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Research ArticleSharp Bounds for the General Sum-Connectivity Indices ofTransformation Graphs

HaiyingWang,1 Jia-Bao Liu,2 Shaohui Wang,3 Wei Gao,4 Shehnaz Akhter,5

Muhammad Imran,5,6 andMohammad R. Farahani7

1School of Science, China University of Geosciences (Beijing), Beijing 100083, China2School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, China3Department of Mathematics, Savannah State University, Savannah, GA 31404, USA4School of Information and Technology, Yunnan Normal University, Kunming 650500, China5Department of Mathematics, School of Natural Sciences (SNS), National University of Sciences and Technology (NUST),Sector H-12, Islamabad, Pakistan6Department of Mathematical Sciences, College of Science, United Arab Emirates University, P.O. Box 15551, Al Ain, UAE7Department of Applied Mathematics, Iran University of Science and Technology (IUST), Narmak, Tehran 16844, Iran

Correspondence should be addressed to Haiying Wang; [email protected]

Received 12 July 2017; Revised 23 October 2017; Accepted 9 November 2017; Published 3 December 2017

Academic Editor: Chris Goodrich

Copyright © 2017 Haiying Wang 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.

Given a graph 𝐺, the general sum-connectivity index is defined as 𝜒𝛼(𝐺) = ∑𝑢V∈𝐸(𝐺)(𝑑𝐺(𝑢) + 𝑑𝐺(V))𝛼, where 𝑑𝐺(𝑢) (or 𝑑𝐺(V))denotes the degree of vertex 𝑢 (or V) in the graph 𝐺 and 𝛼 is a real number. In this paper, we obtain the sharp bounds for generalsum-connectivity indices of several graph transformations, including the semitotal-point graph, semitotal-line graph, total graph,and eight distinct transformation graphs 𝐺𝑢V𝑤, where 𝑢, V, 𝑤 ∈ {+, −}.

1. Introduction

In this paper, we consider simple, undirected, and connectedgraphs. Let 𝐺 be the graph with vertex set 𝑉(𝐺) and edgeset 𝐸(𝐺). The order and size of 𝐺 are denoted by 𝑛 and𝑒, respectively. For a vertex 𝑎 ∈ 𝑉(𝐺), 𝑑𝐺(𝑎) denotes thedegree of 𝑎. Two vertices in 𝐺 are adjacent if and only if theyare end vertices of an edge, and each of the two vertices iscalled incident to the edge. Besides, two edges are adjacentto each other if and only if they share a common vertex. Theminimum and maximum degrees of graph 𝐺 are denoted by𝛿(𝐺) and Δ(𝐺), respectively. We will use the notations 𝑃𝑛, 𝐶𝑛,and 𝐾𝑛 for a path, cycle, and complete graph of order 𝑛 [1],respectively.

The complement of 𝐺, denoted by 𝐺, is the graph with𝑉(𝐺) = 𝑉(𝐺) and two vertices in 𝐺 are adjacent if and only ifthey are not adjacent in 𝐺. Thus, the size of 𝐺 is ( 𝑛2 ) − 𝑒 and if𝑎 ∈ 𝑉(𝐺) then 𝑑𝐺(𝑎) = 𝑛 − 1 − 𝑑𝐺(𝑎).

A topological index is a numeric quantity associatedwith a graph which characterizes the topology of graph.A topological index Top(𝐺) of a graph 𝐺 is equal to thetopological index Top(𝐻) of 𝐻, if and only if two graphs 𝐺and𝐻 are isomorphic. The idea of topological index appearsfrom work done by Wiener in 1947, this index is calledWiener index. The first and second Zagreb indices have beenintroduced by Gutman and Trinajestic [2]. These indices aredefined on the ground of vertex degrees as follows:

𝑀1 (𝐺) = ∑𝑎∈𝑉(𝐺)

(𝑑𝐺 (𝑎))2 ,𝑀2 (𝐺) = ∑

𝑎𝑏∈𝐸(𝐺)

𝑑𝐺 (𝑎) 𝑑𝐺 (𝑏) .(1)

The Randic connectivity index was defined in 1975 by Randic[3]. It has been extended to the general Randic connec-tivity index. The general Randic connectivity index (general

HindawiDiscrete Dynamics in Nature and SocietyVolume 2017, Article ID 2941615, 7 pageshttps://doi.org/10.1155/2017/2941615

https://doi.org/10.1155/2017/2941615

2 Discrete Dynamics in Nature and Society

product-connectivity index) was defined by Bollobas andErdos [4] as follows:

𝑅𝛼 (𝐺) = ∑𝑎𝑏∈𝐸(𝐺)

(𝑑𝐺 (𝑎) 𝑑𝐺 (𝑏))𝛼 , (2)

where 𝛼 is a real number. Then 𝑅−1/2 is the classical Randicconnectivity index.The sum-connectivity indexwas proposedin [5]. This concept was extended to the general sum-connectivity index in [6], which is defined as

𝜒𝛼 (𝐺) = ∑𝑎𝑏∈𝐸(𝐺)

(𝑑𝐺 (𝑎) + 𝑑𝐺 (𝑏))𝛼 , (3)

where 𝛼 is a real number. Then 𝜒−1/2(𝐺) is the classicalsum-connectivity index. The sum-connectivity index and theproduct-connectivity index correlate well with the 𝜋-electronenergy of benzenoid hydrocarbons [7].

The total graph 𝑇(𝐺) of the graph 𝐺 is a graph whosevertex set is the union of 𝑉(𝐺) and 𝐸(𝐺) such that 𝑎𝑏 ∈𝐸(𝑇(𝐺)) if and only if 𝑎 and 𝑏 are either adjacent or incidentin 𝐺 [8]. Let 𝑢, V, and𝑤 be the variables having values + or −.The transformation graph 𝐺𝑢V𝑤 is a graph whose vertex set isthe union of 𝑉(𝐺) and 𝐸(𝐺), and 𝑎𝑏 ∈ 𝐸(𝐺𝑢V𝑤) if and only if

(1) 𝑎, 𝑏 ∈ 𝑉(𝐺); then𝑢 = + or𝑢 = − if 𝑎 and 𝑏 are adjacentor nonadjacent in 𝐺, respectively;

(2) 𝑎, 𝑏 ∈ 𝐸(𝐺); then V = + or V = − if 𝑎 and 𝑏 are adjacentor nonadjacent in 𝐺, respectively;

(3) 𝑎 ∈ 𝑉(𝐺) and 𝑏 ∈ 𝐸(𝐺); then 𝑤 = + or 𝑤 = − if 𝑎 and𝑏 are incident or nonincident in 𝐺, respectively.There are eight different transformations of the given

graph 𝐺. For instance, 𝐺+++ is the total graph 𝑇(𝐺) of 𝐺 withnumber of vertices 𝑛 + 𝑒 and number of edges (1/2)𝑀1(𝐺) +2𝑒, and 𝐺−−− is the complement of total graph 𝐺+++. Forother transformations of graph,𝐺++−,𝐺+−+, and𝐺+−− are thecomplements of 𝐺−−+, 𝐺−+−, and 𝐺−++, respectively.

The concepts of semitotal-point graph and semitotal-linegraph are introduced by Sampathkumar and Chikkodimath[9]. The semitotal-point graph 𝑇1(𝐺) is a graph whose vertexset is the union of 𝑉(𝐺) and 𝐸(𝐺), and 𝑎𝑏 ∈ 𝐸(𝑇1(𝐺)) if andonly if (i) 𝑎 and 𝑏 are adjacent vertices in 𝐺 or (ii) one is avertex of 𝐺 and the other is an edge of 𝐺 incident to it. Thus,semitotal-point graph has 𝑛 + 𝑒 number of vertices and 3𝑒number of edges.

The semitotal-line graph𝑇2(𝐺) is a graphwhose vertex setis the union of𝑉(𝐺) and 𝐸(𝐺), and 𝑎𝑏 ∈ 𝐸(𝑇2(𝐺)) if and onlyif (i) 𝑎 and 𝑏 are adjacent edges in 𝐺 and (ii) one is a vertex of𝐺 and the other is an edge of𝐺 incident to it.Thus, semitotal-line graph has 𝑛 + 𝑒 number of vertices and (1/2)𝑀1(𝐺) + 𝑒number of edges.

Eventually, many properties of these transformationgraphs can be determined. For example, the Zagreb indices oftransformation graphs and total transformation graphs werecalculated by Basavanagoud and Patil [10] andHosamani andGutman [11], respectively. Wu and Meng [12] investigatedthe basic properties (connectedness, graph equations anditeration, and diameter) of total transformation. Xu and Wu[13] determined the connectivity, the Hamiltonian, and the

independence number of 𝐺−+−. Yi and Wu [14] determinedthe connectivity, the Hamiltonian, and the independencenumber of 𝐺++−.

In this paper, we obtain lower and upper bounds forthe general sum-connectivity indices of the above-definedtransformation graphs.

2. Main Results

In this section, we discuss the lower and upper bounds for thegeneral sum-connectivity indices of transformation graphsdefined in Section 1.

Theorem 1. For 𝛼 < 0, we have 𝛾1 ≤ 𝜒𝛼(𝑇1(𝐺)) ≤ 𝛾2, where𝛾1 = 2𝛼𝜒𝛼 (𝐺) + 2𝛼+1𝑒 (Δ (𝐺) + 1)𝛼 ,𝛾2 = 2𝛼𝜒𝛼 (𝐺) + 2𝛼+1𝑒 (𝛿 (𝐺) + 1)𝛼 ; (4)

the equalities hold if and only if 𝐺 is a regular graph.

Proof. Since𝑇1(𝐺) has 𝑛+𝑒 vertices and 3𝑒 edges, it holds that𝜒𝛼 (𝑇1 (𝐺)) = ∑

𝑎𝑏∈𝐸(𝑇1(𝐺))

(𝑑𝑇1(𝐺) (𝑎) + 𝑑𝑇1(𝐺) (𝑏))𝛼

= ∑𝑎𝑏∈𝐸(𝑇1(𝐺)),𝑎,𝑏∈𝑉(𝐺)

(𝑑𝑇1(𝐺) (𝑎) + 𝑑𝑇1(𝐺) (𝑏))𝛼

+ ∑𝑎𝑏∈𝐸(𝑇1(𝐺)),𝑎∈𝑉(𝐺),𝑏∈𝐸(𝐺)

(𝑑𝑇1(𝐺) (𝑎) + 𝑑𝑇1(𝐺) (𝑏))𝛼 .(5)

Note that if 𝑎 ∈ 𝑉(𝐺) then 𝑑𝑇1(𝐺)(𝑎) = 2𝑑𝐺(𝑎) and if 𝑎 ∈𝐸(𝐺) then 𝑑𝑇1(𝐺)(𝑎) = 2. It is clear that 𝛿(𝐺) ≤ 𝑑𝐺(𝑎) andΔ(𝐺) ≥ 𝑑𝐺(𝑎). And these equalities hold if and only if 𝐺 is aregular graph. Therefore,

𝜒𝛼 (𝑇1 (𝐺)) = 2𝛼 ∑𝑎𝑏∈𝐸(𝑇1(𝐺)),𝑎,𝑏∈𝑉(𝐺)

(𝑑𝐺 (𝑎) + 𝑑𝐺 (𝑏))𝛼

+ 2𝛼 ∑𝑎𝑏∈𝐸(𝑇1(𝐺)),𝑎∈𝑉(𝐺),𝑏∈𝐸(𝐺)

(𝑑𝐺 (𝑎) + 1)𝛼

≥ 2𝛼𝜒𝛼 (𝐺) + 2𝛼+1𝑒 (Δ (𝐺) + 1)𝛼 .

(6)

Similarly, we can compute

𝜒𝛼 (𝑇1 (𝐺)) ≤ 2𝛼𝜒𝛼 (𝐺) + 2𝛼+1𝑒 (𝛿 (𝐺) + 1)𝛼 . (7)

The two equalities in (6) and (7) obviously hold if and only if𝐺 and𝐻 are regular, respectively.

Example 2. By Theorem 1, the general sum-connectivityindices of some semitotal-point graphs are given below:

(1) 𝑛(8𝛼 + 2 × 6𝛼) − 3 × 8𝛼 ≤ 𝜒𝛼(𝑇1(𝑃𝑛)) ≤ 22𝛼𝑛(2𝛼 + 2) +2 × 6𝛼 − 3 × 8𝛼 − 22𝛼+1.

Discrete Dynamics in Nature and Society 3

(2) 𝜒𝛼(𝑇1(𝐶𝑛)) = 2𝛼𝑛(4𝛼 + 2 × 3𝛼).(3) 𝜒𝛼(𝑇1(𝐾𝑛)) = 2𝛼𝑛(𝑛 − 1)[2𝛼−1(𝑛 − 1)𝛼 + 𝑛𝛼].

Theorem 3. If 𝛼 < 0 then 𝛾1 ≤ 𝜒𝛼(𝑇2(𝐺)) ≤ 𝛾2, where

𝛾1 = 22𝛼−1𝑀1 (𝐺) Δ𝛼 (𝐺) + 𝑒Δ𝛼 (𝐺) [2 × 3𝛼 − 4𝛼] ,𝛾2 = 22𝛼−1𝑀1 (𝐺) 𝛿𝛼 (𝐺) + 𝑒𝛿𝛼 (𝐺) [2 × 3𝛼 − 4𝛼] ;

(8)


Proof. Since |𝑉(𝑇2(𝐺))| = 𝑛 + 𝑒 and |𝐸(𝑇2(𝐺))| = (1/2)𝑀1(𝐺) + 𝑒, we have

𝜒𝛼 (𝑇2 (𝐺)) = ∑𝑎𝑏∈𝐸(𝑇2(𝐺))

(𝑑𝑇2(𝐺) (𝑎) + 𝑑𝑇2(𝐺) (𝑏))𝛼

= ∑𝑎𝑏∈𝐸(𝑇2(𝐺)),𝑎,𝑏∈𝐸(𝐺)

(𝑑𝑇2(𝐺) (𝑎) + 𝑑𝑇2(𝐺) (𝑏))𝛼

+ ∑𝑎𝑏∈𝐸(𝑇2(𝐺)),𝑎∈𝑉(𝐺),𝑏∈𝐸(𝐺)

(𝑑𝑇2(𝐺) (𝑎) + 𝑑𝑇2(𝐺) (𝑏))𝛼 .

(9)

Note that if 𝑎 ∈ 𝐸(𝐺) then 𝑑𝑇2(𝐺)(𝑎) = 𝑑𝐺(𝑤𝑖) + 𝑑𝐺(𝑤𝑗) and if𝑎 ∈ 𝑉(𝐺) then 𝑑𝑇2(𝐺)(𝑎) = 𝑑𝐺(𝑎). Therefore, we have

𝜒𝛼 (𝑇2 (𝐺)) = ∑𝑤𝑖𝑤𝑗∈𝐸(𝐺),

𝑤𝑗𝑤𝑘∈𝐸(𝐺),

𝑤𝑖 =𝑤𝑘

[(𝑑𝐺 (𝑤𝑖) + 𝑑𝐺 (𝑤𝑗))

+ (𝑑𝐺 (𝑤𝑗) + 𝑑𝐺 (𝑤𝑘))]𝛼 + ∑𝑎𝑏∈𝐸(𝑇2(𝐺)),𝑎∈𝑉(𝐺),𝑏=𝑎𝑥∈𝐸(𝐺),𝑥∈𝑉(𝐺)

[𝑑𝐺 (𝑎)

+ (𝑑𝐺 (𝑎) + 𝑑𝐺 (𝑥))]𝛼 = ∑𝑤𝑖𝑤𝑗∈𝐸(𝐺),


𝑤𝑖 =𝑤𝑘

[𝑑𝐺 (𝑤𝑖)

+ 2𝑑𝐺 (𝑤𝑗) + 𝑑𝐺 (𝑤𝑘)]𝛼 + ∑𝑎∈𝑉(𝐺),𝑏=𝑎𝑥∈𝐸(𝐺),𝑥∈𝑉(𝐺)

[𝑑𝐺 (𝑎)

+ (𝑑𝐺 (𝑎) + 𝑑𝐺 (𝑥))]𝛼 .

(10)

Since 𝑑𝐺(𝑎) ≥ 𝛿(𝐺) and 𝑑𝐺(𝑎) ≤ Δ(𝐺), each equality holds ifand only if 𝐺 is a regular graph.

After simplification we get

𝜒𝛼 (𝑇2 (𝐺)) ≥ [4Δ (𝐺)]𝛼 ∑𝑤𝑖𝑤𝑗∈𝐸(𝐺),


𝑤𝑖 =𝑤𝑘

1

+ [3Δ (𝐺)]𝛼 ∑𝑎𝑏∈𝐸(𝑇2(𝐺)),𝑎∈𝑉(𝐺),𝑏=𝑎𝑥∈𝐸(𝐺),𝑥∈𝑉(𝐺)

1

= [4Δ (𝐺)]𝛼 ⋅ [𝐸 (𝑇2 (𝐺)) − 2𝑒]+ [3Δ (𝐺)]𝛼 ⋅ (2𝑒)

= 22𝛼−1𝑀1 (𝐺) Δ𝛼 (𝐺)+ 𝑒Δ𝛼 (𝐺) [2 × 3𝛼 − 4𝛼] .

(11)

Similarly, we can calculate

𝜒𝛼 (𝑇2 (𝐺)) ≤ 22𝛼−1𝑀1 (𝐺) 𝛿𝛼 (𝐺)+ 𝑒𝛿𝛼 (𝐺) [2 × 3𝛼 − 4𝛼] . (12)

Obviously the equalities in (11) and (12) hold if and onlyif 𝐺 is a regular graph.

Example 4. By Theorem 3, the general sum-connectivityindices of some semitotal-line graphs are given below:

(1) 2𝛼𝑛(4𝛼+2×3𝛼)−2𝛼+1(4𝛼−3𝛼) ≤ 𝜒𝛼(𝑇2(𝑃𝑛)) ≤ 𝑛(4𝛼+2 × 3𝛼) − 2(4𝛼 + 2 × 3𝛼).(2) 𝜒𝛼(𝑇2(𝐶𝑛)) = 2𝛼𝑛(4𝛼 + 2 × 3𝛼).(3) 𝜒𝛼(𝑇2(𝐾𝑛)) = 𝑛𝛼(𝑛 − 1)𝛼+1[22𝛼−1𝑛 + 22𝛼 + 3𝛼].

Theorem 5. Let 𝛼 < 0. Then 𝛾1 ≤ 𝜒𝛼(𝑇(𝐺)) ≤ 𝛾2, where𝛾1 = 2𝛼𝜒𝛼 (𝐺) + 22𝛼−1𝑀1 (𝐺) Δ𝛼 (𝐺) + 4𝛼𝑒Δ𝛼 (𝐺) ,𝛾2 = 2𝛼𝜒𝛼 (𝐺) + 22𝛼−1𝑀1 (𝐺) 𝛿𝛼 (𝐺) + 4𝛼𝑒𝛿𝛼 (𝐺) ; (13)


Proof. Since |𝑉(𝑇(𝐺))| = 𝑛+𝑒 and |𝐸(𝑇(𝐺))| = (1/2)𝑀1(𝐺)+2𝑒, we have𝜒𝛼 (𝑇 (𝐺)) = ∑

𝑎𝑏∈𝐸(𝑇(𝐺))

(𝑑𝑇(𝐺) (𝑎) + 𝑑𝑇(𝐺) (𝑏))𝛼

= ∑𝑎𝑏∈𝐸(𝑇(𝐺)),𝑎,𝑏∈𝑉(𝐺)

(𝑑𝑇(𝐺) (𝑎) + 𝑑𝑇(𝐺) (𝑏))𝛼

+ ∑𝑎𝑏∈𝐸(𝑇(𝐺)),𝑎,𝑏∈𝐸(𝐺)

(𝑑𝑇(𝐺) (𝑎) + 𝑑𝑇(𝐺) (𝑏))𝛼

+ ∑𝑎𝑏∈𝐸(𝑇(𝐺)),𝑎∈𝑉(𝐺),𝑏∈𝐸(𝐺)

(𝑑𝑇(𝐺) (𝑎) + 𝑑𝑇(𝐺) (𝑏))𝛼 .

(14)


Note that 𝑑𝑇(𝐺)(𝑎) = 2𝑑𝐺(𝑎) for 𝑎 ∈ 𝑉(𝐺) and 𝑑𝑇(𝐺)(𝑎) =𝑑𝐺(𝑤𝑖) + 𝑑𝐺(𝑤𝑗) for 𝑎 ∈ 𝐸(𝐺). So𝜒𝛼 (𝑇 (𝐺)) = 2𝛼 ∑

𝑎𝑏∈𝐸(𝐺),𝑎,𝑏∈𝑉(𝐺)

(𝑑𝐺 (𝑎) + 𝑑𝐺 (𝑏))𝛼

+ ∑𝑎=𝑤𝑖𝑤𝑗∈𝐸(𝐺),

𝑏=𝑤𝑗𝑤𝑘∈𝐸(𝐺),

𝑤𝑖 =𝑤𝑘

[(𝑑𝐺 (𝑤𝑖) + 𝑑𝐺 (𝑤𝑗))

+ (𝑑𝐺 (𝑤𝑗) + 𝑑𝐺 (𝑤𝑘))]𝛼 + ∑𝑏=𝑎𝑥∈𝐸(𝐺),𝑎∈𝑉(𝐺),𝑥∈𝑉(𝐺)

[2𝑑𝐺 (𝑎)

+ (𝑑𝐺 (𝑎) + 𝑑𝐺 (𝑥))]𝛼 = 2𝛼 ⋅ 𝜒𝛼 (𝐺)+ ∑𝑎=𝑤𝑖𝑤𝑗∈𝐸(𝐺),


𝑤𝑖 =𝑤𝑘

[𝑑𝐺 (𝑤𝑖) + 2𝑑𝐺 (𝑤𝑗) + 𝑑𝐺 (𝑤𝑘)]𝛼

+ ∑𝑏=𝑎𝑥∈𝐸(𝐺),𝑎∈𝑉(𝐺),𝑥∈𝑉(𝐺)

[2𝑑𝐺 (𝑎) + (𝑑𝐺 (𝑎) + 𝑑𝐺 (𝑥))]𝛼 .

(15)

Note that 𝑑𝐺(𝑎) ≤ Δ(𝐺) and 𝑑𝐺(𝑎) ≥ 𝛿(𝐺). The equalitieshold if and only if 𝐺 is a regular graph.

After simplification, we get

𝜒𝛼 (𝑇 (𝐺)) ≥ 2𝛼𝜒𝛼 (𝐺) + [4Δ (𝐺)]𝛼 [12𝑀1 (𝐺) − 𝑒]+ [4Δ (𝐺)]𝛼 [2𝑒]

= 2𝛼𝜒𝛼 (𝐺) + 22𝛼−1Δ𝛼 (𝐺)𝑀1 (𝐺)+ 22𝛼𝑒Δ𝛼 (𝐺) .

(16)


𝜒𝛼 (𝑇 (𝐺)) ≤ 2𝛼𝜒𝛼 (𝐺) + 22𝛼−1𝛿𝛼 (𝐺)𝑀1 (𝐺)+ 22𝛼𝑒𝛿𝛼 (𝐺) . (17)

Since 𝜒𝛼(𝐺) ≥ 2𝛼𝑒Δ𝛼(𝐺), we can also write the resultsabove as

4𝛼𝛿𝛼 (𝐺) [12𝑀1 (𝐺) + 2𝑒] ≤ 𝜒𝛼 (𝑇 (𝐺))≤ 4𝛼Δ𝛼 (𝐺) [12𝑀1 (𝐺) + 2𝑒] .

(18)

Thus, if 𝐺 is a regular graph, then we obtain the equalityin (16), (17), and (18).

Example 6. By Theorem 5, the general sum-connectivityindices of some total graphs are given below:

(1) 23𝛼+2𝑛 + 2 × 6𝛼 − 7 × 8𝛼 ≤ 𝜒𝛼(𝑇(𝑃𝑛)) ≤ 4𝛼𝑛(2𝛼 + 1) +2𝛼(2 × 3𝛼 − 3 × 4𝛼 + −2𝛼+2).

(2) 𝜒𝛼(𝑇(𝐶𝑛)) = 4 × 8𝛼𝑛.(3) 𝜒𝛼(𝑇(𝐾𝑛)) = 22𝛼−1𝑛(𝑛 − 1)𝛼+1(𝑛 + 1).

Theorem 7. Let 𝛼 < 0. Then 𝛾1 ≤ 𝜒𝛼(𝐺−−−) ≤ 𝛾2, where

𝛾1 = 2𝛼 [12𝑀1 (𝐺) + 2𝑒] (𝑒 + 𝑛 − 1 − 2Δ𝛼 (𝐺))𝛼 ,𝛾2 = 2𝛼 [12𝑀1 (𝐺) + 2𝑒] (𝑒 + 𝑛 − 1 − 2𝛿𝛼 (𝐺))𝛼 ;

(19)


Proof. For a given graph 𝐺, since 𝐺−−− ≅ 𝐺+++ and 𝐺+++ =𝑇(𝐺), then |𝑉(𝐺−−−)| = 𝑛 + 𝑒, |𝐸(𝐺−−−)| = ( 𝑒+𝑛2 ) −(1/2)𝑀1(𝐺) − 2𝑒, and 2Δ(𝐺) = 𝑒 + 𝑛 − 1 − 2Δ(𝐺). Using thesevalues, we can compute the required results.

Theorem 8. Let 𝛼 < 0. Then 𝛾1 ≤ 𝜒𝛼(𝐺++−) ≤ 𝛾2, where

𝛾1 = 2𝛼𝑒𝛼+1 + 2𝛼 [12𝑀1 (𝐺) − 𝑒] [𝑛 − 4 + 2Δ (𝐺)]𝛼

+ [(𝑒 + 𝑛2 ) − (𝑛

2) − (𝑒2) − 2𝑒]

⋅ [𝑒 + 𝑛 − 4 + 2Δ (𝐺)]𝛼 ,𝛾2 = 2𝛼𝑒𝛼+1 + 2𝛼 [12𝑀1 (𝐺) − 𝑒] [𝑛 − 4 + 2𝛿 (𝐺)]𝛼

+ [(𝑒 + 𝑛2 ) − (𝑛

2) − (𝑒2) − 2𝑒]

⋅ [𝑒 + 𝑛 − 4 + 2𝛿 (𝐺)]𝛼 ;

(20)


Proof. Since |𝑉(𝐺++−)| = 𝑛 + 𝑒 and |𝐸(𝐺++−)| = ( 𝑒+𝑛2 ) − ( 𝑛2 ) −( 𝑒2 ) + (1/2)𝑀1(𝐺) − 2𝑚,

𝜒𝛼 (𝐺++−) = ∑𝑎𝑏∈𝐸(𝐺++−)

(𝑑𝐺++− (𝑎) + 𝑑𝐺++− (𝑏))𝛼

= ∑𝑎𝑏∈𝐸(𝐺++−),𝑎,𝑏∈𝑉(𝐺)

(𝑑𝐺++− (𝑎) + 𝑑𝐺++− (𝑏))𝛼

+ ∑𝑎𝑏∈𝐸(𝐺++−),𝑎,𝑏∈𝐸(𝐺)

(𝑑𝐺++− (𝑎) + 𝑑𝐺++− (𝑏))𝛼

+ ∑𝑎𝑏∈𝐸(𝐺++−),𝑎∈𝑉(𝐺),𝑏∈𝐸(𝐺)

(𝑑𝐺++− (𝑎) + 𝑑𝐺++− (𝑏))𝛼 .

(21)


Note that if 𝑎 ∈ 𝑉(𝐺) then 𝑑𝐺++−(𝑎) = 𝑒 and if 𝑎 ∈ 𝐸(𝐺) then𝑑𝐺++−(𝑎) = 𝑑𝐺(𝑤𝑖) + 𝑑𝐺(𝑤𝑗) + 𝑛 − 4𝜒𝛼 (𝐺++−) = ∑

𝑎𝑏∈𝐸(𝐺),𝑎,𝑏∈𝑉(𝐺)

[2𝑒]𝛼

+ ∑𝑎=𝑤𝑖𝑤𝑗∈𝐸(𝐺),


𝑤𝑖 =𝑤𝑘

[(𝑑𝐺 (𝑤𝑖) + 𝑑𝐺 (𝑤𝑗) + 𝑛 − 4)

+ (𝑑𝐺 (𝑤𝑗) + 𝑑𝐺 (𝑤𝑘) + 𝑛 − 4)]𝛼 + ∑𝑏=𝑥𝑦∈𝐸(𝐺),𝑎∈𝑉(G),𝑎∉{𝑥,𝑦}

[𝑒

+ (𝑑𝐺 (𝑥) + 𝑑𝐺 (𝑦) + 𝑛 − 4)]𝛼 = 2𝛼𝑒𝛼+1

+ ∑𝑤𝑖𝑤𝑗∈𝐸(𝐺),

𝑤𝑗𝑤𝑘∈𝐸(𝐺)

[2𝑛 − 8 + 𝑑𝐺 (𝑤𝑖) + 2𝑑𝐺 (𝑤𝑗)

+ 𝑑𝐺 (𝑤𝑘)]𝛼 + ∑𝑏=𝑥𝑦∈𝐸(𝐺),𝑎∈𝑉(𝐺),𝑎∉{𝑥,𝑦}

[𝑒 + 𝑛 − 4 + 𝑑𝐺 (𝑥)

+ 𝑑𝐺 (𝑦)]𝛼 .

(22)

Note that 𝑑𝐺(𝑎) ≤ Δ(𝐺) and 𝑑𝐺(𝑎) ≥ 𝛿(𝐺). The equalitieshold if and only if 𝐺 is a regular graph. After simplification,we get

𝜒𝛼 (𝐺++−) ≥ 2𝛼𝑒𝛼+1 + 2𝛼 [12𝑀1 (𝐺) − 𝑒]⋅ [𝑛 − 4 + 2Δ (𝐺)]𝛼

+ [(𝑒 + 𝑛2 ) − (𝑛

2) − (𝑒2) − 2𝑒]

⋅ (𝑒 + 𝑛 − 4 + 2Δ (𝐺))𝛼 .

(23)


𝜒𝛼 (𝐺++−) ≤ 2𝛼𝑒𝛼+1 + 2𝛼 [12𝑀1 (𝐺) − 𝑒]⋅ (𝑛 − 4 + 2𝛿 (𝐺))𝛼

+ [(𝑒 + 𝑛2 ) − (𝑛

2) − (𝑒2) − 2𝑒]

⋅ (𝑒 + 𝑛 − 4 + 2𝛿 (𝐺))𝛼 .

(24)

The equalities in (23) and (24) obviously hold if and onlyif 𝐺 is a regular graphs.

Theorem 9. Let 𝛼 < 0. Then 𝛾1 ≤ 𝜒𝛼(𝐺−+−) ≤ 𝛾2, where

𝛾1 = 2𝛼 [𝑒 + 𝑛 − 1 − 2Δ (𝐺)]𝛼 [(𝑛2) − 𝑒]

+ 2𝛼 [𝑛 − 4 + 2Δ (𝐺)]𝛼 [12𝑀1 (𝐺) − 𝑒]

+ (𝑒 + 2𝑛 − 5)𝛼 [(𝑒 + 𝑛2 ) − (𝑛

2) − (𝑒2) − 2𝑒] ,

𝛾2 = 2𝛼 [𝑒 + 𝑛 − 1 − 2𝛿 (𝐺)]𝛼 [(𝑛2) − 𝑒]

+ 2𝛼 [𝑛 − 4 + 2𝛿 (𝐺)]𝛼 [12𝑀1 (𝐺) − 𝑒]

+ (𝑒 + 2𝑛 − 5)𝛼 [(𝑒 + 𝑛2 ) − (𝑛

2) − (𝑒2) − 2𝑒] ;

(25)


Proof. Since |𝑉(𝐺−+−)| = 𝑛 + 𝑒 and |𝐸(𝐺−+−)| = ( 𝑒+𝑛2 ) − ( 𝑒2 ) +(1/2)𝑀1(𝐺) − 4𝑒,𝜒𝛼 (𝐺−+−) = ∑

𝑎𝑏∈𝐸(𝐺−+−)

(𝑑𝐺−+− (𝑎) + 𝑑𝐺−+− (𝑏))𝛼

= ∑𝑎𝑏∈𝐸(𝐺−+−),𝑎,𝑏∈𝑉(𝐺)

(𝑑𝐺−+− (𝑎) + 𝑑𝐺−+− (𝑏))𝛼

+ ∑𝑎𝑏∈𝐸(𝐺−+−),𝑎,𝑏∈𝐸(𝐺)

(𝑑𝐺−+− (𝑎) + 𝑑𝐺−+− (𝑏))𝛼

+ ∑𝑎𝑏∈𝐸(𝐺−+−),𝑎∈𝑉(𝐺),𝑏∈𝐸(𝐺)

(𝑑𝐺−+− (𝑎) + 𝑑𝐺−+− (𝑏))𝛼 .

(26)

Note that 𝑑𝐺−+−(𝑎) = 𝑒 + 𝑛 − 1 − 2𝑑𝐺(𝑎) for 𝑎 ∈ 𝑉(𝐺) and𝑑𝐺−+−(𝑎) = 𝑑𝐺(𝑤𝑖) + 𝑑𝐺(𝑤𝑗) + 𝑛 − 4 for 𝑎 ∈ 𝐸(𝐺). Then

𝜒𝛼 (𝐺−+−) = ∑𝑎𝑏∈𝐸(𝐺−+−),𝑎,𝑏∈𝑉(𝐺)

[(𝑒 + 𝑛 − 1 − 2𝑑𝐺 (𝑎))

+ (𝑒 + 𝑛 − 1 − 2𝑑𝐺 (𝑏))]𝛼

+ ∑𝑤𝑖𝑤𝑗∈𝐸(𝐺),


𝑤𝑖 =𝑤𝑘

[(𝑑𝐺 (𝑤𝑖) + 𝑑𝐺 (𝑤𝑗) + 𝑛 − 4)

+ (𝑑𝐺 (𝑤𝑗) + 𝑑𝐺 (𝑤𝑘) + 𝑛 − 4)]𝛼


+ ∑𝑎𝑏∈𝐸(𝐺−+−),𝑎∈𝑉(𝐺),𝑏=𝑥𝑦∈𝐸(𝐺)

[(𝑒 + 𝑛 − 1 − 2𝑑𝐺 (𝑎))

+ (𝑑𝐺 (𝑥) + 𝑑𝐺 (𝑦) + 𝑛 − 4)]𝛼 = 2𝛼 ∑𝑎𝑏∉𝐸(𝐺),𝑎,𝑏∈𝑉(𝐺)

[𝑒 + 𝑛

− 1 − 𝑑𝐺 (𝑎) − 𝑑𝐺 (𝑏)]𝛼 + ∑𝑤𝑖𝑤𝑗∈𝐸(𝐺),


𝑤𝑖 =𝑤𝑘

[2𝑛 − 8

+ 𝑑𝐺 (𝑤𝑖) + 2𝑑𝐺 (𝑤𝑗) + 𝑑𝐺 (𝑤𝑘)]𝛼 + ∑𝑏=𝑥𝑦∈𝐸(𝐺),𝑎∉{𝑥,𝑦},𝑎∈𝑉(𝐺)

[𝑒

+ 2𝑛 − 5 − 2𝑑𝐺 (𝑎) + 𝑑𝐺 (𝑥) + 𝑑𝐺 (𝑦)]𝛼 .(27)

Note that 𝑑𝐺(𝑎) ≤ Δ(𝐺) and 𝑑𝐺(𝑎) ≥ 𝛿(𝐺). The equalitieshold if and only if 𝐺 is a regular graph.

After simplification, we get

𝜒𝛼 (𝐺−+−)≥ 2𝛼 [𝑒 + 𝑛 − 1 − 2Δ (𝐺)]𝛼 [(𝑛

2) − 𝑒]

+ 2𝛼 [𝑛 − 4 + 2Δ (𝐺)]𝛼 [12𝑀1 (𝐺) − 𝑒]

+ (𝑒 + 2𝑛 − 5)𝛼 [(𝑒 + 𝑛2 ) − (𝑛

2) − (𝑒2) − 2𝑒] .

(28)

Similarly, we calculate

𝜒𝛼 (𝐺−+−) ≤ 2𝛼 [𝑒 + 𝑛 − 1 − 2𝛿 (𝐺)]𝛼

⋅ [𝑛 − 4 + 2𝛿 (𝐺)]𝛼 [(𝑛2) − 𝑒]

+ 2𝛼 [12𝑀1 (𝐺) − 𝑒]

+ [(𝑒 + 𝑛2 ) − (𝑛

2) − (𝑒 + 2𝑛 − 5)𝛼 (𝑒2) − 2𝑒] .

(29)

If 𝐺 is a regular graph then we obtain the equalities in (28)and (29).

In fully analogousmanner, we also arrive at the following.

Theorem 10. If 𝛼 < 0 then(1) 𝛾1 ≤ 𝜒𝛼(𝐺+−−) ≤ 𝛾2, where

𝛾1 = 2𝛼𝑒𝛼+1 + 2𝛼 [(𝑒2) − 1

2𝑀1 (𝐺) + 𝑒]⋅ [𝑛 + 𝑒 − 1 − 2Δ (𝐺)]𝛼

+ [(𝑒 + 𝑛2 ) − (𝑛

2) − (𝑒2) − 2𝑒]

⋅ [2𝑒 + 𝑛 − 1 − 2Δ (𝐺)]𝛼 ,𝛾2 = 2𝛼𝑒𝛼+1 + 2𝛼 [(𝑒

2) − 12𝑀1 (𝐺) + 𝑒]

⋅ [𝑛 + 𝑒 − 1 − 2𝛿 (𝐺)]𝛼

+ [(𝑒 + 𝑛2 ) − (𝑛

2) − (𝑒2) − 2𝑒]

⋅ [2𝑒 + 𝑛 − 1 − 2𝛿 (𝐺)]𝛼 ;(30)

(2) 𝛾1 ≤ 𝜒𝛼(𝐺−−+) ≤ 𝛾2, where𝛾1 = 2𝛼 [(𝑛

2) − 𝑒] (𝑛 − 1)𝛼

+ 2𝛼 [(𝑒2) − 1

2𝑀1 (𝐺) + 𝑒] [𝑒 − 3 − 2Δ (𝐺)]𝛼

+ 2𝑒 [𝑒 + 𝑛 + 2 − 2Δ (𝐺)]𝛼 ,𝛾2 = 2𝛼 [(𝑛

2) − 𝑒] (𝑛 − 1)𝛼

+ 2𝛼 [(𝑒2) − 1

2𝑀1 (𝐺) + 𝑒] [𝑒 − 3 − 2𝛿 (𝐺)]𝛼

+ 2𝑒 [𝑒 + 𝑛 + 2 − 2𝛿 (𝐺)]𝛼 ;

(31)

(3) 𝛾1 ≤ 𝜒𝛼(𝐺−++) ≤ 𝛾2, where𝛾1 = 2𝛼 [(𝑛

2) − 𝑒] (𝑛 − 1)𝛼

+ 4𝛼Δ𝛼 (𝐺) [12𝑀1 (𝐺) − 𝑒]+ 2𝑒 [𝑛 − 1 + 2Δ (𝐺)]𝛼 ,

𝛾2 = 2𝛼 [(𝑛2) − 𝑒] (𝑛 − 1)𝛼

+ 4𝛼𝛿𝛼 (𝐺) [12𝑀1 (𝐺) − 𝑒]+ 2𝑒 [𝑛 − 1 + 2𝛿 (𝐺)]𝛼 ;

(32)

(4) 𝛾1 ≤ 𝜒𝛼(𝐺+−+) ≤ 𝛾2, where𝛾1 = 22𝛼𝑒Δ𝛼 (𝐺)

+ [(𝑒2) − 1

2𝑀1 (𝐺) + 𝑒] [3 − 2Δ (𝐺)]𝛼

+ 2𝑒 (𝑒 + 3)𝛼 ,


𝛾2 = 22𝛼𝑒𝛿𝛼 (𝐺)+ [(𝑒

2) − 12𝑀1 (𝐺) + 𝑒] [3 − 2𝛿 (𝐺)]𝛼

+ 2𝑒 (𝑒 + 3)𝛼 .(33)

In all the above cases, the equalities hold if and only if 𝐺 is aregular graph, respectively.

3. Conclusion

In this paper, we obtain the sharp lower and upper boundsfor general sum-connectivity indices of the semitotal-pointgraph, the semitotal-line graph, the total graph, and the eightdistinct transformation graphs𝐺𝑢V𝑤, where 𝑢, V, 𝑤 ∈ {+, −} interms of the order, minimum degree, and maximum degreeof a graph. Moreover, the extremal graphs achieving thesebounds have been described.

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper.

Acknowledgments

This research is supported by the Fundamental ResearchFunds for the Central Universities (nos. 2652015193 and2652017146) and NSFC of China (no. 11701530).

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