓ+α+1+2αβ+11∗ΔΓ+β+11+α1+ΔΓ+β+11+α+2β+12ΔΓ+β+11+α2+ΔΓ+β+11+α+2ΔΓk−β−1ΔΓ+α+1δΓΔΓ+α+1+δΓ+2ΔΓΔΓ+β+11+αδΓΔΓ+β+11+α+δΓ+GAΓ+kl−2−α−ΔΓk−β.
After simplification, we get the required result:(40)GAAαβΓnk,l≥2k−β−1δΓ+α+1αΔΓ+α+2+2ΔΓ+α+3+ΔΓδΓΔΓ+α+1+δΓ+2ΔΓ+β+11+α×αβ+11+ΔΓ+β+11+α+2β+12+ΔΓ+β+11+α+2ΔΓΔΓ+β+11+αδΓδΓ+ΔΓ+β+11+α+kl−2−α−ΔΓk−β+2k23+GAΓ.
Now, again substituting the following inequalities in equation (35),(41)22degs+1+α2+degs+1+α≤22δΓ+1+α2+δΓ+1+α,2degs+α+1degtdegs+α+1+degt≤2δΓ+α+1δΓ+α+1+1,22degs+β+11+α2+degs+β+11+α≤22δΓ+β+11+α2+δΓ+β+11+α,2degs+β+11+α1+degs+β+11+α≤2δΓ+β+11+α1+δΓ+β+11+α,∑st∈Adegu+degv2degs+degtdegs+degt≤GAΓ+2ΔΓk−β−1ΔΓ+1ΔΓΔΓ+1+ΔΓ+2ΔΓΔΓ+1ΔΓΔΓ+1+ΔΓ−2ΔΓk−βΔΓδΓΔΓ+δΓ.
We obtain(42)GAAαβΓnk,l≤2k23+2αk−β−11∗δΓ+α+11+δΓ+α+1+2k−β−12δΓ+α+12+δΓ+α+1+2αβ+11∗δΓ+β+11+α1+δΓ+β+11+α+2β+12δΓ+β+11+α2+δΓ+β+11+α+GAΓ+2ΔΓk−β−1ΔΓ+1ΔΓΔΓ+1+ΔΓ+2ΔΓΔΓ+1ΔΓΔΓ+1+ΔΓ−2ΔΓk−βΔΓδΓΔΓ+δΓ.
After simplification, we get the required result:(43)GAAαβΓnk,l≤2k−β−1δΓ+α+1αδΓ+α+2+2δΓ+α+3+kl−2−α+2k23+2δΓ+β+11+ααβ+11+δΓ+β+11+α+2β+12+δΓ+β+11+α+2ΔΓΔΓk−βΔΓ+12ΔΓ+1−δΓΔΓ+δΓ+GAΓ.
Inequalities (40) and (43) complete the proof.


## 2.1. Graph Transformations

LetHΓ⊂EΓ, Γ′=Γ−H be the new graph generated by removing set edges of HΓ, and Γ″=Γ−V1Γ be the new graph generated by deleting set of vertices V1Γ⊂VΓ. We use the following transformations as used in [27]. These transformations have solid effect over GA of Γnk,l.TransformationA: let wj∈VΓ,degwj≥2, for 1≤j≤k≤n, and paths pendent at wj of the form wjuj1,uj1uj2,uj2uj3,…,ujl−1ujl comprise Γnk,l. Then,(19)AΓnk,l=Γnk,l−∑j=1kuj2uj3,uj3uj4,…,ujl−1ujl+∑j=1kwjuj2,uj2uj3,…,ujl−1ujl.The transformationA is shown in Figure 2.Figure 2
TransformationA.In Theorem2, we discuss the effect of transformation A over the GA index.Theorem 2.
Let graphΓnk,l comprise of n-vertex simple connected graph Γ along with k pendent paths of length l≥2 attached with v∈Γ of degree dv≥2, maximum degree of v∈Γnk,l is ΔΓ+1, and minimum δΓ. Then,(20)GAAαΓnk,l≥2kΔΓ+α+1αΔΓ+α+2+2ΔΓ+α+3+ΔΓδΓΔΓ+1+α+δΓ+2k23+GAΓ+kl−k2+α+ΔΓ.Equality holds for all graphs of type II:(21)GAAαΓnk,l≤2kδΓ+α+1αδΓ+α+2+2δΓ+α+3+2k23+kl−2−α+GAΓ+2kΔΓ3/3ΔΓ+12ΔΓ+1−δΓΔΓ+δΓ.Equality holds for all graphs of the type II andα=0.Proof.
Let a simple graphΓ be of order n, size m, minimum degree δΓ, and maximum ΔΓ. Let Γnk,l be the graph formed by k number of paths of length l pendent at distinct fully connected vertices of Γ. The geometric-arithmetic index of any graph Γ is(22)GAΓ=∑st∈EG2degsdegtdegs+degt.
The construction ofΓnk,l,l≥2, implies EΓnk,l=m+kl. After successive applications of transformation A as Aα, α≤l−1, the edge set of AαΓnk,l is partitioned as Edegs+degtAαΓnk,l, degs+degt∈3,4,degu+α+2,degu+α+3,degu+degv,degu+α+1+degv:(23)E3AαΓnk,l=st∈Γnk,l:degs=1,degt=2,E4AαΓnk,l=st∈Γnk,l:degs=degt=2,Edegu+α+2AαΓnk,l=st∈Γnk,l:δΓ≤degs=degu+α+1≤ΔΓ+α+1,degt=1,Edegu+α+3AαΓnk,l=st∈Γnk,l:δΓ≤degs=degu+α+1≤ΔΓ+α+1,degt=2,Edegu+α+1AαΓnk,l=st∈Γnk,l:δΓ≤degs=degu+α≤ΔΓ,degt=1,Edegu+degvAαΓnk,l=st∈Γnk,l:δΓ≤degs=degu,degt=degv≤ΔΓ,Edegu+α+1+degvAαΓnk,l=st∈Γnk,l:δΓ≤degv,degu≤ΔΓ,degs=degu+α+1,degt=degv,(24)GAAαΓnk,l=∑Edegs+degtAαΓnk,l⊆EAαΓnk,l∑st∈Edegs+degtAαΓnk,l∑2degsdegtdegs+degt.
The cardinality ofA3 is k,i.e.,E3AαΓnk,l=k,|E4AαΓnk,l=kl−α−2, Edegu+α+2AαΓnk,l=kα, and Edegu+α+3AαΓnk,l=k. The function fx=2ax/a+x is decreasing, where a≤x is a constant. So, for δΓ minimum degree of Γ and ΔΓ maximum, for any graph,(25)22∗degs+1+α2+degs+1+α≥22∗ΔΓ+1+α2+ΔΓ+1+α,2ΔΓ+α+1δΓΔΓ+α+1+δΓ≤2degs+α+1degtdegs+α+1+degt∑st∈Adegu+degv2degs+degtdegs+degt≥GAΓ−ΔΓk.
Substituting these changes in equation (24), we have the following inequality:(26)GAAαΓnk,l≥2k23+kl−2−α+2αk1∗ΔΓ+α+11+ΔΓ+α+1+2k2ΔΓ+α+12+ΔΓ+α+1+2kΔΓΔΓ+α+1δΓΔΓ+α+1+δΓ+GAΓ−ΔΓK.
After simplification, we get the required result(27)GAAαΓnk,l≥2kΔΓ+α+1αΔΓ+α+2+2ΔΓ+α+3+ΔΓδΓΔΓ+1+α+δΓ+2k23+GAΓ+kl−k2+α+ΔΓ.
Now, again, from equation (24) and inequalities,(28)∑st∈Adegu+degv2degs+degtdegs+degt≥GAΓ−2ΔΓΔΓδΓΔΓ+δΓ,2degs+α+11+degs+α+1≤2δΓ+α+11+δΓ+α+1,2ΔΓkΔΓ+1ΔΓΔΓ+1+ΔΓ≥∑st∈Adegu+degv+α+12degs+α+1degtdegs+α+1+degt,22∗degs+α+12+degs+α+1≤22∗δΓ+α+12+δΓ+α+1,(29)GAAαΓnk,l≤2k23+2αk1∗δΓ+α+11+δΓ+α+1+2k2∗δΓ+α+12+δΓ+α+1+2ΔΓkΔΓ+1ΔΓΔΓ+1+ΔΓ+kl−2−α+GAΓ−2kΔΓΔΓδΓΔΓ+δΓ.
After simplification, we obtain(30)GAAαΓnk,l≤2kδΓ+α+1αδΓ+α+2+2δΓ+α+3+2k23+kl−2−α+GAΓ+2kΔΓ3/3ΔΓ+12ΔΓ+1−δΓΔΓ+δΓ.
Inequalities (27) and (30) complete the proof.
TransformationB: let wj∈VΓ,degwj≥2, for 1≤j≤k≤n, and paths pendent at wj of the form wjuj1,uj1uj2,uj2uj3,…,ujl−1ujl which comprises Γnk,l. Then, for fixed vertex w1,(31)BΓnk,l=Γnk,l−uj1uj2,uj2uj3,…,ujl−1ujl+w1uj1,uj1uj2,uj2uj3,…,ujl−1ujl.
The transformationB is shown in Figure 3 and Aαβ shown in Figure 4.
TransformationAαβ: let 0≤α≤l−1 and 0≤β≤k−1. The transformation Aαβ is the composition of successive applications of transformation A and B as Aα and Bβ, respectively [27].
In Theorem3, we discuss the effect of transformation Aαβ over the GA index.Figure 3
TransformationB for fixed vertex w1.Figure 4
TransformationAαβ.Theorem 3.
Let graphΓnk,l comprise of n-vertex simple connected graph Γ along with k pendent paths of length l≥2 attached with v∈Γ of degree dv≥2, maximum degree of v∈Γnk,l is ΔΓ+1, and minimum δΓ. Then,(32)GAAαβΓnk,l≥2k−β−1ΔΓ+α+1αΔΓ+α+2+2ΔΓ+α+3+ΔΓδΓΔΓ+α+1+δΓ+2ΔΓ+β+11+ααβ+11+ΔΓ+β+11+α+2β+12+ΔΓ+β+11+α+2ΔΓΔΓ+β+11+αδΓδΓ+ΔΓ+β+11+α+kl−2−α−ΔΓk−β+2k23+GAΓ,GAAαβΓnk,l≤2k−β−1δΓ+α+1αδΓ+α+2+2δΓ+α+3+kl−2−α+2k23+2δΓ+β+11+ααβ+11+δΓ+β+11+α+2β+12+δΓ+β+11+α+2ΔΓΔΓk−βΔΓ+12ΔΓ+1−δΓΔΓ+δΓ+GAΓ.Equality holds for graph of the type II withα=0 and β=0.Proof.
Let a simple graphΓ of order n, size m, minimum degree δΓ, and maximum ΔΓ. Let Γnk,l be the graph formed by k number of paths of length l pendent at distinct fully connected vertices of Γ. The geometric-arithmetic index of any graph Γ is(33)GAΓ=∑st∈EG2degsdegtdegs+degt.
The construction ofΓnk,l,l≥2, implies EΓnk,l=m+kl. Let u∗ be the fixed vertex. Applications of transformation Aαβ has an effect over the edge set partition as Edegs+degtAαβΓnk,l, degs+degt∈3,4,degu+α+2,degu+α+3,degu+degv,degu+α+1+degv,degu∗+β+1α+1+degv,degu∗+β+1α+1+1,degu∗+β+1α+1+2:(34)E3AαβΓnk,l=st∈Γnk,l:degs=1,degt=2,E4AαβΓnk,l=st∈Γnk,l:degs=degt=2,Edegu+α+2AαβΓnk,l=st∈Γnk,l:δΓ≤degs=degu+α+1≤ΔΓ+α+1,degt=1,Edegu+α+3AαβΓnk,l=st∈Γnk,l:δΓ≤degs=degu+α+1≤ΔΓ+α+1,degt=2,Edegu+degvAαβΓnk,l=st∈Γnk,l:δΓ≤degs=degu,degt=degv≤ΔΓ,Edegu+α+1+degvAαβΓnk,l=st∈Γnk,l:δΓ≤degv=degt,degu≤ΔΓ,degs=degu+α+1,Edegu∗+β+1α+1+degvAαβΓnk,l=st∈Γnk,l:δΓ≤degv=degt,degu≤ΔΓ,degs=degu+β+1α+1,Edegu∗+β+1α+1+1AαβΓnk,l=st∈Γnk,l:δΓ≤degv,degu≤ΔΓ,degs=degu+β+1α+1,degt=1,Edegu∗+β+1α+1+2AαβΓnk,l=st∈Γnk,l:δΓ≤degv,degu≤ΔΓ,degs=degu+β+1α+1,degt=2,(35)GAAαΓnk,l=∑Edegs+degtAαΓnk,l⊆EAαβΓnk,l∑st∈Edegs+degtAαΓnk,l∑2degsdegtdegs+degt.
The cardinality ofE3 is k,i.e.,E3AαβΓnk,l=k,E4AαβΓnk,l=kl−α−2, Edegu+α+2AαβΓnk,l=αk−β−1, Edegu+α+3AαβΓnk,l=k−β−1, Edegu∗+β+1α+1+1AαβΓnk,l=αβ+1, and(36)Edegu∗+β+1α+1+2AαβΓnk,l=β+1.
The functionfx=2ax/a+x is decreasing, where a≤x is a constant. So, for δΓ minimum degree of Γ and ΔΓ maximum, we have(37)22degs+1+α2+degs+1+α≥22ΔΓ+1+α2+ΔΓ+1+α,2ΔΓ+α+1∗1ΔΓ+α+1+1≤2degs+α+1degtdegs+α+1+degt,22∗degs+β+11+α2+degs+β+11+α≥22∗ΔΓ+β+11+α2+ΔΓ+β+11+α,2ΔΓ+β+11+αδΓΔΓ+β+11+α+δΓ≤2degs+β+11+αdegtdegs+β+11+α+degt,(38)∑st∈Adegu+degv2degs+degtdegs+degt≥GAΓ+2ΔΓk−β−1ΔΓ+α+1δΓΔΓ+α+1+δΓ+2ΔΓΔΓ+β+11+αδΓΔΓ+β+11+α+δΓ−ΔΓk−β.
Substituting these changes in equation (35), we obtained the following inequality:(39)GAAαβΓnk,l≥2k23+2αk−β−11∗ΔΓ+α+11+ΔΓ+α+1+2k−β−12ΔΓ+α+12+ΔΓ+α+1+2αβ+11∗ΔΓ+β+11+α1+ΔΓ+β+11+α+2β+12ΔΓ+β+11+α2+ΔΓ+β+11+α+2ΔΓk−β−1ΔΓ+α+1δΓΔΓ+α+1+δΓ+2ΔΓΔΓ+β+11+αδΓΔΓ+β+11+α+δΓ+GAΓ+kl−2−α−ΔΓk−β.
After simplification, we get the required result:(40)GAAαβΓnk,l≥2k−β−1δΓ+α+1αΔΓ+α+2+2ΔΓ+α+3+ΔΓδΓΔΓ+α+1+δΓ+2ΔΓ+β+11+α×αβ+11+ΔΓ+β+11+α+2β+12+ΔΓ+β+11+α+2ΔΓΔΓ+β+11+αδΓδΓ+ΔΓ+β+11+α+kl−2−α−ΔΓk−β+2k23+GAΓ.
Now, again substituting the following inequalities in equation (35),(41)22degs+1+α2+degs+1+α≤22δΓ+1+α2+δΓ+1+α,2degs+α+1degtdegs+α+1+degt≤2δΓ+α+1δΓ+α+1+1,22degs+β+11+α2+degs+β+11+α≤22δΓ+β+11+α2+δΓ+β+11+α,2degs+β+11+α1+degs+β+11+α≤2δΓ+β+11+α1+δΓ+β+11+α,∑st∈Adegu+degv2degs+degtdegs+degt≤GAΓ+2ΔΓk−β−1ΔΓ+1ΔΓΔΓ+1+ΔΓ+2ΔΓΔΓ+1ΔΓΔΓ+1+ΔΓ−2ΔΓk−βΔΓδΓΔΓ+δΓ.
We obtain(42)GAAαβΓnk,l≤2k23+2αk−β−11∗δΓ+α+11+δΓ+α+1+2k−β−12δΓ+α+12+δΓ+α+1+2αβ+11∗δΓ+β+11+α1+δΓ+β+11+α+2β+12δΓ+β+11+α2+δΓ+β+11+α+GAΓ+2ΔΓk−β−1ΔΓ+1ΔΓΔΓ+1+ΔΓ+2ΔΓΔΓ+1ΔΓΔΓ+1+ΔΓ−2ΔΓk−βΔΓδΓΔΓ+δΓ.
After simplification, we get the required result:(43)GAAαβΓnk,l≤2k−β−1δΓ+α+1αδΓ+α+2+2δΓ+α+3+kl−2−α+2k23+2δΓ+β+11+ααβ+11+δΓ+β+11+α+2β+12+δΓ+β+11+α+2ΔΓΔΓk−βΔΓ+12ΔΓ+1−δΓΔΓ+δΓ+GAΓ.
Inequalities (40) and (43) complete the proof.


## 3. Conclusion

The study of mathematical aspect regarding topological indices is a partially open problem [20, 28, 29]. For which member family of graphs the certain index has a minimal or maximal value? In this work, we discussed, for this fundamental question, general graphs with pendent paths for the most studied index named geometric-arithmetic index GA and developed tight bounds by characterizing graphs. In Theorems 2 and 3, for the first time, we defined tight bonds for the transformed graphs under the effect of transformations defined in [27].



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