# Upper Bounds of AZI and ABC Index for Transformed Families of Graphs

Authors: Muhammad Hussain; Muhammad Asif; Ashit Kumar Dutta; Sultan Almotairi
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Topological index is a mapping which corresponds underlying graph with a numeric value and invariant up to all the isomorphisms of graph. Our study is based on a partial open question regarding topological indices: for which members ofn-vertex graph family, certain index has minimum or maximum value? In this work, we answered the above-mentioned question regarding AZI and ABC for transformed families of graphs Γnk,l and AαΓnk,l. We investigated the fact of pendent paths and the transformation Aα over these indices and developed the tight upper bounds regarding these families of graphs. Moreover, we characterized transformed graphs associated with maximum values of these indices.


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## Body

## 1. Introduction

Nowadays, graph theory has potential applications in different fields of science. It is especially used for theoretical study of chemical compounds in chemistry. This area of study named as chemical graph theory deals with the problems related to the properties in chemistry. In the middle of last century, theoretical study of chemical compounds attracted the researchers due to its effective applications such as prediction of physiochemical properties of substances in cheminformatics, pharmaceutical sciences, materials science, engineering, and so forth [1]. Cheminformatics is comparatively the latest area of information technology which comprises chemistry, mathematics, and other informational sciences that concentrate on gathering, storing, treating, and examining chemical data. There are many theoretical molecular descriptors in literature used to predict properties of chemical compounds. Among these molecular descriptors, topological indices have an impact in chemistry due to the prediction of physiochemical properties of underlying substance. Its role in “quantitative-structure property relationship” (QSPR) or “quantitative-structure activity relationship” (QSAR) investigation models is also remarkable [2, 3].In 1947, Wiener for the first time introduced the use of topological index during his work on paraffin’s boiling points [4] and provided that it has best correlation with the boiling points of alkanes. The discovery of the Wiener index provided emerging research platform to the research community. In the later years, researchers of different communities proposed many other topological indices and used them for approximation of the chemical properties of their own interest.In the race for better prediction, Randic [5] in 1975 introduced degree-based topological index named Randić connectivity index which was the best predictive invariant in those days. The Randić index was reported as the first degree-based index in QSPR study because Zagreb indices by Gutman and Trinajstic̀ [6] were used for totally different purpose before Randić index. In 1998, parallel to the work of Bollobás and Erdös [7], Estrada et al. [8] defined atom bond connectivity (ABC) index as(1)ABCΓ=∑uv∈EΓdegu+degv−2degu×degv,which has a good correlation with the heat of formation of alkanes. Star graph among trees and complete graph in general for fixed number of vertices have maximal value for ABC index [9]. For more details, one can see [10, 11]. Furtula et al. [12] made a generalization of ABC index as(2)ABCλΓ=∑uv∈EΓdegu+degv−2degu×degv−λ,by replacing 1/2 with −λ. The augmented Zagreb index AZI is ABCλ,λ=3 as,(3)AZIΓ=∑uv∈EΓdegu×degvdegu+degv−23.Its correlation potential reported is even better than that of other indices in [13–15].Mathematical study ofABC index and AZI [16–35] encouraged us to answer the fundamental question regarding characterization of transformed families of graphs with maximum and minimum values for ABC index and AZI. Gupta et.al. determined bounds for symmetric division degree index in [36]. In this work, we studied ABC index and AZI for transformed graphs Γnk,l and AαΓnk,l under the fact of transformation Aα, 0≤α≤l−2. We characterize extremal graphs of these transformed families of graphs for AZI and ABC and established their bounds for Γnk,l and AαΓnk,l. When a path is attached with the fully connected vertex (vertex with degree greater than one) of the graph, then it has an impact over the increase and decrease of the index under study. Throughout this work, consider graph Γnk,l [37]. It comprises n-vertex simple connected graph Γ along with k pendent paths of length l≥2 attached with v∈Γ having degree 2≤dv≤ΔΓ. Let degu1=δΓ≤degu2≤degu3≤…≤ΔΓ+1 be the degree sequence of Γnk,l. Γnk,l is shown in Figure 1.Figure 1
GraphΓnk,l.


### 1.1. 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 define following transformations.LetAα;0≤α≤l−2 be the transformation defined over pendent paths attached with the graph [38]. Aα has solid effect over increase and decrease of AZI and ABC.


#### 1.1.1. TransformationA

Letwj∈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,(4)AΓnk,l=Γnk,l−∑j=1kuj2uj3,uj3uj4,…,ujl−1ujl+∑j=1kwjuj2,uj2uj3,…,ujl−1ujl.The transformationA is shown in Figure 2.Figure 2
TransformationA.


#### 1.1.2. TransformationAα

Aα is the α≥0 time repetition of transformation A.Let graphΓ=ΓV,E with degree of vertex u∈Γ, δΓ≤degu≤ΔΓ and δΓ≤degv≤ΔΓ+1 be the degree of v∈Γnk,l.


## 1.1. 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 define following transformations.LetAα;0≤α≤l−2 be the transformation defined over pendent paths attached with the graph [38]. Aα has solid effect over increase and decrease of AZI and ABC.


### 1.1.1. TransformationA

Letwj∈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,(4)AΓnk,l=Γnk,l−∑j=1kuj2uj3,uj3uj4,…,ujl−1ujl+∑j=1kwjuj2,uj2uj3,…,ujl−1ujl.The transformationA is shown in Figure 2.Figure 2
TransformationA.


### 1.1.2. TransformationAα

Aα is the α≥0 time repetition of transformation A.Let graphΓ=ΓV,E with degree of vertex u∈Γ, δΓ≤degu≤ΔΓ and δΓ≤degv≤ΔΓ+1 be the degree of v∈Γnk,l.


## 1.1.1. TransformationA

Letwj∈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,(4)AΓnk,l=Γnk,l−∑j=1kuj2uj3,uj3uj4,…,ujl−1ujl+∑j=1kwjuj2,uj2uj3,…,ujl−1ujl.The transformationA is shown in Figure 2.Figure 2
TransformationA.


## 1.1.2. TransformationAα

Aα is the α≥0 time repetition of transformation A.Let graphΓ=ΓV,E with degree of vertex u∈Γ, δΓ≤degu≤ΔΓ and δΓ≤degv≤ΔΓ+1 be the degree of v∈Γnk,l.


## 2. Upper Bounds forAZIΓnk,l and AZIAαΓnk,l

Initially, we proved Proposition1, which is helpful to prove the main results for AZI.Proposition 1.
Let(5)fη,ζ=fζ,η=ηζη+ζ−23.
Then, fora≥b and ζ≥2, fa,ζ≥fb,ζ.Proof.
Let(6)fη,ζ=fζ,η=ηζη+ζ−23,fa,ζ−fb,ζ=aζa+ζ−23−bζb+ζ−23=a3ζ3b+ζ−23−b3ζ3a+ζ−23a+ζ−23b+ζ−23=ζ3a3b3+ζ−23+3b2ζ−2+3bζ−22−b3a3+ζ−23+3a2ζ−2+3aζ−22a+ζ−23b+ζ−23=ζ3a3−b3ζ−23+3a2b2a−bη−2+3aba2−b2η−22a+ζ−23b+ζ−23=ζ3a−ba2+ab+b2ζ−23+3a2b2a−bζ−2+3aba+ba−bζ−22a+ζ−23b+ζ−23=ζ3a−ba2+ab+b2ζ−23+3a2b2ζ−2+3aba+bζ−22a+ζ−23b+ζ−23≥0.
This implies(7)fa,ζ≥fb,ζ.Lemma 1 (see [15]).
Let(8)Φu,v=Φv,u=uvu+v−23.
Then,(1)
Φ1,vis decreasing forv≥2.(2)
Φ2,v=8for any real numberv.(3)
For fixedv≥3,Φu,vis increasing andΦu,v>8foru>2.
In Theorem1, we discuss the effect of pendent paths over AZI and determine its upper bound.Theorem 1.
Let graphΓnk,l comprise n-vertex graph Γ having m edges and p≥0 pendent vertices. Then,(9)AZIΓnk,l≤kΔΓ2ΔΓ+122ΔΓ3+ΔΓΔΓ+12ΔΓ−13+AZIΓ−kδΓ2δΓδΓ+12δΓ−13+δΓ22δΓ−23+8kl+p8−ΔΓ+1ΔΓ3.Equality holds for a complete graphΓ of size n with pendent paths of length l at each vertex, i.e., k=n.Proof.
LetΓnk,l be the graph formed by k number of paths having length l pendent at distinct vertices u∈Γ such that 2≤degu≤δΓ+1. Then,(10)AZIΓ=∑uv∈EΓdegudegvdegu+degv−23.
The construction ofΓnk,l,l≥2, implies EΓnk,l=m+kl, and for(11)uv∈EΓnk,l,degu+degv∈3,4,degu+3,degu+degv,degu+degv+1.
The edge set ofΓnk,l is partitioned as(12)A3=uv∈Γnk,l:degu=1,degv=2,A4=uv∈Γnk,l:degu=degv=2,Adegu+3=uv∈Γnk,l:δΓ≤degu=degu≤ΔΓ+1,degv=2,Adegu+degv=uv∈Γnk,l:δΓ≤degu=degu,degv=degv≤ΔΓ,Adegu+degv+1=uv∈Γnk,l:δΓ≤degu=degu,degv=degv+1≤ΔΓ+1,AZIΓnk,l=∑uvare edges of pendent pathsdegudegvdegu+degv−23+∑uvare edges ofΓdegudegvdegu+degv−23,AZIΓnk,l=∑uv∈A3degudegvdegu+degv−23+∑uv∈A4degudegvdegu+degv−23+∑uv∈Adegu+3degudegvdegu+degv−23+∑uv∈Adegu+degu+1degu,degu≥2degudegvdegu+degv−23+∑uvare pendent edges ofΓdegudegvdegu+degv−23+∑uv∈Adegu+degvdegudegvdegu+degv−23.
The construction ofAZIΓnk,l implies that the cardinality of A3 is k,i.e.,A3=k,A4=kl−2, Adegu+3=k, Adegu+3=k,Adegu+degu+1degu,degu≥2≤kΔΓ−p, and A1+degu=p. For δΓ minimum degree of vertices of Γ and maximum degree ΔΓ, using Proposition 1 and Lemma 1, we have(13)∑uv are pendent edges ofΓdegudegvdegu+degv−23≤8p,∑uv∈Adegu+degu+1degu,degu≥2degudegvdegu+degv−23≤kΔΓ2ΔΓ+122ΔΓ3+ΔΓΔΓ+12ΔΓ−13,∑uv∈Adegu+degudegu,degu≥2degudegvdegu+degv−23≤AZIΓ−pΔΓ+1ΔΓ3−kδΓ2δΓδΓ+12δΓ−13+δΓ22δΓ−23.
Now, from equation (12), we get(14)AZIΓnk,l≤kΔΓ2ΔΓ+122ΔΓ3+ΔΓΔΓ+12ΔΓ−13+AZIΓ−kδΓ2δΓδΓ+12δΓ−13+δΓ22δΓ−23−pΔΓ+1ΔΓ3+8k+8kl−2+8k+8p.
After simplification, we get(15)AZIΓnk,l≤kΔΓ2ΔΓ+122ΔΓ3+ΔΓΔΓ+12ΔΓ−13+AZIΓ−kδΓ2δΓδΓ+12δΓ−13+δΓ22δΓ−23+8kl+p8−ΔΓ+1ΔΓ3.
Inequality (15) completes the proof.
In Theorem2, we discussed the effect of successive applications of transformation A as shown in Figure 2 over AZI.Theorem 2.
Let graphΓnk,l comprise n-vertex simple connected graph Γ. Then,(16)AZIAαΓnk,l≤kΔΓ2ΔΓ+α+122ΔΓ+α3+ΔΓΔΓ+α+12ΔΓ+α−13−kδΓ2δΓδΓ+α+12δΓ+α−13+δΓ22δΓ−23+p8−ΔΓ+1ΔΓ3+8kl−8kα+AZIΓ.Equality holds for a complete graphΓ of size n with pendent paths of length l at each vertex, i.e., k=n.Proof.
Let a simple graphΓ of order n , Size m having p≥0 pendent vrtives. The augmented Zagreb index of any graph Γ is(17)AZIΓ=∑uv∈EΓdegudegvdegu+degv−23.
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 Edegu+degvAαΓnk,l where(18)degu+degv∈3,4,degu+α+2,degu+α+3,degu+degv,degu+α+1+degv,which implies(19)E3AαΓnk,l=uv∈Γnk,l:degu=1,degv=2,E4AαΓnk,l=uv∈Γnk,l:degu=degv=2,Edegu+α+2AαΓnk,l=uv∈Γnk,l:δΓ≤degu=degu+α+1≤ΔΓ+α+1,degv=1,Edegu+α+3AαΓnk,l=uv∈Γnk,l:δΓ≤degu=degu+α+1≤ΔΓ+α+1,degv=2,Edegu+degvAαΓnk,l=uv∈Γnk,l:δΓ≤degu=degu,degv=degv≤ΔΓ,Edegu+α+1+degvAαΓnk,l=uv∈Γnk,l:δΓ≤degv=degv,degu≤ΔΓ,degu=degu+α+1,AZIAαΓnk,l=∑uvare edges of pendent pathsdegudegvdegu+degv−23+∑uvare edges ofΓdegudegvdegu+degv−23,AZIAαΓnk,l=∑uv∈A3degudegvdegu+degv−23+∑uv∈A4degudegvdegu+degv−23+∑uv∈Adegu+α+2degudegvdegu+degv−23+∑uv∈Adegu+α+3degudegvdegu+degv−23+∑uvare pendent edges ofΓdegudegvdegu+degv−23+∑uv∈Adegu+degvdegudegvdegu+degv−23+∑uv∈Adegu+degu+α+1degu,degu≥2degudegvdegu+degv−23.
The construction ofAZIΓnk,l implies that the cardinality is(20)E3AαΓnk,l=k,E4AαΓnk,l=kl−α−2,Edegu+α+2AαΓnk,l=kα,|Edegu+α+3AαΓnk,l|=k,|Adegu+degu+1degu,degu≥2|≤kΔΓ−p,and A1+degu=p. For δΓ minimum degree of vertices of Γ and maximum degree ΔΓ, using Lemma 1, we have(21)∑uv are pendent edges ofΓdegudegvdegu+degv−23≤8p,∑uv∈Adegu+degv+α+1degu,degu≥2degudegvdegu+degv−23≤kΔΓ2ΔΓ+α+122ΔΓ+2α3+ΔΓΔΓ+α+12ΔΓ+α−13,∑uv∈Adegu+α+2degudegvdegu+degv−23≤kαΔΓ+α+1ΔΓ+α3,∑uv∈Adegu+α+3degudegvdegu+degv−23≤8k.
Substituting these changes in equation (19), we have following inequality.(22)AZIAαΓnk,l≤8k+8kl−2−α+8k+kαΔΓ+α+1ΔΓ+α3+8p+kΔΓ2ΔΓ+α+122ΔΓ+α3+ΔΓΔΓ+α+12ΔΓ+α−13+AZIΓ−kδΓ2δΓδΓ+α+12δΓ+α−13+δΓ22δΓ−23−pΔΓ+α+1ΔΓ+α3.
After simplification, we get(23)AZIAαΓnk,l≤kΔΓ2ΔΓ+α+122ΔΓ+α3+ΔΓΔΓ+α+12ΔΓ+α−13−kδΓ2δΓδΓ+α+12δΓ+α−13+δΓ22δΓ−23+p8−ΔΓ+1ΔΓ3+8kl−8kα+AZIΓ.
Inequality (23) completes the proof.


## 3. Upper Bounds forABCΓnk,l and ABCAαΓnk,l

Lemma 2 (see [39, 40]).
Let(24)Φx,y=Φy,x=x+y−2xy.Then,(1)
Φ1,yis increasing fory.(2)
Φ2,y=2/2for any real numbery.(3)
For fixedy≥2,Φx,yis decreasing forx.Proposition2 is related to the ABC index.Proposition 2.
Let(25)Φη,ζ=Φζ,η=η+ζ−2ηζ.
Forη,ζ,Δ∈ℜ,η,ζ≤Δ and η,ζ,Δ≥2.(26)Φη,ζ≥ΦΔ,Δ.Proof.
Let(27)Φη,ζ=Φζ,η=η+ζ−2ηζ.
By Lemma2, for η=2 or ζ=2 and Δ=2,(28)Φη,ζ=ΦΔ,Δ=ΦΔ,ζ=Φη,Δ=22.
Now forη≥3 or ζ≥3 and Δ=ζ, let α,β,γ≥1, α≤γ, β=γ and η=2+α or ζ=2+β, Δ=2+γ,(29)Φη,ζ−ΦΔ,Δ=η+ζ−2ηζ−Δ+Δ−2ΔΔ=α+2+β+2−2α+2β+2−2γ+2−2γ+22=α+2+βα+2β+2−2γ+2−2γ+22=1α+2+β/α+2β+2+2γ+2−2/γ+22α+2+βα+2β+2−2γ+2−2γ+22=1α+2+β/α+2β+2+2γ+2−2/γ+22=1β+2+βα+2β+2−2γ+2+2γ+22=1α+2+β/α+2β+2+2γ+2−2/γ+22γ−ββ+2+βα+2β+2−1γ+2+2γ+22=1α+2+β/α+2β+2+2γ+2−2/γ+22γ−ββ+2+β/α+2β+2−γ/γ+22=1α+2+β/α+2β+2+2γ+2−2/γ+22γ−ββ+2+βγ+22−γα+2β+2α+2β+2γ+22=1α+2+β/α+2β+2+2γ+2−2/γ+22γ−ββ+2+βγγ−α+2γβ−α+4β−4γα+2β+2γ+22=1α+2+β/α+2β+2+2γ+2−2/γ+22γ−ββ+2+βγγ−α+2γβ−α+4β−γα+2β+2γ+22=1α+2+β/α+2β+2+2γ+2−2/γ+22βγγ−α+2γβ−αα+2β+2γ+22+γ−βα+2γ+22−4α+2β+2γ+22=1η+ζ−2/ηζ+Δ+Δ−2/ΔΔΔ−2ζ−2Δ−η+2ζ−ηηζΔ2+Δ−ζηΔ2−4ηζΔ2.
SinceΦη,ζ is a symmetric function, one can let η≥ζ or ζ≥η, so the factor ζ−η≥0 along with η−2≥0, ζ−2≥0, Δ−2≥0Δ−η≥0, and Δ−ζ≥0. All the factors involved in equation (29) are positive. This implies(30)Φη,ζ−ΦΔ,Δ≥0.
Hence, for allη,ζ≥2 and η,ζ≤Δ,(31)Φη,ζ≥ΦΔ,Δ.
In Theorem3, we discuss the effect of pendent paths over ABC index and determine its upper bound.Theorem 3.
Let graphΓnk,l and Γ having order n , size m , and p≥0 pendent vertices. Then,(32)ABCΓnk,l≤kΔΓ12−kδΓ22ΔΓΔΓ+12+2ΔΓ−1ΔΓΔΓ+1+22kl+pΔΓΔΓ+1−22+ABCΓ.Equality holds forΓ=Cn with pendent paths at each vertex, i.e., k=n.Proof.
LetΓnk,l be the graph. ABCΓ is(33)ABCΓ=∑uv∈EΓdegu+degv−2degudegv.
The construction ofΓnk,l,l≥2, implies EΓnk,l=m+kl, and for(34)uv∈EΓnk,l,degu+degv∈3,4,degu+3,degu+degv,degu+degv+1.
We use edge set partition ofΓnk,l defined in Theorem 1:(35)ABCΓnk,l=∑uv are edges of pendent pathsdegu+degv−2degudegv+∑uv are edges ofΓdegu+degv−2degudegv,ABCΓnk,l=∑uv∈A3degu+degv−2degudegv+∑uv∈A4degu+degv−2degudegv+∑uv∈Adegu+3degu+degv−2degudegv+∑uv∈Adegu+degu+1degu,degu≥2degu+degv−2degudegv+∑uv are pendent edges ofΓdegu+degv−2degudegv+∑uv∈Adegu+degvdegu+degv−2degudegv.
The construction ofABCΓnk,l implies that the cardinality of A3 is k,i.e.,A3=k,A4=kl−2, Adegu+3=k, Adegu+3=k,Adegu+degu+1degu,degu≥2≤kΔΓ, and A1+degu=p. For δΓ minimum degree of vertices of Γ and maximum degree ΔΓ, using Lemma 2 and Proposition 2, we have(36)∑uv are pendent edges ofΓdegu+degv−2degudegv≤ΔΓΔΓ+1p,∑uv∈Adegu+degu+1deguordegu≥2degu+degv−2degudegv≤kΔΓ12,∑uv∈Adegu+degudegu,degu≥2degu+degv−2degudegv≤ABCΓ−p12−kδΓ22ΔΓΔΓ+12+2ΔΓ−1ΔΓΔΓ+1.
Now, from equation (35),(37)ABCΓnk,l≤ABCΓ+pΔΓΔΓ+1+kΔΓ12+22k−kδΓ22ΔΓΔΓ+12+2ΔΓ−1ΔΓΔΓ+1−22p+22kl−2+22k.
After simplification,(38)ABCΓnk,l≤kΔΓ12−kδΓ22ΔΓΔΓ+12+2ΔΓ−1ΔΓΔΓ+1+22kl+pΔΓΔΓ+1−22+ABCΓ.
Inequality (38) completes the proof.
Theorem4 gives the discussion about the effect of successive applications of transformation A as shown in Figure 2 over ABC index.Theorem 4.
Let graphΓnk,l with maximum degree of ΔΓ+1 and minimum δΓ. Then,(39)ABCAαΓnk,l≤12kl+kαΔΓ+αΔΓ+α+1+kΔΓ2δΓ+2α−1δΓ+αδΓ+α+1−kδΓ22ΔΓ+2αΔΓ+α+12+2ΔΓ+2α−1ΔΓ+αΔΓ+α+1+pΔΓ+αΔΓ+α+1−12+ABCΓ.Equality holds forΓ a complete graph of size n with pendent paths of length l at each vertex, i.e., k=n.Proof.
LetΓnk,l graph having pendent paths and Aα be the α time repetition of transformation A. ABCΓ is(40)ABCΓ=∑uv∈EΓdegu+degv−2degudegv.
After successive applications of transformationA as Aα, α≤l−1, the edge set of AαΓnk,l is partitioned as Edegu+degvAαΓnk,l where(41)degu+degv∈3,4,degu+α+2,degu+α+3,degu+degv,degu+α+1+degv.
The construction ofAαΓnk,l shows(42)E3AαΓnk,l=uv∈Γnk,l:degu=1,degv=2,E4AαΓnk,l=uv∈Γnk,l:degu=degv=2,Edegu+α+2AαΓnk,l=uv∈Γnk,l:δΓ+1≤degu=degu+α+1≤ΔΓ+α+1,degv=1,Edegu+α+3AαΓnk,l=uv∈Γnk,l:δΓ+1≤degu=degu+α+1≤ΔΓ+α+1,degv=2Edegu+degvAαΓnk,l=uv∈Γnk,l:δΓ≤degu=degu,degv=degv≤ΔΓ,Edegu+α+1+degvAαΓnk,l=uv∈Γnk,l:δΓ≤degv=degv,degu≤ΔΓ,degu=degu+α+1,ABCAαΓnk,l=∑uvare edges of pendent pathsdegu+degv−2degudegv+∑uvare edges ofΓdegu+degv−2degudegv,ABCAαΓnk,l=∑uv∈A3degu+degv−2degudegv+∑uv∈A4degu+degv−2degudegv+∑uv∈Adegu+α+2degu+degv−2degudegv+∑uv∈Adegu+α+3degu+degv−2degudegv+∑uvare pendent edges ofΓdegu+degv−2degudegv+∑uv∈Adegu+degvdegu+degv−2degudegv+∑uv∈Adegu+degu+α+1degu,degu≥2degu+degv−2degudegv.
The construction ofAαΓnk,l implies that(43)E3AαΓnk,l=k,E4AαΓnk,l=kl−α−2,Edegu+α+2AαΓnk,l=kα,Edegu+α+3AαΓnk,l=k,Adegu+degu+1degu,degu≥2≤kΔΓ−p,and A1+degu=p. For δΓ minimum degree of vertices of Γ and maximum degree ΔΓ, using Lemma 2 and Proposition 2, we have(44)∑uvare pendent edges ofΓdegu+degv−2degudegv≤pΔΓ+αΔΓ+α+1∑uv∈Adegu+degv+α+1degu,degu≥2degu+degv−2degudegv≤kΔΓ2δΓ+2α−1δΓ+αδΓ+α+1,∑uv∈Adegu+α+2degu+degv−2degudegv≤kαΔΓ+αΔΓ+α+1,∑uv∈Adegu+α+3degu+degv−2degudegv=12k,∑uv∈Adegu+degudegu,degu≥2degu+degv−2degudegv≤ABCΓ−p12−kδΓ22ΔΓ+2αΔΓ+α+12−kδΓ22ΔΓ+2α−1ΔΓ+αΔΓ+α+1.
Substituting these changes in equation (42), we have following inequality.(45)ABCAαΓnk,l≤pΔΓ+αΔΓ+α+1+12kl−2−α+12k+kαΔΓ+αΔΓ+α+1+kΔΓ2δΓ+2α−1δΓ+αδΓ+α+1+ABCΓ−p12−kδΓ22ΔΓ+2αΔΓ+α+12+2ΔΓ+2α−1ΔΓ+αΔΓ+α+1.
After simplification, we get(46)ABCAαΓnk,l≤12kl+kαΔΓ+αΔΓ+α+1+kΔΓ2δΓ+2α−1δΓ+αδΓ+α+1−kδΓ22ΔΓ+2αΔΓ+α+12+2ΔΓ+2α−1ΔΓ+αΔΓ+α+1+pΔΓ+αΔΓ+α+1−12+ABCΓ.
Inequality (46) completes the proof.


## 4. Conclusion

The study of mathematical aspect regarding topological indices is a partially open problem: for which members of graph family, certain index has minimal or maximal value? In this work, we deal with this fundamental question. We considered graphs of familyΓnk,l with 0≤k≤n pendent paths of length l≥2 and transformed family AαΓnk,l where transformation Aα is the graph transformation. The concluding key points concerning our study are given as follows. We studied the fact of pendent paths over the increase and decrease of AZI and ABC index in addition to defining upper bounds for these indices and mentioned graphs with maximum values of these indices. We discussed the fact of defined transformation as Aα; 0≤α≤l−2 , 0≤k≤n over AZI and ABC indices. We determined upper bounds and characterized extremal graphs for such bounds.



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*Source: 4152854-2022-01-25.xml*