# Zagreb Connection Indices of Subdivision and Semi-Total Point Operations on Graphs

Authors: Jiang-Hua Tang; Usman Ali; Muhammad Javaid; Khurram Shabbir
Journal: Journal of Chemistry
 (2019)

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## Abstract

Representation or coding of the molecular graphs with the help of numerical numbers plays a vital role in the studies of physicochemical and structural properties of the chemical compounds that are involved in the molecular graphs. For the first time, the modified first Zagreb connection index appeared in the paper by Gutman and Trinajstic (1972) to compute total electron energy of the alternant hydrocarbons, but after that, for a long time, it has not been studied. Recently, Ali and Trinajstic (2018) restudied the first Zagreb connection indexZC1, the second Zagreb connection index ZC2, and the modified first Zagreb connection index ZC1∗ to find entropy and acentric factor of the octane isomers. They also reported that the values provided by the International Academy of Mathematical Chemistry show better chemical capability of the Zagreb connection indices than the ordinary Zagreb indices. Assume that S1 and S2 denote the operations of subdivision and semitotal point, respectively. Then, the S-sum graphs Q1+QS2 are obtained by the cartesian product of SQ1 and Q2, where S∈S1,S2, Q1 and Q2 are any connected graphs, and SQ1 is a graph obtained after applying the operation S on Q1. In this paper, we compute the Zagreb connection indices (ZC1, ZC2, and ZC1∗) of the S-sum graphs in terms of various topological indices of their factor graphs. At the end, as an application of the computed results, the Zagreb connection indices of the S-sum graphs obtained by the particular classes of alkanes are also included.


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## Body

## 1. Introduction

The importance of graph theory [1] is increased day by day in the various disciplines of science, especially in the field of chemistry and information science [2–5]. In view of Todeschini et al. [6], a topological invariant (TI) is the final outcome of a logical, systematic, and mathematical process that transforms features encoded in a molecular graph to a fixed real number. These TIs predict the chemical and physical features of the molecular graphs just on solubility, weight, connectivity, polarizability, surface tension, critical temperature, freezing, melting and boiling point, heat of evaporation, and heat of formation. A number of drugs, micromaterials, and crystalline materials which are strongly applied in several industrial networks and pharmaceutical fields are studied and evaluated on the basis or frame provided by the TIs (see [7–10]).Moreover, TIs are presented into useful identity in the study of the quantitative structure activity relationships (QSARs) and quantitative structure property relationships (QSPRs), which connect the molecular graphs to their biological behaviour [11–16]. Mathematically, we represent this link as T=χU, where χ is the real-valued function, U is the molecular structure, and T is a real value that depends upon U. For more details, we refer to [17, 18, 19, 20]. The first TI was defined by Wiener in 1947 working on the boiling point of paraffin [21]. In 1972, Gutman and Trinajstić [22] defined degree-based TIs named as first Zagreb index and second Zagreb index. In the same paper, another TI is also defined, which got less tension of the researchers. Furtula and Gutman [23] reinvestigated this index with its various basic properties and called it by the name of forgotten index. Nowadays, many extended studies have been delivered on these invariants. For more study, we refer to [6, 24–27]. In brief, to date, many TIs with various properties have been defined (see [6, 27]). These TIs have been classified into various classes, but degree-based TIs are more popular than others (see [7–9, 28, 29]).Also another TI is defined in the same paper [22] of Gutman and Trinajstić, but there is no more attention or study on this predictor for a long time. In 2018, Ali and Trinajstic [30] reinvestigated this topological index and called it modified first Zagreb connection index ZC1∗. They also reported (as per data provided by the International Academy of Mathematical Chemistry) that its chemical capability has more precise correlation coefficient values for the following thirteen physicochemical properties of octane isomers: density, heat capacity at temperature, standard enthalpy of vaporization, boiling point, heat capacity at pressure, enthalpy of vaporization, entropy, acentric factor, molar volume, total surface area, enthalpy of formation, standard enthalpy of formation, and octanol water partition coefficient. Du et al. [31] determined extremal alkanes by using modified first Zagreb connection index. In the same year, Ducoffe et al. [32] also characterized extremal graphs with respect to the modified first Zagreb connection index for trees and unicyclic graphs with or without a fixed girth. Shao et al. [33] also used the same molecular descriptor to find the extremal graphs in the class of alkanes and cycloalkanes under certain conditions.In the computational graph theory, the operations on graphs played an important role in the studies of graph invariants in the complex structures into their factors. For the purpose, Yan et al. [34] defined the operations of subdivision and semitotal point on graphs and obtained the Wiener index of the resultant graphs. Eliasi and Taeri [35] used these operations to construct new families of S-sum graphs. Deng and Akhter [36, 37] computed the first Zagreb index M1, second Zagreb index M2, and forgotten index M3 of the S-sum graphs. Liu et al. [38] derived the exact formulas for the first general Zagreb index Mα on the S-sum graphs, where α is a real number. For more details about TIs and their applications, we refer to [39–48] and the references therein.In this paper, we extend this study and compute Zagreb connection indices such as first Zagreb connection indexZC1, second Zagreb connection index ZC2, and modified first Zagreb connection index ZC1∗ of the S-sum graphs Q1+QS2, which are obtained by the cartesian product of SQ1 and Q2, where S∈S1,S2, Q1 and Q2 are any connected graphs, and SQ1 is a graph obtained after applying the operations S on Q1. The rest of the paper is organized as follows: Section 2 covers the preliminary definitions of the Zagreb indices and Zagreb connection indices. Section 3 presents the main results, and Section 4 includes the conclusion.


## 2. Preliminaries

LetQ=VQ,EQ be a simple and connected graph with vertex set VQ and edge set EQ⊆VQ×VQ such that their cardinalities are called order and size of Q, respectively. Todesehini et al. [6] defined the generalized form of the degrees of vertices of the graph Q as fkQb=NkQb, where NkQb=a∈VQ:da,b=k. For k = 1 and k = 2, f1Qb=dQb and f2Qb=τQb are called the degree and connection number of the vertex b in the graph Q. In chemical modeling of molecular descriptors [49], chemical terms atom and bond are equal to graphical terms vertex and edge, respectively. For further basic terminologies of graphs, we refer to [50].Definition 1.
For a graphQ, the first Zagreb index M1Q, second Zagreb index M2Q, and forgotten index FQ are defined as(1)M1Q=∑b∈VQdQb2=∑ab∈EQdQa+dQb,M2Q=∑ab∈EQdQa×dQb,FQ=∑b∈VQdQb3=∑ab∈VQdQ2a+dQ2b.
These degree-based indices are defined by Gutman, Trinajstic, Ruscic, and Furtula (see [22, 23, 51]). These are frequently used to predict better outcomes in molecular structures such as entropy, heat capacity, ZE isomerism, acentric factor, and absolute value of correlation coefficient [52]. These indices also played an important role in the study of QSPR and QSAR (see [18, 19, 49]).
Corresponding to these degree-based TIs, the connection-based TIs are defined in Definition2. For further studies of connection-based TIs, see [30, 53].Definition 2.
For a graphQ, the first Zagreb connection index ZC1Q, second Zagreb connection index ZC2Q, and modified first Zagreb connection index ZC1∗Q are defined as(2)ZC1Q=∑b∈VQτQb2,ZC2Q=∑ab∈EQτQa×τQb,ZC1∗Q=∑b∈VQdQbτQb=∑ab∈EQτQa+τQb.
Now, we describe the subdivision and semitotal point operations on graphs as follows: (a)S1Q is a graph that is obtained by including a new vertex between each edge of the graph Q; (b) S2Q is a graph that is obtained from S1Q by adding the edges between old vertices which are adjacent in Q. More details of these operations can be found in [34, 35, 54, 55]. For further understanding, see Figure 1. The concept of S-sum graphs is defined as in Definition 3.Figure 1
(a)Q, (b) S1Q, and (c) S2Q.Definition 3.
Fori∈1,2 and S∈Si, assume that Qi are connected graphs and the graph SQ1 with vertex set VSQ1 and edge set ESQ1 is obtained after applying the operation S on Q1. Then, the graphs Q1+Qs2 with vertex set VQ1+QS2=VSQ1×V2=V1∪E1×V2 and edge set EQ1+QS2 are called S-sum graphs if for a1,b1a2,b2∈EQ1+Qs2 either a1=a2 in VQ1 and b1b2∈EQ2 or b1=b2 in VQ2 and a1a2∈ESQ1, where a1,b1, a2,b2∈VQ1+SQ2. For more clearance, see Figures 2 and 3.
Now, we present some important results which are used in the main results.Figure 2
(a)Q1≅C5, (b) Q2≅P2, (c) S1Q1, and (d) Q1+Qs2 iff S=S1.Figure 3
(a)S2Q1 and (b) Q1+Qs2 iff S=S2.Lemma 1 (see [56]).
LetQ be a connected graph with n vertices and m edges. Then, τQa+dQa≤∑b∈NQadQb, where equality holds if and only if Q is a C3,C4−free graph.Lemma 2.
LetQ be a connected and C3,C4−free graph with n vertices and m edges. Then, ∑a∈VQτQa=M1Q−2m.Proof.
Proof is obvious using Hand-Shaking Lemma and Lemma1.


## 3. Main Results

This section contains the main results of the first Zagreb connection indexZC1, second Zagreb connection index ZC2, and modified first Zagreb connection index ZC1∗ on the S-sum graphs under the operations of subdivision and semitotal point.Theorem 1.
LetQ1 and Q2 be two connected and C3,C4−free graphs. Then, the first Zagreb connection indices of their S-sum graphs under the operations of subdivision and semitotal point are(3)ZC1Q1+QS12=4e1ZC1∗Q2+n1ZC1G2+n2FQ1+2n2M2Q1+3M1Q14e2−n2+M1Q28e1+M1Q1+4e1n2−6e2,(4)ZC1Q1+QS22=16e2ZC1∗Q1+8e1ZC1∗Q2+4n2ZC1Q1+n1ZC1Q2+n2FQ1+2n2M2Q1−M1Q1+4M1Q22M1Q1−e1+e1n2+8e2.Proof.
(a) Letτa,b=τQ1+QS12a,b be a connection number of the vertex a,b in the graph Q1+QS12. Then,(5)ZC1Q1+QS12=∑a,b∈VQ1+QS12τa,b2=∑a∈VQ1∑b∈VQ2τa,b2+∑b∈VQ2∑a1a2∈EQ1τa,b2.
Consider(6)∑a∈VQ1∑b∈VQ2τa,b2=∑a∈VQ1∑b∈VQ2dQ1a+dQ1adQ2b+τQ2b2=∑a∈VQ1∑b∈VQ2dQ12a+dQ12adQ22b+τQ22b+2dQ12adQ2b+2dQ1adQ2bτQ2b+2dQ1aτQ2b.
By Lemma2 ∑b∈VQ2τQ2b=M1Q2−2e2, we have(7)=n2M1Q1+M1Q1M1Q2+n1ZC1Q2+4e2M1Q1+4e1ZC1∗Q2+4e1M1Q2−2e2.
Sincera,b=dQ1a1+2dQ2b−1+dQ1a22, we have(8)∑b∈VQ2∑a1a2∈EQ1τa,b2=∑b∈VQ2∑a1a2∈EQ1dQ12a1+4dQ2b−12+dQ12a2+4dQ1a1dQ2b−1+4dQ1a2dQ2b−1+2dQ1a1dQ1a2=∑b∈VQ2∑a1a2∈EQ1dQ12a1+dQ12a2+4dQ22b−8dQ2b+4dQ2bdQ1a1+dQ1a2−4dQ1a1+dQ1a2+2dQ1a1dQ1a2+4.
By Definition1 FQ1=∑a1a2∈EQ1dQ12a1+dQ12a2, we have(9)=∑b∈VQ2FQ1+4e1dQ22b−8e1dQ2b+4dQ2bM1Q1−4M1Q1+2M2Q1+4e1=n2FQ1+4e1M1Q2−16e1e2+8e2M1Q1−4n2M1Q1+2n2M2Q1+4n2e1.
Finally, we obtain(10)ZC1Q1+QS12=4e1ZC1∗Q2+n1ZC1Q2+n2FQ1+2n2M2Q1+3M1Q14e2−n2+M1Q28e1+M1Q1+4e1n2−6e2.Proof.
(b) Letτa,b=τQ1+QS22a,b be a connection number of the vertex a,b in the graph Q1+QS22. Then,(11)ZC1Q1+QS22=∑a,b∈VQ1+QS22τa,b2=∑a∈VQ1∑b∈VQ2τa,b2+∑b∈VQ2∑a1a2∈ES2Q1τa,b2.
Consider(12)∑a∈VQ1∑b∈VQ2τa,b2=∑a∈VQ1∑b∈VQ22τQ1a+τQ2b+2dQ1adQ2b2=∑a∈VQ1∑b∈VG24τQ12a+τQ22b+4dQ12adQ22b+4τQ1aτQ2b+4dQ1adQ2bτQ2b+8dQ2bdQ1aτQ1a.
Using Definition2 and Lemma 2, we have(13)=4n2ZC1Q1+n1ZC1Q2+4M1Q1M1Q2+4∑a∈VQ1τQ1a∑b∈VQ2τQ2b+8e1ZC1∗Q2+8e2ZC1∗Q1=4n2ZC1Q1+n1ZC1Q2+8M1Q1M1Q2−8e2M1Q1−8e1M1Q2+16e1e2+8e1ZC1∗Q2+16e2ZC1∗Q1.
Also, consider(14)∑b∈VQ2∑a1a2∈ES2Q1τa,b2=∑b∈VQ2∑a1a2∈EQ1dQ1a1+2dQ2b−1+dQ1a22=∑b∈VQ2∑a1a2∈EQ1dQ12a1+2dQ2b−12+dQ12a2+2dQ1a12dQ2b−1+2dQ1a22dQ2b−1+2dQ1a1dQ1a2=∑b∈VQ2FQ1+4e1dQ22b−4e1dQ2b+4M1Q1dQ2b−2M1Q1+2M2Q1+e1=n2FQ1+4e1M1Q2−8e1e2+8e2M1Q1−2n2M1Q1+2n2M2Q1+n2e1.
Finally, we obtain(15)ZC1Q1+QS22=16e2ZC1∗Q1+8e1ZC1∗Q2+4n2ZC1Q1+n1ZC1Q2+n2FQ1+2n2M2Q1−M1Q1+4M1Q22M1Q1−e1+e1n2+8e2.Theorem 2.
LetQ1 and Q2 be two connected and C3,C4−free graphs. Then, the second Zagreb connection indices of their S-sum graphs under the operations of subdivision and semitotal point are(16)ZC2G1+QS12=6e1ZC1∗Q2+n1ZC2Q2+n2+2e2M3Q1+2M2Q1−2n2+3e2M1Q1−4e1M1Q2+5M1Q2+M2Q2M1Q1+8e1e2+2e1∑b1b2∈EQ2dQ2b1τQ2b2+dQ2b2τQ2b1,(17)ZC2Q1+QS22=6M1Q2+4e2−2n2ZC1∗Q1+4M1Q1ZC1∗Q2+n1ZC2Q2+4n2ZC2Q1+4e2ZC1Q1+e1ZC1Q2+4e2M3Q1+8e2M2Q1−M1Q1−2e1M1Q2+2M1Q12M2Q2+3M1Q2+4M1Q2M2Q1+4e1∑b1b2∈EQ2dQ2b2τQ2b1+dQ2b1τQ2b2+2n2+4e2∑a1a2∈EQ1dQ1a2τQ1a1+dQ1a1τQ1a2+2n2∑a1a2∈EQ1dQ1a1τQ1a1+dQ1a2τQ1a2+4e1e2.Proof.
(a) Letτa,b=τQ1+QS12a,b be a connection number of the vertex a,b in the graph Q1+S1Q2. Then,(18)ZC2Q1+QS12=∑a1,b1a2,b2∈EQ1+QS12τa1,b1×τa2,b2=∑a∈VQ1∑b1b2∈EQ2τa,b1×τa,b2+∑b∈VQ2∑a1a2∈ES1G1τa1,b×τa2,b.
Consider(19)∑a∈VQ1∑b1b2∈EQ2τa,b1×τa,b2=∑a∈VQ1∑b1b2∈EQ2dQ1a+dQ1adQ2b1+τQ2b1×dQ1a+dQ1adQ2b2+τQ2b2=∑a∈VQ1∑b1b2∈EQ2dQ12a+dQ12adQ2b2+dQ1aτQ2b2+dQ12adQ2b1+dQ12adQ2b1dQ2b2+dQ1adQ2b1τQ2b2+dQ1aτQ2b1+dQ1aτQ2b1dQ2b2+τQ2b1τQ2b2=∑a∈VQ1∑b1b2∈EQ2dQ12a+dQ12adQ2b1+dQ2b2+dQ1aτQ2b1+τQ2b2+dQ12adQ2b1dQ2b2+dQ1adQ2b1τQ2b2+dQ2b2τQ2b1+τQ2b1τQ2b2=∑a∈VQ1e2dQ12a+M1Q2dQ12a+ZC1∗Q2dQ1a+M2Q2dQ12a+dQ1a∑b1b2∈EQ2dQ2b1τQ2b2+dQ2b2τQ2b1+ZC2Q2=e2M1Q1+M1Q1M1Q2+2e1ZC1∗Q2+M1Q1M2Q2+n1ZC2Q2+2e1∑b1b2∈EQ2dQ2b1τQ2b2+dQ2b2τQ2b1.
Also, consider(20)∑b∈VQ2∑a1a2∈ES1Q1τa1,b×τa2,b=∑b∈VQ2∑a1a2∈EQ1dQ1a1+dQ1a1dQ2b+2τQ2b+dQ1a2+dQ1a2dQ2b×dQ1a1+dQ1a2+2dQ2b−2=∑b∈VQ2∑a1a2∈EQ1dQ12a1+dQ1a1dQ1a2+2dQ1a1dQ2b−2dQ1a1+dQ12a1dQ2b+dQ1a1dQ1a2dQ2b+2dQ1a1dQ22b−2dQ1a1dQ2b+2dQ1a1τQ2b+2dQ1a2τQ2b+4dQ2bτQ2b−4τQ2b+dQ1a1dQ1a2+dQ12a2+2dQ1a2dQ2b−2dQ1a2+dQ1a1dQ1a2dQ2b+dQ12a2dQ2b+2dQ1a2dQ22b−2dQ1a2dQ2b=∑b∈VQ2∑a1a2∈EQ1dQ12a1+dQ12a2+dQ1a1dQ1a2−2dQ1a1+dQ1a2+dQ2bdQ12a1+dQ12a2+2dQ1a1dQ1a2dQ2b+2dQ22bdQ1a1+dQ1a2+2τQ2bdQ1a1+dQ1a2+4dQ2bτQ2b−4τQ2b+dQ1a1dQ1a2=∑b∈VQ2M3Q1+M2Q1−2M1Q1+M3Q1dQ2b+2M2Q1dQ2b+2M1Q1dQ22b+2M1Q1τQ2b+4e1dQ2bτG2b−4e1τQ2b+M2Q1=n2M3Q1+n2M2Q1−2n2M1Q1+2e2M3Q1+4e2M2Q1+2M1Q1M1Q2+2M1Q1M1Q2−2e2+4e1ZC1∗Q2−4e1M1Q2−2e2+n2M2Q1=n2M3Q1+2n2M2Q1−2n2M1Q1+2e2M3Q1+4e2M2Q1+4M1Q1M1Q2−4e2M1Q1+4e1ZC1∗Q2−4e1M1Q2+8e1e2.
Therefore,(21)ZC2G1+QS12=6e1ZC1∗Q2+n1ZC2Q2+n2+2e2M3Q1+2M2Q1−2n2+3e2M1Q1−4e1M1Q2+5M1Q2+M2Q2M1Q1+8e1e2+2e1∑b1b2∈EQ2dQ2b1τQ2b2+dQ2b2τQ2b1.proof.
(b) Letτa,b=τQ1+QS22a,b be a connection number of the vertex a,b in the graph Q1+QS22. Then,(22)ZC2Q1+S2Q2=∑a1,b1a2,b2∈EQ1+S2Q2τa1,b1×τa2,b2=∑a∈VQ1∑b1b2∈EQ2τa,b1×τa,b2+∑b∈VQ2∑a1a2∈ES2G1τa1,b×τa2,b.
Consider(23)∑1=∑a∈VQ1∑b1b2∈EQ2τa,b1×τa,b2=∑a∈VQ1∑b1b2∈EG22τQ1a+τQ2b1+2dQ1adQ2b1×2τQ1a+τQ2b2+2dQ1adQ2b2=∑a∈VQ1∑b1v2∈EQ24τQ12a+2τQ1aτQ2b2+4dQ1aτQ1adQ2b2+2τQ1aτQ2b1+τQ2b1τG2b2+2dQ1adQ2b2τQ2b1+4dQ1aτQ1adQ2b1+2dQ1adQ2b1τQ2b2+4dQ12adQ2b1dQ2b2=∑a∈VQ1∑b1v2∈EQ24τQ12a+2τQ1aτQ2b1τQ2b2+4dQ1aτQ1adQ2b1+dQ2b2+τQ2b1τQ2b2+2dQ1udQ2v2τQ2v1+2dQ1udQ2v1τQ2b2+4dQ12adQ2b1dQ2b2=∑a∈VQ14e2τQ12a+2ZC1∗Q2τQ1a+4M1Q2dQ1aτQ1a+ZC2Q2+2dQ1a∑b1b2∈EQ2dQ2b2τQ2b1+2dQ1a∑b1b2∈EQ2dQ2b1τQ2b2+4M2Q2dQ12a=4e2ZC1Q1+2ZC1∗Q2M1Q1−2e1+4M1Q2ZC1∗Q1+n1ZC2Q2+4e1∑v1v2∈EQ2dQ2b2τQ2b1+4e1∑b1b2∈EQ2dQ2b1τQ2b2+4M1Q1M2Q2=4e2ZC1Q1+2M1Q1ZC1∗Q2−4e1ZC1∗Q2+4M1Q2ZC1∗Q1+n1ZC2Q2+4M1Q1M2Q2+4e1∑b1b2∈EQ2dQ2b2τQ2b1+dQ2b1τQ2b2.
Also, consider(24)∑b∈VQ2∑a1a2∈ES2G1τa1,b×τa2,b=∑2+∑3∑2=∑b∈VQ2∑a1a2∈EQ12τQ1a1+τQ2b+2dQ1a1dQ2b×2τQ1a2+τQ2b+2dQ1a2dQ2b=∑b∈VQ2∑a1a2∈EG14τQ1a1τQ1a2+2τQ1a1τQ2b+4dQ1a2τQ1a1dQ2b+2τQ1a2τQ2b+τQ22b+2dQ1a2dQ2bτQ2b+4dQ1a1τQ1a2dQ2b+2dQ1a1dQ2bτQ2b+4dQ1a1dQ1a2dQ22b=∑b∈VQ2∑a1a2∈EQ14τQ1a1τQ1a2+2τQ2bτQ1a1+τQ1a2+τQ22b+2dQ2bτQ2bdQ1a1+dQ1a2+4dQ1a1dQ1a2dQ22b+4dQ1a2τQ1a1dQ2b+4dQ1a1τQ1a2dQ2b=∑b∈VQ24ZC2Q1+2ZC1∗Q1τQ2b+e1τQ22b+2M1Q1dQ2bτQ2b+4M2Q1dQ22b+4dQ2b∑a1a2∈EQ1dQ1a2τQ1a1+4dQ2b∑a1a2∈EQ1dQ1a1τQ1a2=4n2ZC2Q1+2ZC1∗Q1M1Q2−2e2+e1ZC1Q2+2M1Q1ZC1∗Q2+4M2Q1M1Q2+8e2∑u1u2∈EQ1dQ1a2τQ1a1+8e2∑a1a2∈EQ1dQ1a1τQ1a2=4n2ZC2Q1+2M1Q2ZC1∗Q1−4e2ZC1∗Q1+e1ZC1Q2+2M1Q1ZC1∗Q2+4M2Q1M1Q2+8e2∑a1a2∈EQ1dQ1a2τQ1a1+dQ1a1τQ1a2,∑3=∑b∈VQ2∑a1a2∈EQ12τQ1a1+2τQ2b+2dQ1a1dQ2b+2τQ1a2+2dQ1a2dQ2b×dQ1a1+dQ1a2+2dQ2b−1=∑b∈VQ2∑a1a2∈EQ12dQ1a1τQ1a1+2dQ1a2τQ1a1+4τQ1a1dQ2b−2τQ1a1+2dQ1a1τQ2b+2dQ1a2τQ2b+4dQ2bτQ2b−2τQ2b+2dQ12a1dQ2b+2dQ1a1dQ1a2dQ2b+4dQ1a1dQ22b−2dQ1a1dQ2b+2dQ1a1τQ1a2+2dQ1a2τQ1a2+4τQ1a2dQ2b−2τQ1a2+2dQ1a1dQ1a2dQ2b+2dQ12a2dQ2b+4dQ1a2dQ22b−2dQ1a2dQ2b=∑b∈VQ2∑a1a2∈EG14dQ2bτQ1a1+τQ1a2−2τQ1a1+τQ1a2+2τQ2bdQ1a1+dQ1a1+4dQ2bτQ2b−2τQ2b+2dQ2bdQ12a1+dQ12a2+4dQ1a1dQ1a2dQ2b+4dQ22bdQ1a1+dQ1a2−2dQ2bdQ1a1+dQ1a2+2dQ1a1τQ1a1+2dQ1a2τQ1a1+2dQ1a1τQ1a2+2dQ1a2τQ1a2=∑b∈VQ24ZC1∗G1dQ2b−2ZC1∗Q1+2M1Q1τQ2v+4e1dQ2bτQ2b−2e1τQ2b+2M3Q1dQ2b+4M2Q1dQ2b+4M1Q1dQ22b−2M1Q1dQ2b+2∑a1a2∈EQ1dQ1a1τG1a1+2∑a1a2∈EG1dG1a2τG1a1+2∑a1a2∈EQ1dQ1a1τQ1a2+2∑a1a2∈EQ1dQ1a2τQ1a2=8e2ZC1∗Q1−2n2ZC1∗Q1+6M1Q1M1Q2−8e2M1Q1+4e1ZC1∗Q2−2e1M1Q2+4e1e2+4e2M3Q1+8e2M2Q1+2n2∑a1a2∈EQ1dQ1a1τQ1a1+dQ1a2τQ1a2+2n2∑a1a2∈EQ1dQ1a2τQ1a1+dQ1a1τQ1a2.
Consequently,(25)ZC2Q1+QS22=∑1+∑2+∑3=6M1Q2+4e2−2n2ZC1∗Q1+4M1Q1ZC1∗Q2+n1ZC2Q2+4n2ZC2Q1+4e2ZC1Q1+e1ZC1Q2+4e2M3Q1+8e2M2Q1−M1Q1−2e1M1Q2+2M1Q12M2Q2+3M1Q2+4M1Q2M2Q1+4e1∑b1b2∈EQ2dQ2b2τQ2b1+dQ2b1τQ2b2+2n2+4e2∑a1a2∈EQ1dQ1a2τQ1a1+dQ1a1τQ1a2+2n2∑a1a2∈EQ1dQ1a1τQ1a1+dQ1a2τQ1a2+4e1e2.Theorem 3.
LetQ1 and Q2 be two connected and C3,C4−free graphs. Then, the modified first Zagreb connection indices of their S-sum graphs under the operations of subdivision and semitotal point are(26)ZC1∗Q1+QS12=n1ZC1∗Q2+M1Q13n2+2e2+4e1M1Q2+8e1e2−4n2e1,(27)ZC1∗Q1+QS22=4n2ZC1∗Q1+n1ZC1∗Q2+2M1Q1n2+6e2+8e1M1Q2−2e1n2+4e2.Proof.
(a) Letτa,b=τQ1+QS12a,b be a connection number of the vertex a,b in the graph Q1+QS12. Then,(28)ZC1∗Q1+QS12=∑a1,b1a2,b2∈EQ1+QS12τa1,b1+τa2,b2=∑a∈VQ1∑b1b2∈EQ2τa,b1+τa,b2+∑b∈VQ2∑a1b2∈ES1Q1τa1,b×τa2,b.
Consider(29)∑a∈VQ1∑b1b2∈EQ2τa,b1+τa,b2=∑a∈VQ1∑b1b2∈EQ2dG1a+dQ1adQ2b1+τQ2b1+dQ1a+dQ1adQ2b2+τQ2b2=∑a∈VQ1∑b1b2∈EQ22dQ1a+dQ1adQ2b1+dQ2b2+τQ2b1+τQ2b2=∑a∈VQ12e2dQ1u+M1Q2dQ1u+ZC1∗Q2=n1ZC1∗Q22e1M1Q2+4e1e2.
Also, consider(30)∑b∈VQ2∑a1a2∈ES1Q1τa1,b+τa2,b=∑b∈VQ2∑a1a2∈EQ1dQ1a1+dQ1a1dQ2b+τQ2b+2dQ1a1+dQ1a2+2dQ2b−2+dQ1a2+dQ1a2dQ2b+τQ2b=∑b∈VQ2∑a1a2∈EQ1dQ1a1+dQ1a2+dQ2bdQ1a1+dQ1a2+2τQ2b+2dQ1a1+dQ1a2+4dQ2b−4.=∑b∈VQ2M1Q1+M1Q1dQ2b+2e1τQ2b+2M1Q1+4e1dQ2b−4e1=n2M1Q1+M1Q12e2+2e1M1Q2−2e2+2n2M1Q1+4e12e2−4n2e1=M1Q13n2+2e2+2e1M1Q2+4e1e2−4n2e1.
Finally, we obtain(31)ZC1∗Q1+QS12=n1ZC1∗Q2+M1Q13n2+2e2+4e1M1Q2+8e1e2−4n2e1.proof.
(b) Letτa,b=τQ1+S2Q2a,b be a connection number of the vertex a,b in the graph Q1+S2Q2. Then,(32)ZC1∗Q1+QS22∑a1,b1a2,b2∈EQ1+QS22τa1,b1+τa2,b2=∑a∈VQ1∑b1b2∈EQ2τa,b1+τa,b2+∑b∈VQ2∑a1a2∈ES2Q1τa1,b+τa2,b.
Consider(33)∑A=∑a∈VG1∑b1b2∈EQ2τa,b1+τa,b2=∑a∈VQ1∑b1b2∈EQ22τQ1a+τQ2b1+2dQ1adQ2b1+2τQ1a+τQ2b2+2dQ1adQ2b2=∑a∈VQ1∑b1b2∈EQ24τQ1a+τQ2b1+τQ2b2+2dQ1adQ2b1+dQ2b2=∑a∈VQ14e2τQ1a+ZC1∗Q2+2M1Q2dQ1a=4e2M1Q1−2e1+n1ZC1∗Q2+2M1Q22e1=n1ZC1∗Q2+4e2M1Q1+4e1M1Q2−8e1e2.
Also, consider(34)∑b∈VQ2∑a1a2∈ES2Q1τa1,b+τa2,b=∑B+∑C,∑B=∑b∈VQ2∑a1a2∈EQ12τQ1a1+τQ2b+2dQ1a1dQ2b+2τQ1a2+τQ2b+2dQ1a2dQ2b=∑b∈VQ2∑a1a2∈EQ12τQ1a1+τQ1a2+2τQ2b+2dQ2bdQ1a1+dQ1a2=∑b∈VQ22ZC1∗Q1+2e1τQ2b+2M1Q1dQ2b=2n2ZC1∗Q1+4e2M1Q1+2e1M1Q2−4e1e2,∑C=∑b∈VQ2∑a1a2∈EQ12τQ1a1+τQ2b+2dQ1a1dQ2b+2τQ1a2+τQ2b+2dQ1a2dQ2b+2dQ1a1+dQ1a2+2dQ2b−1=∑b∈VQ2∑a1a2∈EQ12τQ1a1+τQ1a2+2τQ2b+2dQ2bdQ1a1+dQ1a2+2dQ1a1+dQ1a2+4dQ2b−2=∑b∈VQ22ZC1∗Q1+2e1τQ2b+2M1Q1dQ2b+2M1Q1+4e1dQ2b−2e1=2n2ZC1∗Q1+2n2M1Q1+4e2M1Q1+2e1M1Q2+4e1e2−2n2e1.
Consequently,(35)ZC1∗Q1+QS22=∑A+∑B+∑C=4n2ZC1∗Q1+n1ZC1∗Q2+2M1Q1n2+6e2+8e1M1Q2−2e1n2+4e2.


## 4. Applications and Conclusion

LetQ1≅Pm and Q2≅Pn be two particular alkanes called by paths, where m,n≥5. Then, the Zagreb connection indices of their S-sum graphs as the consequences of the obtained main results are as follows:(1)
(a)ZC1Pm+PS1n=100mn−148m−136n+172,(b)
ZC1Pm+PS2n=245mn−332m−443n+512.(2)
(a)ZC2Pm+Ps1n=160mn−274m−236n+358,(b)
ZC2Pm+PS2n=588mn−906m−1196n+1628.(3)
(a)ZC1∗Pm+PS1n=44mn−50m−50n+44,(b)
ZC1∗Pm+PS2n=98mn−98m−146n+112.In this paper, we obtained theS-sum graphs using the operations of subdivision, semitotal point and cartesian product on two connected graphs in a certain order. Finally, we computed the first Zagreb connection index, second Zagreb connection index, and modified first Zagreb connection index of the obtained S-sum graphs in the terms of Zagreb indices and Zagreb connection indices of their factor graphs. Our results are interesting for pharmaceuticals [57–61]. Also, the problem is still open to compute the Zagreb connection indices for other families of graphs obtained by various operations on graphs.



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*Source: 9846913-2019-12-28.xml*