# Zagreb Connection Indices of Molecular Graphs Based on Operations

Authors: Jinde Cao; Usman Ali; Muhammad Javaid; Chuangxia Huang
Journal: Complexity
 (2020)

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## Abstract

Topological index (numeric number) is a mathematical coding of the molecular graphs that predicts the physicochemical, biological, toxicological, and structural properties of the chemical compounds that are directly associated with the molecular graphs. The Zagreb connection indices are one of the TIs of the molecular graphs depending upon the connection number (degree of vertices at distance two) appeared in 1972 to compute the total electron energy of the alternant hydrocarbons. But after that, for a long period, these are not studied by researchers. Recently,Ali and TrinajsticMol.Inform.372018,1−7 restudied the Zagreb connection indices and reported that the Zagreb connection indices comparatively to the classical Zagreb indices provide the better absolute value of the correlation coefficient for the thirteen physicochemical properties of the octane isomers (all these tested values have been taken from the website http://www.moleculardescriptors.eu). In this paper, we compute the general results in the form of exact formulae & upper bounds of the second Zagreb connection index and modified first Zagreb connection index for the resultant graphs which are obtained by applying operations of corona, Cartesian, and lexicographic product. At the end, some applications of the obtained results for particular chemical structures such as alkanes, cycloalkanes, linear polynomial chain, carbon nanotubes, fence, and closed fence are presented. In addition, a comparison between exact and computed values of the aforesaid Zagreb indices is also included.


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## Body

## 1. Introduction

Graph theory has provided a variety of useful tools in which one of the best tools is a topological index (TI). Molecules and molecular compounds are often modeled by molecular graphs. The topological indices (TIs) predict hydrocarbon, physicochemical, and structural properties of the molecular graphs such as critical temperature, ZE-isomerism, chirality, solubility, molecular mass, and connectivity, see [1–4]. Medical behaviours of the drugs, crystallin materials, and nanomaterials which are very important for chemical and pharmaceutical industries are also studied by TIs, see [5–8]. Todeschini et al. [9] also reported that TIs are widely used in the study of quantitative structure-activity relationships (QSARs) and quantitative structure-property relationships (QSPRs). These relationships play a vital role in the subject of cheminformatics, see [9–13].TIs have been divided into different classes, but degree-based are studied more, see [1, 4, 7, 14, 15, 16]. Gutman and Trinajstić [17] investigated the correlation value between the total π-electron energy and the structure of a molecule using the first Zagreb index. Gutman et al. [18] developed their work and established another TI for molecular structures called the second Zagreb index. After that, many extended works have been appeared on these invariants. For more study, we refer to [9, 10, 19, 20]. Another TI was studied by Gutman and Trinajstić in the same paper [17], but there was not more attention on this index by other researchers up to 2017. Ali and Trinajstić [21] restudied this TI and renamed it as the modified first Zagreb connection index (ZCI). They also reported that it has more precise values of the correlation coefficients of various octane isomers. Du et al. [22], Ducoffe et al. [23], and Shao et al. [24] determined extremal alkanes and cycloalkanes under different conditions using this ZCI. Zhu et al. [25] established the lower bound by using the modified first ZCI of trees in terms of their order and maximum degree. Tang et al. [26] computed the first and second (ZCI) and modified first (ZCI) of the S-sum graphs.A simple graph can be molded into a chemical structure by using some operations. First of all, Graovac and Pisanski [27] computed different results of the Wiener index using product based on operations. So, many chemical graphs can be generated by using simple graphs based on operations such as alkane C3H6 is the corona product of P3&N2, a type of cycloalkanes is cyclohexane C6H12 that is the corona product of C6&N2, a polynomial chain and nanotube TUC4m,n are the Cartesian product of Pm&P2 and Cm&P2, and a fence and closed fence are the lexicographic product of Pm&P2 and Cm&P2. Up till now, many results of the various TIs have been presented under different molecular graphs based on operations, see [26, 28–38].In this paper, we compute the second ZCI and modified first ZCI of the resultant graphs which are obtained by applying various operations of corona product, Cartesian product, and lexicographic product (composition) in the form of exact formulae and upper bounds. The rest of the paper is settled as Section2 represents the preliminary definitions and results, Section 3 covers the general results of molecular graphs based on operations, and Section 4 includes the applications and conclusion.


## 2. Preliminaries

LetQ=VQ,EQ be a simple and connected molecular graph with a vertex set VQ and an edge set EQ⊆VQ×VQ. A null graph (N) has at least two vertices and no edge. It becomes a trivial graph K1 if it has exactly one vertex and no edge. Todesehini et al. [9] defined fkQb=NkQb for NkQb=a∈VQ:da,b=k such that f1Qb=dGb and f2Qb=τGb are called the degree and connection number of the vertex b∈Q. Now, throughout the paper, we assume that Q1 and Q2 are two connected graphs such that VQ1=n1, VQ2=n2, EQ1=e1, and EQ2=e2.Alkanes are the simplest organic compounds containing a single bound between carbon atoms. They are also called hydrocarbon compounds. Its simple and Lewis structures are shown in Figure1.Figure 1
(a) Simple structure of alkanesP2, P3, and P4, respectively, and (b) Lewis structure of alkanes P2, P3, and P4, respectively.
(a)
(b)The some examples of alkanes are methaneCH4, ethane H3C‐CH3, and propane H3C‐CH2‐CH3, and their Lewis structures are shown in Figure 2.Figure 2
Lewis structure of (a) methane, (b) ethane and (c) propane.
(a)
(b)
(c)Cycloalkanes are cyclic organic compounds containing the closed chain of carbon atoms. In other words, a cycloalkane is arranged into a chemical structure obtained a single ring (sometime side chains may be attached), and all of the carbon-carbon bonds are single. These are classified into two classes as homocyclic and heterocyclic compounds. If cyclo-organic compounds containing between carbons [1 to 5], [6 to 10], and [11 to on wards] are called small, mediam, and large cyclo organic compounds, respectively. Cycloalkanes are the isomers of alkene, e.g.,C3H6 is used as the same chemical formula of cyclopropane as well as propene. The general formula of cycloalkanes is CnH2n. The sets of Cyclopropane,cyclobutane,cyclopentane,cyclohexane etc.& pyrol,thiophene etc. are examples of cycloalkanes that are classified by homocyclic and heterocyclic compounds, respectively. Furthermore, the following Lewis structures of these cycloalkanes are shown in Figure 3.Figure 3
(a) The set of Lewis structure of cyclopropane, cyclobutane, cyclopentane, and cyclohexane, (b) the set of Lewis structure of pyrrole and thiophene.
(a)
(b)Definition 1.
For a graphQ, the first Zagreb index M1Q, second Zagreb index M2Q, and their coindices are defined as(1)M1Q=∑b∈VQdQb2=∑ab∈EQdQa+dQb,M2Q=∑ab∈EQdQa×dQb,M1¯Q=∑ab∉EQdQa+dQb,M2¯Q=∑ab∉EQdQa×dQb.
Gutman et al. [17, 18, 39] defined the different degree-based TIs which are frequently used in the studies of QSPR and QSAR [40–43]. Corresponding to these degree-based TIs, the connection-based TIs are defined in Definition 2. For further studies of connection-based TIs, see [21, 22, 44].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.Definition 3.
The corona productQ1∘Q2 of two graphs Q1 and Q2 is obtained by taking one copy of Q1 and n1 copies of Q2 (i.e., Q2i:1≤i≤n1) and then by joining each vertex of the ith copy of Q2 to the ith vertex of one copy of Q1, where 1≤i≤n1. Also, a number of vertex set and edge set are defined as: VQ1∘Q2=n1n2+n1 and EQ1∘Q2=e1+n1e2+n1n2. For more detail, see Figure 4.Figure 4
(a)Q1≅C4, (b) Q2≅C3, and (c) CoronaProductC4∘C3.
(a)
(b)
(c)Definition 4.
The Cartesian productQ1⊙Q2 and lexicographic product or composition Q1Q2 of two graphs Q1 and Q2 are obtained by taking the vertex set VQ1×Q2=VQ1×VQ2 and the edge set EQ1×Q2=a1,b1a2,b2wherea1,b1,a2,b2∈VQ1×VQ2, where with conditions(i)
eithera1=a2∈VQ1∧b1b2∈EQ2 or b1=b2∈VQ2∧a1a2∈EQ1,(ii)
eithera1=a2∈VQ1∧b1b2∈EQ2 or b1,b2∈VQ2∧a1a2∈EQ1, respectively.
For more detail, see Figures5 and 6.Figure 5
(a)Q1≅C5, (b) Q2≅P2, and (c) CartesianProductC5⊙P2.
(a)
(b)
(c)Figure 6
(a)Q1≅C4, (b) Q2≅P2, and (c) LexicographicProductC4P2.
(a)
(b)
(c)Lemma 1 (see [45]).
LetQ be a connected graph and Q¯ be its complement. Then, (i) ∑b∈VQdQb=2e, (ii) dQ¯b=n−1−dQb, and (iii) M1Q¯=M1Q−4, where VQ=n and EQ=e.Lemma 2 (see [46]).
LetQ be a connected graph with n vertices and e edges. Then, τQa+dQa≤∑b∈NQadQb, where equality holds if and only if Q is a C3,C4− free graph.Lemma 3 (see [26]).
LetQ be a connected and C3,C4− free graph with n vertices and e edges. Then, ∑a∈VQτQa=M1Q−2e.


## 3. Main Results

This section consists on the main results.Theorem 1.
LetQ1 and Q2 be two connected and C3,C4− free graphs. Then, ZC2 and ZC1∗ of the corona product of Q1 and Q2 are as follows: (3)aZC2G1∘Q2=n2ZC1∗Q1+ZC2Q1+n22M2Q1+n1M2Q2+n22+e2M1Q1−n1n2−1M1Q2+n1n2−12e2+2e12n2−1e2−M1Q2+n2n2−1−2e2M1G1−2e1+2e1n22n2−1−2n2e2+n2∑ab∈EQ1dQ1aτQ1b+dQ1bτQ1a.bZC1∗Q1∘Q2=ZC1∗Q1+2n2M1Q1−n1M1Q2+2n1e2n2−2+2n22e1+n1n2n2−1+4e1e2.Proof.
(a)
If for anyb∈VQ1∘Q2 either b∈VQ1 or b∈VQ2i, where 1≤i≤n1 and
(i)
Case I: ifb∈VQ1, then τQ1∘Q2b=τQ1b+n2dQ1b.
(ii)
Case II: ifb∈VQ2i, then τQ1∘Q2b=n2−1−dQ2ib+dQ1bi.
(4)
Z
C
2
Q
1
∘
Q
2
=
∑
a
b
∈
E
Q
1
∘
Q
2
τ
Q
1
∘
Q
2
a
×
τ
Q
1
∘
Q
2
b
=
∑
a
b
∈
E
Q
1
∘
Q
2
a
,
b
∈
V
Q
1
τ
Q
1
a
×
τ
Q
1
b
+
∑
a
b
∈
E
Q
1
∘
Q
2
a
,
b
∈
V
Q
2
τ
Q
2
a
×
τ
Q
2
b
+
∑
a
b
∈
E
Q
1
∘
Q
2
a
∈
V
Q
1
∧
b
∈
V
Q
2
τ
Q
1
a
×
τ
Q
2
b
.
Take
(5)
∑
u
v
∈
E
Q
1
∘
Q
2
a
,
b
∈
V
Q
1
τ
Q
1
a
×
τ
Q
1
b
=
∑
a
b
∈
E
Q
1
τ
Q
1
a
+
n
2
d
Q
1
a
×
τ
Q
1
b
+
n
2
d
Q
1
b
=
∑
a
b
∈
E
Q
1
τ
Q
1
a
τ
Q
1
b
+
n
2
d
Q
1
b
τ
Q
1
a
+
n
2
d
Q
1
a
τ
Q
1
b
+
n
2
2
d
Q
1
a
d
Q
1
b
=
Z
C
2
Q
1
+
n
2
2
M
2
Q
1
+
n
2
∑
a
b
∈
E
Q
1
d
Q
1
a
τ
Q
1
b
+
d
Q
1
b
τ
Q
1
a
.
Also take
(6)
∑
a
b
∈
E
Q
1
∘
Q
2
a
,
b
∈
V
Q
2
τ
Q
2
a
×
τ
Q
2
b
=
∑
i
=
1
n
1
∑
a
b
∈
E
Q
2
i
n
2
−
1
−
d
Q
2
i
a
+
d
Q
1
b
i
×
n
2
−
1
−
d
Q
2
i
b
+
d
Q
1
b
i
=
∑
i
=
1
n
1
∑
a
b
∈
E
Q
2
i
n
2
−
1
2
−
n
2
−
1
d
Q
2
i
a
+
d
Q
2
i
b
+
2
n
2
−
1
d
Q
1
b
i
+
d
Q
2
i
u
d
Q
2
i
b
−
d
Q
1
b
i
d
Q
2
i
a
−
d
Q
1
b
i
d
Q
2
i
b
+
d
Q
1
2
b
i
=
n
1
n
2
−
1
2
e
2
−
n
1
n
2
−
1
M
1
Q
2
+
2
e
1
2
n
2
−
1
e
2
−
M
1
Q
2
+
n
1
M
2
Q
2
+
e
2
M
1
Q
1
.
Similarly,
(7)
∑
a
b
∈
E
Q
1
∘
Q
2
a
∈
V
Q
1
∧
b
∈
V
Q
2
τ
Q
1
a
×
τ
Q
2
b
=
∑
i
=
1
n
1
∑
b
∈
V
Q
2
i
τ
Q
1
a
i
+
n
2
d
Q
1
a
i
×
n
2
−
1
−
d
Q
2
i
b
+
d
Q
1
a
i
=
n
2
n
2
−
1
−
2
e
2
+
M
1
Q
1
−
2
e
1
+
n
2
Z
C
1
∗
Q
1
+
2
e
1
n
2
2
n
2
−
1
−
2
n
2
e
2
+
n
2
2
M
1
Q
1
.
Consequently,
(8)
Z
C
2
Q
1
∘
Q
2
=
n
2
Z
C
1
∗
Q
1
+
Z
C
2
Q
1
+
n
2
2
M
2
Q
1
+
n
1
M
2
Q
2
+
n
2
2
+
e
2
M
1
Q
1
−
n
1
n
2
−
1
M
1
Q
2
+
n
1
n
2
−
1
2
e
2
+
2
e
1
2
n
2
−
1
e
2
−
M
1
Q
2
+
n
2
n
2
−
1
−
2
e
2
M
1
G
1
−
2
e
1
+
2
e
1
n
2
2
n
2
−
1
−
2
n
2
e
2
+
n
2
∑
a
b
∈
E
Q
1
d
Q
1
a
τ
Q
1
b
+
d
Q
1
b
τ
Q
1
a
.
(b)
(9)
Z
C
1
∗
Q
1
∘
Q
2
=
∑
a
b
∈
E
Q
1
∘
Q
2
τ
Q
1
∘
Q
2
a
+
τ
Q
1
∘
Q
2
b
=
∑
a
b
∈
E
Q
1
∘
Q
2
a
,
b
∈
V
Q
1
τ
Q
1
a
+
τ
Q
1
b
+
∑
a
b
∈
E
G
1
∘
G
2
a
,
b
∈
V
Q
2
τ
Q
2
a
+
τ
Q
2
b
+
∑
a
b
∈
E
Q
1
∘
Q
2
a
∈
V
Q
1
∧
b
∈
V
Q
2
τ
Q
1
a
+
τ
Q
2
b
.
Take
(10)
∑
a
b
∈
E
Q
1
∘
Q
2
a
,
b
∈
V
Q
1
τ
Q
1
a
+
τ
Q
1
b
=
∑
a
b
∈
E
Q
1
τ
Q
1
a
+
n
2
d
Q
1
a
+
τ
Q
1
b
+
n
2
d
Q
1
b
=
=
Z
C
1
∗
Q
1
+
n
2
M
1
Q
1
.
Also take
(11)
∑
a
b
∈
E
Q
1
∘
Q
2
a
,
b
∈
V
Q
2
τ
Q
2
a
+
τ
Q
2
b
=
∑
i
=
1
n
1
∑
a
b
∈
E
Q
2
i
n
2
−
1
−
d
Q
2
i
u
+
d
Q
1
b
i
+
n
2
−
1
−
d
Q
2
i
b
+
d
Q
1
b
i
=
2
n
1
n
2
−
1
e
2
−
n
1
M
1
Q
2
+
4
e
1
e
2
.
Similarly,
(12)
∑
a
b
∈
E
Q
1
∘
Q
2
a
∈
V
Q
1
∧
b
∈
V
Q
2
τ
Q
1
a
+
τ
Q
2
b
=
n
2
M
1
Q
1
−
2
e
1
+
2
n
2
2
e
1
+
n
1
n
2
n
2
−
1
−
2
n
1
e
2
+
2
n
2
e
1
.
Consequently,
(13)
Z
C
1
∗
Q
1
∘
Q
2
=
Z
C
1
∗
Q
1
+
2
n
2
M
1
Q
1
−
n
1
M
1
Q
2
+
2
n
1
e
2
n
2
−
2
+
2
n
2
2
e
1
+
n
1
n
2
n
2
−
1
+
4
e
1
e
2
.Theorem 2.
LetQ1 and Q2 be two connected and C3,C4− free graphs. Then, ZC2 and ZC1∗ of the Cartesian product of Q1 and Q2 are as follows:(14)aZC2Q1⊙Q2=2M1Q2−e2ZC1∗Q1+2M1Q1−e1ZC1∗Q2+n2ZC2Q1+n1ZC2Q2+e2ZC1Q1+e1ZC1Q2+M1Q1M2Q2+M1Q2M2Q1+2e2∑a1a2∈EQ1dQ1u1τQ1a2+dQ1a2τQ1a1+2e1∑b1b2∈EQ2dQ2b1τQ2b2+dQ2b2τQ2b1,bZC1∗Q1⊙Q2=n2ZC1∗Q1+n1ZC1∗Q2+4e2M1Q1+4e1M1Q2−8e1e2.Proof.
(a)
Fora∈VQ1, b∈VQ2, and a,b∈VQ1⊙Q2, we have(15)τQ1⊙Q2a,b=τQ1a+dQ1adQ2b+τQ2b,ZC2Q1⊙Q2=∑a1,b1a2,b2∈EQ1⊙Q2τQ1⊙Q2a1,b1×τQ1⊙Q2a2,b2=∑a∈VQ1∑b1b2∈EQ2τQ1⊙Q2a,b1×τQ1⊙Q2a,b2+∑b∈VQ2∑a1a2∈EQ1τQ1⊙Q2a1,b×τQ1⊙Q2a2,b.Take(16)∑a∈VQ1∑b1b2∈EQ2τQ1⊙Q2a,b1×τQ1⊙Q2a,b2=∑a∈VQ1∑b1b2∈EQ2τQ1a+dQ1adQ2b1+τQ2b1×τQ1a+dQ1adQ2b2+τQ2b2=e2ZC1Q1+M1Q2ZC1∗Q1+ZC1∗Q2M1Q1−2e1+M1Q1M2Q2+n1ZC2Q2+2e1∑b1b2EQ2dQ2b1τQ2b2+dQ2b2τQ2b1.Similarly,(17)∑b∈VQ2∑a1a2∈EQ1τQ1⊙Q2a1,b×τQ1⊙Q2a2,b=n2ZC2Q1+ZC1∗Q1M1Q2−2e2+2e2∑a1a2∈EQ1dQ1a1τQ1a2+dQ1a2τQ1a1+M1Q2M2Q1+M1Q1ZC1∗Q2+e1ZC1∗Q2.Consequently,(18)ZC2Q1⊙Q2=2M1Q2−e2ZC1∗Q1+2M1Q1−e1ZC1∗Q2+n2ZC2Q1+n1ZC2G2+e2ZC1Q1+e1ZC1Q2+M1Q1M2Q2+M1Q2M2Q1+2e2∑a1a2∈EQ1dQ1u1τQ1a2+dQ1a2τQ1a1+2e1∑b1b2∈EQ2dQ2b1τQ2b2+dQ2b2τQ2b1.(b)
Consider(19)ZC1∗Q1⊙Q2=∑a1,b1a2,b2∈EQ1⊙Q2τQ1⊙Q2a1,b1+τQ1⊙Q2a2,b2=∑a∈VQ1∑b1b2∈EQ2τQ1⊙Q2a,b1+τQ1⊙Q2a,b2+∑b∈VQ2∑a1a2∈EQ1τQ1⊙Q2a1,b+τQ1⊙Q2a2,b.Take(20)∑a∈VQ1∑b1b2∈EQ2τQ1⊙Q2a,b1+τQ1⊙Q2a,b2=∑a∈VQ1∑b1b2∈EQ2τQ1a+dQ1adQ2b1+τQ2b1+τQ1a+dQ1adQ2b2+τQ2b2=2e2M1Q1−2e1+2e1M1Q2+n1ZC1∗Q2.Similarly,(21)∑b∈VQ2∑a1a2∈EQ1τQ1⊙Q2a1,b+τQ1⊙Q2a2,b=n2ZC1∗Q1+2e2M1Q1+2e1M1Q2−4e1e2.Consequently,(22)ZC1∗Q1⊙Q2=n2ZC1∗Q1+n1ZC1∗Q2+4e2M1Q1+4e1M1Q2−8e1e2.Theorem 3.
LetQ1 and Q2 be two connected graphs. Then, ZC2 and ZC1∗ of the composition (or lexicographic product) of Q1 and Q2 are as follows:(23)aZC2Q1Q2≤n2n2n2−1−4e2+2n2e2+n2δ2−δ2ZC1∗Q1+n22n2+δ2+2e2ZC2Q1+n22e2ZC1Q1+n1−n1n2−2n2e1+3e1M1Q2+n1+2e1M2Q2+2n2n2−1e2−n2M1Q2M1Q1−2e1+n2−1n1e2n2−1+e1e2n2−2+δ2n2+n2e2−e1−1−n2−1e1M1¯Q2+e1M2¯Q2−2n2∑a1a2∈EQ1∑b1b2∈EQ2dQ2b1τQ1a2+dQ2b2τQ1a1−n2∑a1a2∈EQ1∑b1b2∉EQ2dQ2b1τQ1a2+dQ2b2τQ1a1,bZC1∗Q1Q2≤n2n2+δ2+2e2ZC1∗Q1+2n2e2M1Q1−n1+2e1M1Q2+n2−12n1e2+2n2e1+2δ2e1+4e1e2−4n2+1e1e2−e1M1¯Q2.Proof.
(a)
Fora∈VQ1, b∈VQ2, and a,b∈VQ1Q2, we have(24)τQ1Q2a,b=n2τQ1a+dQ2¯b=n2τQ1a+n2−1−dQ2b,ZC2Q1Q2=∑a1,b1a2,b2∈EQ1Q2τQ1Q2a1,b1×τQ1Q2a2,b2=∑a∈VQ1∑b1b2∈EQ2τQ1Q2a,b1×τQ1Q2a,b2+∑b∈VQ2∑a1a2∈EQ1τQ1Q2a1,b×τQ1Q2a2,b+∑a1a2∈EQ1∑b1b2∈EQ2τQ1Q2a1,b1×τQ1Q2a2,b2+∑a1a2∈EQ1∑b1b2∉EQ2τQ1Q2a1,b1×τQ1Q2a2,b2.Take(25)∑a∈VQ1∑b1b2∈EQ2τQ1Q2a,b1×τQ1Q2a,b2=∑a∈VQ1∑b1b2∈EQ2n2τQ1a+n2−1−dQ2b1×n2τQ1a+n2−1−dQ2b2=n22e2ZC1Q1+2n2n2−1e2−n2M1Q2M1Q1−2e1+n1n2−12e2−n1n2−1M1Q2+n1M2Q2.Also take(26)∑a1a2∈EQ1∑b1b2∉EQ2τQ1Q2a1,b1×τQ1Q2a2,b2≤∑a1a2∈EQ1∑b1b2∉EQ2n2τQ1a1+n2−1−dQ2b1×n2τQ1a2+n2−1−dQ2b2=∑a1a2∈EQ1∑b1b2∉EQ2n22τQ1a1τQ1a2+n2n2−1τQ1a1+τQ1a2−n2dQ2b1τQ1a2+dQ2b2τQ1a1+n2−12−n2−1dQ2b1+dQ2b2+dQ2b1dQ2b2.Let’s suppose that(27)∑b1b2∉EQ2=n2n2−1−2e2=δ2=n22δ2ZC2Q1+n2n2−1δ2ZC1∗Q1−n2∑a1a2∈EQ1∑b1b2∉EQ2dQ2b1τQ1a2+dQ2b2τQ1a1+n2−12δ2e1−n2−1e1M1¯Q2+e1M2¯Q2.Similarly,(28)∑b∈VQ2∑a1a2∈EQ1τQ1Q2a1,b×τQ1Q2a2,b=n2n2n2−1−2e2ZC1∗Q1+n23ZC2Q1+e1M1Q2+n2−1e1n2n2−1−4e2.a1a2∈EQ1∑a1a2∈EQ1∑b1b2∈EQ2τQ1Q2a1,b1×τQ1Q2a2,b2,=2∑a1a2∈EQ1∑b1b2∈EQ2n2τQ1a1+n2−1−dQ2b1×n2τQ1a2+n2−1−dQ2b2=2n22e2ZC2Q1+2n2n2−1e2ZC1∗Q1−2n2∑a1a2∈EQ1∑b1b2∈EQ2dQ2b1τQ1a2+dQ2b2τQ1a1+2n2−12e1e2−2n2−1e1M1Q2+2e1M2Q2.Consequently,(29)ZC2Q1Q2≤n2n2n2−1−4e2+2n2e2+n2δ2−δ2ZC1∗Q1+n22n2+δ2+2e2ZC2Q1+n22e2ZC1Q1+n1−n1n2−2n2e1+3e1M1Q2+n1+2e1M2Q2+2n2n2−1e2−n2M1Q2M1Q1−2e1+n2−1n1e2n2−1+e1e2n2−2+δ2n2+n2e2−e1−1,−n2−1e1M1¯Q2+e1M2¯Q2−2n2∑a1a2∈EQ1∑b1b2∈EQ2dQ2b1τQ1a2+dQ2b2τQ1a1−n2∑a1a2∈EQ1∑b1b2∉EQ2dQ2b1τQ1a2+dQ2b2τQ1a1.(b)
(30)ZC1∗Q1Q2=∑a1,b1a2,b2∈EQ1Q2τQ1Q2a1,b1+τQ1Q2a2,b2=∑a∈VQ1∑b1b2∈EQ2τQ1Q2a,b1+τQ1Q2a,b2+∑b∈VQ2∑a1a2∈EQ1τQ1Q2a1,b+τQ1Q2a2,b+∑a1a2∈EQ1∑b1b2∈EQ2τQ1Q2a1,b1+τQ1Q2a2,b2+∑a1a2∈EQ1∑b1b2∉EQ2τQ1Q2a1,b1+τQ1Q2a2,b2.Take(31)∑a∈VQ1∑b1b2∈EQ2τQ1G2a,b1+τQ1Q2a,b2=∑a∈VQ1∑b1b2∈EQ2n2τQ1a+n2−1−dQ2b1+n2τQ1a+n2−1−dQ2b2=2n2e2M1Q1−2e1+2n1n2−1e2−n1M1Q2.Also take(32)∑a1a2∈EQ1∑b1b2∉EQ2τQ1Q2a1,b1+τQ1Q2a2,b2≤∑a1a2∈EQ1∑b1b2∉EQ2n2τQ1a1+n2−1−dQ2b1+n2τQ1a2+n2−1−dQ2b2=n2δ2ZC1∗Q1+2n2−1δ2e1−e1M1¯Q2.Similarly,(33)∑b∈VQ2∑a1a2∈EQ1τQ1Q2a1,b+τQ1Q2a2,b=n22ZC1∗Q1+2n2n2−1e1−4e1e2,∑a1a2∈EQ1∑b1b2∈EQ2τQ1Q2a1,b1+τQ1Q2a2,b2=2∑a1a2∈EQ1∑b1b2∈EQ2n2τQ1a1+n2−1−dQ2b1+n2τQ1a2+n2−1−dQ2b2=2n2e2ZC1∗Q1+4n2−1e1e2−2e1M1Q2.Consequently,(34)ZC1∗Q1Q2≤n2n2+δ2+2e2ZC1∗Q1+2n2e2M1Q1−n1+2e1M1Q2+n2−12n1e2+2n2e1+2δ2e1+4e1e2−4n2+1e1e2−e1M1¯Q2.


## 4. Applications and Conclusion

In this section, we present some applications of the obtained results for particular chemical structures such as alkanes (see Figure7), cycloalkanes (see Figure 8), linear polynomial chain (see Figure 9), carbon nanotubes (see Figure 10), fence (see Figure 11), and closed fence (see Figure 12). We also give the both exact and computed values of the obtained results for the aforesaid particular chemical structures to develop an easy understanding. Let N2 be a null graph, P2, P3, and P5 be three particular alkanes called by paths, and C4, C5, and C6 be cycles.Figure 7
(a)Q1≅P3, (b) Q2≅N2, and (c) alkane (P3∘N2≅C3H6).
(a)
(b)
(c)Figure 8
(a)Q1≅C6, (b) Q2≅N2, and (c) cyclohexane (C6∘N2≅C6H12).
(a)
(b)
(c)Figure 9
(a)Q1≅P5, (b) Q2≅P2, and (c) (P5⊙P2).
(a)
(b)
(c)Figure 10
(a)Q1≅P3, (b) Q2≅C5, and (c) carbon nanotube (TUC4m,n) P3⊙C5.
(a)
(b)
(c)Figure 11
(a)Q1≅P5, (b) Q2≅P2, and (c) fence (P5P2).
(a)
(b)
(c)Figure 12
(a)Q1≅C4, (b) Q2≅P2, and (c) closed fence (C4P2).
(a)
(b)
(c)


### 4.1. Corona Product

Example 1.
Alkane (C3H6).(i)
Exact value ofZC2P3∘N2=72,(ii)
Exact value ofZC1∗P3∘N2=48,(iii)
Computed value ofZC2P3∘N2=72,(iv)
Computed value ofZC1∗P3∘N2=48.Example 2.
Cyclohexane (C6H12).(i)
Exact value ofZC2C6∘N2=432,(ii)
Exact value ofZC1∗C6∘N2=180,(iii)
Computed value ofZC2C6∘N2=432,(iv)
Computed value ofZC1∗C6∘N2=180.


### 4.2. Cartesian Product

Example 3.
Polynomial chain.(i)
Exact value ofZC2P5⊙P2=114,(ii)
Exact value ofZC1∗P5⊙P2=76,(iii)
Computed value ofZC2P5⊙P2=114,(iv)
Computed value ofZC1∗P5⊙P2=76.Example 4.
Carbon nanotube (TUC4m,n).(i)
Exact value ofZC2P3⊙C5=730,(ii)
Exact value ofZC1∗P3⊙C5=270,(iii)
Computed value ofZC2P3⊙C5=730,(iv)
Computed value ofZC1∗P3⊙C5=270.


### 4.3. Lexicographic Product

Example 5.
Fence (P5P2).(i)
Exact value ofZC2P5P2=128,(ii)
Exact value ofZC1∗P5P2=104,(iii)
Computed value ofZC2P3P3=128,(iv)
Computed value ofZC1∗P5P2=104.Example 6.
Closed fence (C4P2).(i)
Exact value ofZC2C4P2=72,(ii)
Exact value ofZC1∗C4P2=72,(iii)
Computed value ofZC2C4P2≤80,(iv)
Computed value ofZC1∗C4P2≤88.
In this paper, we have computed the general results related to the second ZCI and modified first ZCI of the resultant graphs which are obtained with the help of various operations of product on graphs such as corona product, Cartesian product, and lexicographic product (composition). The obtained results also illustrated with the help of particular class of molecular graphs. However, the problem is still open to compute the ZCI of the molecular graphs under the different operations of subdivision, addition and product, etc.


## 4.1. Corona Product

Example 1.
Alkane (C3H6).(i)
Exact value ofZC2P3∘N2=72,(ii)
Exact value ofZC1∗P3∘N2=48,(iii)
Computed value ofZC2P3∘N2=72,(iv)
Computed value ofZC1∗P3∘N2=48.Example 2.
Cyclohexane (C6H12).(i)
Exact value ofZC2C6∘N2=432,(ii)
Exact value ofZC1∗C6∘N2=180,(iii)
Computed value ofZC2C6∘N2=432,(iv)
Computed value ofZC1∗C6∘N2=180.


## 4.2. Cartesian Product

Example 3.
Polynomial chain.(i)
Exact value ofZC2P5⊙P2=114,(ii)
Exact value ofZC1∗P5⊙P2=76,(iii)
Computed value ofZC2P5⊙P2=114,(iv)
Computed value ofZC1∗P5⊙P2=76.Example 4.
Carbon nanotube (TUC4m,n).(i)
Exact value ofZC2P3⊙C5=730,(ii)
Exact value ofZC1∗P3⊙C5=270,(iii)
Computed value ofZC2P3⊙C5=730,(iv)
Computed value ofZC1∗P3⊙C5=270.


## 4.3. Lexicographic Product

Example 5.
Fence (P5P2).(i)
Exact value ofZC2P5P2=128,(ii)
Exact value ofZC1∗P5P2=104,(iii)
Computed value ofZC2P3P3=128,(iv)
Computed value ofZC1∗P5P2=104.Example 6.
Closed fence (C4P2).(i)
Exact value ofZC2C4P2=72,(ii)
Exact value ofZC1∗C4P2=72,(iii)
Computed value ofZC2C4P2≤80,(iv)
Computed value ofZC1∗C4P2≤88.
In this paper, we have computed the general results related to the second ZCI and modified first ZCI of the resultant graphs which are obtained with the help of various operations of product on graphs such as corona product, Cartesian product, and lexicographic product (composition). The obtained results also illustrated with the help of particular class of molecular graphs. However, the problem is still open to compute the ZCI of the molecular graphs under the different operations of subdivision, addition and product, etc.



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