# Properties of Total Transformation Graphs for General Sum-Connectivity Index

Authors: Anam Rani; Muhammad Imran; Asima Razzaque; Usman Ali
Journal: Complexity
 (2021)

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## Abstract

The study of networks and graphs through structural properties is a massive area of research with developing significance. One of the methods used in studying structural properties is obtaining quantitative measures that encode structural data of the whole network by the real number. A large collection of numerical descriptors and associated graphs have been used to examine the whole structure of networks. In these analyses, degree-related topological indices have a significant position in theoretical chemistry and nanotechnology. Thus, the computation of degree-related indices is one of the successful topics of research. The general sum-connectivityGSC index of graph Q is described as χρQ=∑qq′∈EQdq+dq′ρ, where dq presents the degree of the vertex q in Q and ρ is a real number. The total graph TQ is a graph whose vertex set is VQ∪EQ, and two vertices are linked in TQ if and only if they are either adjacent or incident in Q. In this article, we study the general sum-connectivity index χρQ of total graphs for different values of ρ by using Jensen’s inequality.


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## Body

## 1. Introduction

Chemical graph theory [1] has given a massive area of research to chemists in which they apply graph theory to mathematical modeling of chemical phenomena. This area of research necessitates characteristics of structures, in which researchers deal with their grouping, properties, and ordering. The molecular graph furnishes a graph-theoretical depiction of the molecule and lays out gainful knowledge into the chemical phenomena. The atoms of a molecule are described as vertices, and links signify the covalent bonds between the atoms. A topological index is represented by a single number corresponding with a graph. There are a lot of categories of topological indices, some of them being distance-, counting-, eccentricity-, and degree-related indices and polynomials of graphs. Among the abovementioned categories of indices, the degree-dependent indices show a beneficial character in the chemical graph theory.Here, we examine only simple and connected graphs. LetQ be the graph with vertex and edge sets, VQ and EQ, respectively. For a vertex q of Q, dq denotes the degree of q which is described as the number of its incident edges. The Randić index was described in [2] as(1)ℛQ=∑qq′∈EQ1dqdq′.It is one of the most popular molecular invariants in the study of structure-property and activity relationships [3, 4]. The general Randić index is described as follows:(2)ℛρQ=∑qq′∈EQdqdq′ρ,where ρ is a real number. The sum-connectivity index χ−1/2Q=∑qq′∈EQdq+dq′−1/2 is introduced by Zhou and Trinajstić [5]. Zhou and Trinajstić [6] investigated the idea of the general sum-connectivity index with the help of sum-connectivity index and defined it in following way:(3)χρQ=∑qq′∈EQdq+dq′ρ,where ρ is a real number. The Randić and sum-connectivity indices relate with the π-electron energy of benzenoid hydrocarbons [7]. A lot of extremal characteristics of the general sum-connectivity index for certain graph families were given [8–16]. For additional studies on extremal graphs with topological indices, we refer [17–23].The totalTQ graph is the graph with VTQ=VQ∪EQ and qq′∈EVQ∪EQ if and only if q and q′ are either incident or linked in Q. The total graphs of paths Pn and cycle C6 are shown in Figure 1. Some characteristics of total graphs are described in the literature [24–29].Figure 1
Pn, TsPn, C6, and TC6.A graph whose vertex set is converted into two linked vertex setsQ1 and Q2, such that every vertex in Q1 is attached to every vertex in Q2, is recognized as a complete bipartite Ka,b graph. A star graph Sn is a complete bipartite K1,n graph. The n-hypercube graph Ql can be defined as a graph with vertices being a set of ordered l-tuples of 0′s and 1′s and whose vertices are joined if and only if ordered l-tuples differ in exactly one partition.


## 2. Parametric Inequalities

This section provides certain important lemmas that will be beneficial in Section3 to derive the smallest and largest general sum-connectivity index for different graphs corresponding to their total transformations.Lemma 1.
Letv≥3 be a positive integer; then, the function defined by(4)fv=2ρ−1vv−1ρ+11−2ρv+1,is a strictly decreasing and a strictly increasing function for ρ>−3 and ρ<−3, respectively.Proof.
However, we have(5)f′v=2ρ−1v−1ρ2v+ρv−1−2ρ3v2+ρv2+ρv−1=2ρ−1v−1ρgv,where the function gv is(6)gv=2v+ρv−1−2ρ3v2+ρv2+ρv−1.
Thus, we obtain(7)g″v=−2ρ+13+ρ<0,for ρ>−3, and(8)g″v=−2ρ+13+ρ>0,for ρ<−3. Consequently, we have f′v<0 and f′v>0 when ρ>−3 and ρ<−3, respectively. Therefore fv is a strictly decreasing function when ρ>−3 and is a strictly increasing function when ρ<−3, which finishes the proof.Lemma 2.
Leto,t≥2 be positive integers; then, the function defined by(9)go,t=to+tρ−1o+t+ρo.is a strictly increasing function for ρ>0.Proof.
We obtain(10)∂go,t∂o=ρto+tρ−2o+2t+ρo>0,for every ρ>0, and(11)∂go,t∂t=o+tρ−21+ρo2+ρt2+1+ρot+ρ2o>0,if and only if we have 1+ρo2+ρt2+1+ρot+ρ2o>0 for o,t≥2 and ρ>0. Hence, go,t is a strictly increasing function when ρ>0.Lemma 3.
Leto,t≥2 be positive integers; then, the function defined as(12)θo,t=ot2ρo+tρ,is a strictly increasing function for ρ>0.Proof.
By taking derivative ofθo,t, we acquire(13)∂θo,t∂o=2ρto+tρ−1o+t+ρo=2ρto+tρ−1ψo,t,where ψo,t=o+t+ρo. The function ψo,t is increasing if and only if ρ>0. Thus, we obtain ∂θo,t/∂o>0 and, similarly, we obtain ∂θo,t/∂t>0 when ρ>0. Therefore, the function θo,t is strictly increasing function when ρ>0.Lemma 4.
Let the function be defined by(14)ϕo,t=oto+tρ1−2ρo+2,for o,t≥2, where o and t are the integers. Then, ϕo,t is a decreasing function when ρ>0.Proof.
It is necessary to show that∂ϕo,t/∂o<0 and ∂ϕo,t/∂t<0 for each o,t≥2. We acquire(15)∂ϕo,t∂o=to+tρ−1o+t+ρo1−2ρo+2−oto+tρ2ρ=go,tho−θo,t,where go,t=to+tρ−1o+t+ρo, ho=1−2ρo+2, and to,t=oto+tρ2ρ.
By Lemma2, go,t is a strictly increasing function when ρ>0. Also, h′o=1−2ρ<0 for each ρ>0; therefore, ho is also a strictly decreasing function when ρ>0.
From Lemma3, θo,t is an increasing function for ρ>0, which implies that ∂ϕo,t/∂o<0 and, similarly, we have ∂ϕo,t/∂t<0 where ρ>0 and o,t≥2. Hence, the function ϕo,t is a decreasing function for ρ>0.Lemma 5.
Letu≥2 be a positive integer; then,(16)f1u=2u+ρ−1uρ+11−2ρu+2,is the strictly decreasing function for ρ>−3.Proof.
It is easily seen that(17)f1′u=2u+ρ−1uρuln2+ρ+11−2ρu+2−2ρu=2u+ρ−1uρgu,where gu=uln2+ρ+11−2ρu+2−2ρu. Then, g″u=−2ρ+1ln2<0 for ρ>−3. Consequently, we obtain that f1′u<0; thus, f1u is the strictly decreasing function for ρ>−3.Lemma 6.
(see [30]). Let w>0 be a real number. If ρ<0 or ρ>1, then 1+wρ>1+ρw; but for 0<ρ<1, 1+wρ<1+ρw.Lemma 7.
Letu≥4 be a positive integer; then, the function defined as(18)fu=u−11−2ρ−1uuρ−u+2ρ−3u−2ρ,is a strictly decreasing function for 2≤ρ<3.Proof.
Fromfu, we obtain(19)f′u=u−1uρ−1ρ+1u−ρ−2ρ−1u2ρ+1+2ρ−1uρ−1−u+2ρ−1u+2+ρ−3u−2ρ−13u−2+ρ.
Sinceuρ−1 is strictly convex, by using Jensen’s inequality, we acquire that(20)3u−2ρ−1>4uρ−1−u+2ρ−1,which yields that(21)f′u<u−1uρ−1ρ−11u−5ρ+8−2ρ−1u2ρ+1+2ρ−1uρ−1−2u+2ρ−12−u=u−1uρ−1ρ−11u−5ρ+8−2ρ−1u2ρ+1+2ρ−1uρ−1−22−u1+2uρ−1.
Since1≤ρ−1<2, by Lemma 6, we obtain that 1+2/uρ−1>1+2ρ−1/u. This implies that f′u<u−1uρ−1gu,ρ, where we obtain the function(22)gu,ρ=1uaρu3+bρu2+cρu+dρ.
In this result, an easy routine computation yields thataρ=ρ+2ρ−1ρ+1<0, bρ=2ρ−1ρ−1+ρ−9<0, cρ=−ρ<0, and dρ=−8ρ−1<0. Consequently, the function fu is a strictly decreasing function for 2≤ρ<3.


## 3. Main results

The present section focuses on finding the general sum-connectivityχρ of total graphs of path Pn and cycle Cn, complete Kn graph, complete bipartite Ka,b graph, hypercube graph Qn, star graph Sn, and unicyclic graph with a fixed pendant vertex for different values of ρ.In the next theorem, we determine the graphs having the smallest and largest general sum-connectivityGSC index in the case of total transformation of path Pn.Theorem 1.
LetTPn be the total graph of path Pn, where n≥4. Then, Pn has the smallest and largest general sum-connectivity GSC index when ρ>−2 and ρ<−2, respectively.Proof.
It can be easily seen that(23)χρPn−χρTPn=n−34ρ−4n−138ρ+2⋅3ρ−2⋅5ρ−2⋅6ρ−4⋅7ρ.For u≥4, the function defined as θu=u−34ρ−4u−138ρ is strictly decreasing when ρ>−2. Therefore,(24)χρPn−χρTPn≤θu≤θ4and(25)θ4=4ρ−3⋅8ρ+2⋅3ρ−2⋅5ρ−2⋅6ρ−4⋅7ρ=23ρ−5ρ−6ρ−3⋅8ρ+4⋅7ρ−4ρ<0.This implies that χρPn<χρTPn. Similarly, we obtain χρPn>χρTPn for ρ<−2, which finishes the proof.
In Theorem2, we investigate the graphs with the smallest and largest general sum-connectivity GSC index in the case of total transformation of Cn.Theorem 2.
For everyn≥3, let TCn be the total graph of cycle Cn. Then, the cycle Cn has a smallest and largest general sum-connectivity GSC index when ρ>−2 and ρ<−2, respectively. Moreover,(26)χρCn=χρTCn,if and only if ρ=−2.Proof.
We have(27)χρCn−χρTCn=n⋅4ρ−4n⋅8ρ.For every u≥3, the function ψu=4ρ−4⋅8ρu. Then, we acquire that ψ′u=4ρ−4⋅8ρ<0, which shows that ψu is a decreasing function. Thus,(28)χρCn−χρTCn≤ψu≤ψ3.
Also,ψ3=3⋅4ρ−12⋅8ρ<0 if and only if 1/2ρ<4 which is proved for ρ>−2. Then, we obtain χρCn<χρTCn finally.
Moreover, it can be easily seen thatψ3=3⋅4ρ−12⋅8ρ>0 if and only if 1/2ρ>4, which is obeyed for ρ<−2. Then, we obtain in this case that χρCn>χρTCn. For ρ=−2, we have χρCn=χρTCn, which finishes the proof.
In the upcoming theorem, we determine graphs with the smallest and largestGSC index for the total transformation of the complete graph Kn.Theorem 3.
Forn≥3, let TKn be the total transformation of Kn. Then, Kn has the smallest and largest GSC index for ρ>−3 and ρ<−3, respectively. Moreover,(29)χρKn=χρTKn,if and only if we have ρ=−3 and n=3.Proof.
As we have(30)χρKn−χρTKn=2ρ−1nn−1ρ+11−2ρn+1,for every u≥3, the function φu=2ρ−1uu−1ρ+11−2ρu+1 is a decreasing function when ρ>−3 and increasing function for ρ<−3 by Lemma 1. Then, we acquire(31)χρKn−χρTKn≤φu≤φ3.
For allρ, we have φ3=34ρ−4⋅8ρ<0 if and only if 1/2ρ<4, which is proved for ρ>−3. Then, χρKn<χρTKn. Similarly, φ3=34ρ−4⋅8ρ>0 if and only if 1/2ρ>4, which is obeyed for ρ<−3. Thus, we finally obtain that χρKn>χρTKn with equality if and only if we have ρ=−3 and n=3.
In the upcoming theorem, we derive graphs with the smallestGSC index for the total transformation of Ka,b.Theorem 4.
Fora,b≥2, let TKa,b be the total transformation of Ka,b. Then, the complete bipartite graph Ka,b has the smallest GSC index for ρ>0 with equality if and only if ρ=−2 and a,b=2.Proof.
We have(32)χρKa,b−χρTKa,b=aba+bρ1−2ρa+2.
Let the functionϕu,v=uvu+vρ1−2ρu+2 for u,v≥2. By Lemma 4, we obtain that ϕu,v is a decreasing function for ρ>0. Thus, we have(33)χρKa,b−χρTKa,b≤ϕx,y≤ϕ2,2.
This is because for allρ, we obtain ϕ2,2=44ρ−4⋅8ρ<0 if and only if 1/2ρ<4, which is satisfied for ρ>0. Thus, we obtain that(34)χρKa,b<χρTKa,b,for x,y≥2 and ρ>0 with equality if and only if ρ=−2 and a=b=2.
In the next theorem, we derive graphs with the smallest general sum-connectivityGSC index for the total transformation of the hypercubes Qn.Theorem 5.
LetQn be the hypercube and TQn be the total transformation of the hypercube. For every n≥2 and ρ>−3, the hypercube Qn has the smallest general sum-connectivity GSC index and also(35)χρQn=χρTQn,if and only if n=2 and ρ=−2.Proof.
As we know,(36)χρQn−χρTQn=2n+ρ−1nρ+11−2ρn+2.
As the function defined byf1u=2u+ρ−1uρ+11−2ρu+2 is the strictly decreasing function by Lemma 5, f12=44ρ−4⋅8ρ<0 if and only if 1/2ρ<4, where u≥2 and ρ>−3. Thus, we acquire(37)χρQn−χρTQn≤f1u≤f12,
For allρ, we have f12=44ρ−4⋅8ρ<0 if and only if 1/2ρ<4, which is satisfied for ρ>−3. Therefore,(38)χρQn<χρTQn,with equality if and only if we have n=2 and ρ=−2, which finishes the proof.
Next, we present graphs with the smallestGSC index for the total transformation of the star graph Sn.Theorem 6.
For everyn≥4, let TSn be the total transformation of star graph Sn. Then, the star graph Sn has the smallest GSC index where 2≤ρ<3.Proof.
We easily acquire that(39)χρSn−χρTSn=n−11−2ρ−1nnρ−n+2ρ−3n−2ρ.
For everyu≥4, let fu=u−11−2ρ−1uuρ−u+2ρ−3u−2ρ. Then, by using Lemma 7, the function fu is a strictly decreasing function. Hence,(40)χρSn−χρTSn≤fu≤f4<0.
This is because we havef4=3⋅4ρ−3⋅6ρ−6⋅8ρ−3⋅10ρ<0 if and only if 4ρ−6ρ/2⋅8ρ+10ρ<1, which is satisfied by 2≤ρ<3. This implies that the star Sn has the smallest general sum-connectivity GSC index when 2≤ρ<3.
In the next theorem, we obtain graphs with the smallest and largest general sum-connectivityGSC index for the total transformation of the Un1, where Un1 is the unicyclic graph acquired by connecting a pendant vertex to a unique vertex of Cn.Theorem 7.
LetTUn1 be the total transformation of Un1. Then, for every n≥3, Un1 has the smallest and largest general sum-connectivity GSC index when ρ>−2 and ρ<−2, respectively.Proof.
We acquire(41)χρUn1−χρTUn1=n−14ρ−4n−88ρ+2⋅5ρ−6ρ−6⋅9ρ−4⋅10ρ−2⋅11ρ.
For everyu≥3, let fu=u−14ρ−4u−88ρ be a strictly decreasing function for ρ>−2. Then,(42)χρUn1−χρTUn1≤fu≤f3.
Forρ>−2, we acquire(43)f3=2⋅4ρ−8⋅8ρ+2⋅5ρ−6ρ−6⋅9ρ−4⋅10ρ−2⋅11ρ=24ρ+5ρ−11ρ−24⋅8ρ+3⋅9ρ+2⋅10ρ−6ρ<0.
It follows thatχρUn1<χρTUn1. Similarly, one can also prove that(44)χρUn1>χρTUn1,for ρ<−2, which completes the proof.


## 4. Conclusion

In this manuscript, we focus on finding the general sum-connectivity index in case of total graphs for a pathPn and cycle Cn, complete graph, complete bipartite graph Ka,b, hypercube graph Qn, star graphSn, and unicyclic graph with a fixed pendant vertex for different values of ρ. It would be most appealing to compute beneficial results for the extremal general sum-connectivity index of different graphs with fixed parameters and also to work on the Mostar, edge Mostar and weighted Mostar indices for total graphs and certain transformations of graphs.



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