# Computing Analysis for First Zagreb Connection Index and Coindex of Resultant Graphs

Authors: Muhammad Javaid; Usman Ali; Jia-Bao Liu
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

A numeric parameter which studies the behaviour, structural, toxicological, experimental, and physicochemical properties of chemical compounds under several graphs’ isomorphism is known as topological index. In 2018, Ali and Trinajstić studied the first Zagreb connection indexZC1 to evaluate the value of a molecule. This concept was first studied by Gutman and Trinajstić in 1972 to find the solution of π-electron energy of alternant hydrocarbons. In this paper, the first Zagreb connection index and coindex are obtained in the form of exact formulae and upper bounds for the resultant graphs in terms of different indices of their factor graphs, where the resultant graphs are obtained by the product-related operations on graphs such as tensor product, strong product, symmetric difference, and disjunction. At the end, an analysis of the obtained results for the first Zagreb connection index and coindex on the aforesaid resultant graphs is interpreted with the help of numerical values and graphical depictions.


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## Body

## 1. Introduction

Topological indices (TIs) are used in the study of cheminformatics which has key role in the formational structure of quantitative structures’ activity and property relationships to examine the chemical reactivity and experimental activity of a chemical compound in a molecular graph [1]. So, TIs predict both physical and chemical features that are defined in the molecular graphs such as surface tension, solubility, connectivity, freezing point, boiling point, melting point, critical temperature, polarizability, heat of evaporation, and formation [2]. In addition, the medical behaviours and a number of drug particles of different compounds have formed with the help of various TIs in the pharmaceutical networks (see [3]). In particular, the TIs called by connection number-based Zagreb indices are used to compute the correlation values among various octane isomers such as heat of evaporation, acentric factor, molecular weight, connectivity, critical temperature, stability, and density (see [4, 5]).Product graphs play an essential part to develop new molecular graphs from simple graphs. For this purpose, Ashrafi et al. [6] defined the concept of coindices for several products on graphs. Das et al. [7] computed upper bounds for multiplicative Zagreb indices of operations such as join, corona, Cartesian, disjunction, and composition. The reformulated, multiplicative, hyper, first, second, and third Zagreb coindices with certain properties are defined in [8–13]. Relations between Zagreb coindices and some distance-based indices are computed in [14]. For more details, see [15–18]. For this purpose, we can define some operations such as tensor product, strong product, symmetric difference, and disjunction in the following definitions.Definition 1.
Tensor (or Kronecker or conjunctive or direct) product of two graphsG1 and G2 is a graph G1⊘G2 with vertex set VG1⊘G2=VG1⊘VG2 and edge set EG1⊘G2=a1,b1a2,b2, where a1,b1,a2,b2∈VG1⊘VG2 and(1)a1a2∈EG1∧b1b2∈EG2.
The connection number of a vertexa,b of G1⊘G2 is defined by τQ1⊘Q2a,b=τG1a+τG2b+τG1aτG2b. For more details, see Figure 1.Figure 1
(a)G1≅P4, (b) G2≅P4, and (c) tensor productP4⊘P4.
(a)(b)(c)Definition 2.
Strong product or normal productG1⊗G2 of two graphs G1 and G2 is obtained by taking the vertex set and edge set as VQ1⊗Q2=VQ1⊗VQ2 and EQ1⊗Q2=a1,b1a2,b2, where a1,b1,a2,b2∈VQ1⊗VQ2 with condition(2)eithera1=a2∈VG1∧b1b2∈EQ2orb1=b2∈VQ2∧a1a2∈EQ1ora1a2∈EQ1∧b1b2∈EQ2.
The connection number of a vertexa,b of Q1⊗Q2 is defined by(3)τG1⊗G2a,b=τG1a+τG2b+dG1aτG1a+dG2bτG2b+τG1aτG2b=τG1a1+dG1a+τG2b+τG2b1+dG2b+τG1a,if n1,n2≥4. For more details, see Figure 2.Figure 2
(a)G1≅C4 (b) G2≅P3, and (c) strong productC4⊗P3.
(a)(b)(c)Definition 3.
Symmetric difference of two graphsQ1 and Q2 is a graph G1⊖G2 with vertex set VQ1⊖Q2=VQ1⊖VQ2 and edge set EQ1⊖Q2=a1,b1a2,b2, where a1,b1,a2,b2∈VQ1⊖VQ2 and(4)a1a2∈EQ1orb1b2∈EQ2,but not both hold at the same time, respectively.
The connection number of a vertexa,b of G1⊖G2 is defined by(5)τG1⊖G2a,b=n1n2−1−n2dG1a−n1dG2b+2dG1adG2b=n2n1−dG1a+dG2b2dG1a−n1−1.
For more details, see Figure3.Figure 3
(a)G1≅C3, (b) G2≅P2, and (c) symmetric differenceC3⊖P2.
(a)(b)(c)Definition 4.
Disjunction of two graphsG1 and G2 is a graph G1⊕G2 with vertex set VQ1⊕Q2=VQ1⊕VQ2 and edge set EQ1⊕Q2=a1,b1a2,b2, where a1,b1,a2,b2∈VQ1⊕VQ2 and(6)a1a2∈EQ1orb1b2∈EQ2.
The connection number of a vertexa,b of G1⊕G2 is defined by(7)τG1⊕G2a,b=n1n2−1−n2dG1a−n1dG2b+dG1adG2b=n2n1−dG1a+dG2bdG1a−n1−1.
For more details, see Figure4.
The graph theory provides the significant tools in the field of modern chemistry that is exploited to develop several types of molecular graphs and also predicts their chemical properties. In 1972, Gutman and Trinajstić [19] defined the first degree-based (number of vertices at distance one) TI called the first Zagreb index to compute the total π-electron energy of the molecules in molecular graphs. There are several TIs in literature, but degree-based TIs are studied more than others (see [20–29]).
Ali and Trinajstić [30] restudied the concept of connection number-based (number of vertices at distance two) TIs that were also defined by Gutman and Trinajstić in the same paper in 1972 to compute the total electron energy of the alternant hydrocarbons. They recalled them as Zagreb connection indices and reported that the chemical capability of the Zagreb connection indices is better than the classical Zagreb indices for the thirteen physicochemical properties of octane isomers. After two years, a few works have delivered on these connection number-based descriptors. Ali et al. [31] computed the analysis of Zagreb connection indices and coindices for some chemical structures of operations on graphs. For further studies and properties of the Zagreb connection indices, we refer to [32–40]. Recently, Gong et al. [41, 42] developed blocking lot-streaming flow shop scheduling problems and dynamic interval multiobjective optimization problems. These problems have been considered in various studies which have a close relation with the topic considered in this paper. For more details, see [43, 44].
In this paper, we compute the analysis for the first Zagreb connection index and coindex of the resultant graphs in the shape of exact formulae and upper bounds in terms of their factor graphs, where resultant graphs are obtained by operations such as tensor product, strong product, symmetric difference, and disjunction. Moreover, at the end, an analysis of the first Zagreb connection index and coindex on the aforesaid operations is included with the help of their numerical values and graphical depictions.
The rest of the paper is as follows: Section2 represents the preliminary definitions and key results which are used in the main results, Section 3 contains the main results of product on graphs, and Section 4 includes the analysis and conclusions.Figure 4
(a)G1≅C3, (b) G2≅P2, and (c) disjunctionC3⋅P2.
(a)(b)(c)


## 2. Preliminaries

LetG=VG,EG be a simple graph such that the order and size of G are VG=n and EG=e. The distance da,b between any two vertices a and b of a graph G is equal to the length of a shortest path connecting them. For b∈VG and a positive integer p, Npb/G=u∈VG:da,b=p denotes the open p-neighborhood of b in a graph G, where dpb/G=Npb/G is called p-distance degree of a vertex b in any graph G. The degree of a vertex b in a graph G is the number of edges incident on it, and it is denoted by dGb. In particular:Ifp=1, d1b/G=N1b/G=dGb=degree ofb (number of vertices at distance one from b)Ifp=2, d2b/G=N2b/G=τGb= connection number of  b (number of vertices at distance two from b)The complement of a graphG is denoted by G¯. It is also simple with the same vertex set as of G, but edge set is defined as EG¯=ab:a,b∈VG∧ab∉EG. Thus, EG∪EG¯=EKn, where Kn is a complete graph of order n and size EKn=n2. Moreover, if EG=e, then EG¯=n2−e=μ and dG¯b=n−1−dGb, where dGb and dG¯b are the degrees of the vertex b in G and G¯, respectively. Now, throughout the paper, for two graphs G1 and G2, we assume that VG1=n1, VG2=n2, EG1=e1, and EG2=e2. Finally, it is important to note that Zagreb connection coindices of G are not Zagreb connection indices of G¯ because the connection number works according to G. For any terminology or notion which are not mentioned here, we refer to [45, 46].Definition 5.
For a graphG, the first Zagreb index M1G and second Zagreb index M2G are defined as(8)M1G=∑ab∈EGdGa+dGb,M2G=∑ab∈EGdGa×dGb.
These degree-based indices are defined by Gutman, Trinajstić, and Ruscic (see [19, 47]) which are frequently used to predict better outcomes of the various parameters related to the molecular networks such as chirality, complexity, entropy, heat energy, ZE-isomerism, heat capacity, absolute value of correlation coefficient, chromatographic, retention times in chromatographic, pH, and molar ratio [19, 48]. Symmetrical to these degree-based TIs, the connection-based TIs are discussed in Definitions 6 and 7.Definition 6.
For a graphG, the first Zagreb connection index ZC1G is defined as(9)ZC1G=∑b∈VGτGb2.Definition 7.
For a graphG, the modified first Zagreb connection index ZC1∗G and second Zagreb connection index ZC2G are defined as(10)ZC1∗G=∑ab∈EGτGa+τGb,ZC2G=∑ab∈EGτGa×τGb.Definition 8.
For a graphG, the first Zagreb coindex M¯1G and second Zagreb coindex M¯2G are defined as(11)M¯1G=∑ab∉EGdGa+dGb,M¯2G=∑ab∉EGdGa×dGb.
These coindices associated with the degree-based classical Zagreb indices are defined by Ashrafi et al. (see [6]). The coindices associated with the connection-based Zagreb indices are defined in Definition 9.Definition 9.
For a graphG, the first Zagreb connection coindex Z¯C1G and second Zagreb connection coindex Z¯C2G are defined as(12)Z¯C1G=∑ab∉EGτGa+τGb,Z¯C2G=∑ab∉EGτGa×τGb.
The modified coindices associated with the connection number-based Zagreb indices are defined in Definition10. These modified coindices associated with the connection number-based Zagreb indices are defined by Ali et al. (see [49]).Definition 10.
For a graphG, the modified first Zagreb connection coindex MZ¯C1G and the modified second Zagreb connection coindex MZ¯C2G are defined as(13)MZ¯C1K=∑uv∉EKdKuτKv+dKvτKu,MZ¯C2K=∑uv∉EKdKuτKu+dKvτKv.
The degree/connection-based coindices defined in Definitions8–10 study the various physicochemical and isomer properties of molecules on the basis of the adjacency and nonadjacency pairs of vertices in the molecular networks. For more details, see [6, 30, 31, 36].
Now, we present some important results which are used in the main results.Lemma 1 (see [50]).
LetG be a connected graph and G¯ be its complement with n vertices and e edges. Then,(a)
∑b∈VGdGb=2e(b)
M1G¯≥M1G−4Lemma 2 (see [36]).
LetG be a connected and C3,C4-free graph with n vertices and e edges. Then,(a)
∑b∈VG1τG1b=M1G1−2e1=A(b)
∑b∈VG2τG2b=M1G2−2e2=B


## 3. Main Results

This section contains the main results for the first Zagreb connection indexZC1 and first Zagreb connection coindex Z¯C1 of the product on graphs obtained under the operations of tensor product, strong product, symmetric difference, and disjunction.Theorem 1.
LetG1 and G2 be two connected and C3,C4-free graphs. Then, ZC1 and Z¯C1 of the tensor product G1⊘G2 are(14)ZC1G1⊘G2=n2ZC1G1+n1ZC1G2+ZC1G1ZC1G2+2AB+2AZC1G2+2BZC1G1,(15)Z¯C1G1⊘G2=2μ2Z¯C1G1+2μ1Z¯C1G2+2∑a1a2∉EG1∑b1b2∉EG2τG1a1τG2b1+τG1a2τG2b2.Proof.
SinceτQ1⊘Q2a,b=τG1a+τG2b+τG1aτG2b, where a∈VG1, b∈VG2, and a,b∈G1⊘G2,(16)ZC1G1⊘G2=∑a,b∈VG1⊘G2τG1⊘G2a,b2=∑a∈VG1∑b∈VG2τG1a+τG2b+τG1aτG2b2=∑a∈VG1∑b∈VG2τG12a+τG22b+τG12aτG22b+2τG1a+τG2b+2τG1aτG22b+2τG12aτG2b=∑a∈VG1n2τG12a+ZC1G2+τG12aZC1G2+2τG1aM1G2−2e2+2τG1aZC1G2+2τG12aM1G2−2e2.
Using Lemma2 (a) and (b),(17)ZC1G1⊘G2=n2ZC1G1+n1ZC1G2+ZC1G1ZC1G2+2AB+2AZC1G2+2BZC1G1,(18)Z¯C1G1⊘G2=∑a1,b1a2,b2∉EG1⊘G2τG1⊘G2a1,b1+τG1⊘G2a2,b2=2∑a1a2∉EG1∑b1b2∉EG2τG1a1+τG2b1+τG1a1τG2b1+τG1a2+τG2b2+τG1a2τG2b2=2∑a1a2∉EG1∑b1b2∉EG2τG1a1+τG1a2+τG2b1+τG2b2+τG1a1τG2b1+τG1a2τG2b2Z¯C1G1⊘G2=2μ2Z¯C1G1+2μ1Z¯C1G2+2∑a1a2∉EG1∑b1b2∉EG2τG1a1τG2b1+τG1a2τG2b2.Theorem 2.
LetG1 and G2 be two connected and C3,C4-free graphs. Then, ZC1 and Z¯C1 of the strong product G1⊗G2 are(19)ZC1G1⊗G2=2BZC1∗G1+2AZC1∗G2+2ZC1∗G1ZC1∗G2+n2+2BZC1G1+n1+2AZC1G2+ZC1G1ZC1G2+2AB+2n2+B∑a∈VG1dG1aτG12a+2n1+A∑b∈VG2dG2bτG22b+n2∑a∈VG1dG12aτG12a+n1∑b∈VG2dG22bτG22b,(20)Z¯C1G1⊗G2≤n2+2μ2MZ¯C2G1+n1+2μ1MZ¯C2G2+n2+2μ2+BZ¯C1G1+n1+2μ1+AZ¯C1G2+2μ2ZC1∗G1+μ1ZC1∗G2+2μ2A+μ1B+2∑a1a2∉EG1a1≠a2∧a1∼a2∑b1b2∉EG2b1≠b2∧b1∼b2τG1a1τG2b1+τG1a2τG2b2.Proof.
SinceτG1⊗G2a,b=τG1a1+dG1a+τG2b1+dG2b+τG1a, where a∈VG1, b∈VG2, and a,b∈G1⊗G2,(21)ZC1G1⊗G2=∑a,b∈VG1⊗G2τG1⊗G2a,b2=∑a∈VG1∑b∈VG2τG1a1+dG1a+τG2b1+dG2b+τG1a2=∑a∈VG1∑b∈VG2τG12a1+dG1a2+τG22b1+dG2b+τG1a2+2τG1aτG2b1+dG1a1+dG2b+τG1a=∑a∈VG1∑b∈VG2τG12a1+dG12a+2dG1a+τG22b1+dG2b2+τG12a+21+dG2bτG1a+2τG1aτG2b1+dG2b+τG1a+dG1a+dG1adG2b+dG1aτG1a=∑a∈VG1∑b∈VG2τG12a+dG12aτG12a+2dG1aτG12+τG22b+dG22bτG22b+dG2bτG22b+τG12aτG22b+2τG1aτG22b+2dG2bτG1aτG22b+2τG1aτG2b+2dG2bτG1aτG2b+2τG12aτG2b+2dG1aτG1aτG2b+2dG1aτG1adG2bτG2b+2dG1aτG12aτG2b.
Using Lemma2 (a) and (b),(22)ZC1G1⊗G2=2BZC1∗G1+2AZC1∗G2+2ZC1∗G1ZC1∗G2+n2+2BZC1G1+n1+2AZC1G2+ZC1G1ZC1G2+2AB+2n2+B∑a∈VG1dG1aτG12a+2n1+A∑b∈VG2dG2bτG22b+n2∑a∈VG1dG12aτG12a+n1∑b∈VG2dG22bτG22b,(23)Z-C1G1⊗G2=∑a1,b1a2,b2∉EG1⊗G2τG1⊗G2a1,b1+τG1⊗G2a2,b2=∑a∈VG1∑b1b2∉EG2τG1⊗G2a,b1+τG1⊗G2a,b2+∑b∈VG2∑a1a2∉EG1τG1⊗G2a1,b+τG1⊗G2a2,b+∑a1a2∉EG1a1≠a2∧a1∼a2∑b1b2∉EG2b1≠b2∧b1∼b2τG1⊗G2a1,b1+τG1⊗G2a2,b2.
We take(24)∑a∈VG1∑b1b2∉EG2τG1⊗G2a,b1+τG1⊗G2a,b2=∑a∈VG1∑b1b2∉EG2τG1a+τG2b1+dG1aτG1a+dG2b1τG2b1+τG1aτG2b1+τG1a+dτG2b2G1aτG1a+dG2b2τG2b2+τG1aτG2b2=∑a∈VG1∑b1b2∉EG22τG1a+τG2b1+τG2b2+2dG1aτG1a+dG2b1τG2b1+dG2b2τG2b2+τG1aτG2b1+τG2b2.
We know that(25)∑b1b2∉EG2=n22−e2=μ2=2μ2A+n1Z¯C1G2+2μ2ZC1∗G1+n1MZ¯C2G2+AZ¯C1G2.
Also, we take(26)∑b∈VG2∑a1a2∉EG1τG1⊗G2a1,b+τG1⊗G2a2,b∑b∈VG2∑a1a2∉EG1τG1a1+τG2b+dG1a1τG1a1+dG2bτG2b+τG1a1τG2b+τG1a2+τG2b+dG1a2τG1a2+dG2bτG2b+τG1a2τG2b=n2Z¯C1G1+2μ1B+n2MZ¯C2G1+2μ1ZC1∗G2+BZ¯C1G1.
Again, we take(27)∑a1a2∉EG1a1≠a2∧a1∼a2∑b1b2∉EG2b1≠b2∧b1∼b2τG1⊗G2a1,b1+τG1⊗G2a2,b2≤2∑a1a2∉EG1a1≠a2∧a1∼a2∑b1b2∉EG2b1≠b2∧b1∼b2τG1a1+τG2b1+dG1a1τG1a1+dG2b1τG2b1+τG1a1τG2b1+τG1a2+τG2b2+dG1a2τG1a2+dG2b2τG2b2+τG1a2τG2b2.Similarly=2μ2Z¯C1G1+2μ1Z¯C1G2+2μ2MZ¯C2G1+2μ1MZ¯C2G2+2∑a1a2∉EG1a1≠a2∧a1∼a2∑b1b2∉EG2b1≠b2∧b1∼b2τG1a1τG2b1+τG1a2τG2b2.
Consequently,(28)Z¯C1G1⊗G2≤n2+2μ2MZ¯C2G1+n1+2μ1MZ¯C2G2+n2+2μ2+BZ¯C1G1+n1+2μ1+AZ¯C1G2+2μ2ZC1∗G1+μ1ZC1∗G2+2μ2A+μ1B+2∑a1a2∉EG1a1≠a2∧a1∼a2∑b1b2∉EG2b1≠b2∧b1∼b2τG1a1τG2b1+τG1a2τG2b2.Theorem 3.
LetG1 and G2 be two connected and C3,C4-free graphs. Then, ZC1 and Z¯C1 of the symmetric difference G1⊖G2 are(29)ZC1G1⊖G2=n2n22−8e2M1G1+n1n12−8e1M1G2+4M1G1M1G2+n1n2n12n22−4n22e1−4n12e2−2n1n2+24e1e2+1+4n12e2+n22e1−4e1e2,(30)Z¯C1G1⊖G2=4e2−n22M¯1G1+4e1−n12M¯1G2+2μ1n1n22−n2−2n1e2+2μ2n12n2−n1−2n2e1.Proof.
SinceτG1⊖G2a,b=n2n1−dG1a+dG2b2dG1a−n1−1, where a∈VG1, b∈VG2, and a,b∈G1⊖G2,(31)ZC1G1⊖G2=∑a,b∈VG1⊖G2τG1⊖G2a,b2=∑a∈VG1∑b∈VG2n2n1−dG1a+dG2b2dG1a−n1−12=∑a∈VG1∑b∈VG2n22n1−dG1a2+dG22b2dG1a−n12+1+2n2dG2bn1−dG1a2dG1a−n1−2dG2b2dG1a−n1−2n2n1−dG1a=n13n23+n23M1G1−4n1n23e1+4M1G1M1G2+n13M1G2−8n1e1M1G2+n1n2+16n1n2e1e2−4n13n2e2−8n2e2M1G1+8n1n2e1e2−16e1e2+4n12e2−2n12n22+4n22e1,ZC1G1⊖G2=n2n22−8e2M1G1+n1n12−8e1M1G2+4M1G1M1G2+n1n2n12n22−4n22e1−4n12e2−2n1n2+24e1e2+1+4n12e2+n22e1−4e1e2.(32)Z¯C1G1⊖G2=∑a1,b1a2,b2∉EG1⊖G2τG1⊖G2a1,b1+τG1⊖G2a2,b2=∑a∈VG1∑b1b2∉EG2τG1⊖G2a,b1+τG1⊖G2a,b2+∑b∈VG2∑a1a2∉EG1τG1⊖G2a1,b+τG1⊖G2a2,b∑a∈VG1∑b1b2∉EG2τG1⊖G2a,b1+τG1⊖G2a,b2∑b∈VG2∑a1a2∉EG1τG1⊖G2a1,b+τG1⊖G2a2,bτG1⊖G2a1,b1+τG1⊖G2a2,b2.
We take(33)∑a∈VG1∑b1b2∉EG2τG1⊖G2a,b1+τG1⊖G2a,b2=∑a∈VG1∑b1b2∉EG2n1n2−1−n2dG1a−n1dG2b1+2dG1adG2b1+n1n2−1−n2dG1a−n1dG2b2+2dG1adG2b2.Similarly=2n12n2μ2−2n1μ2−4n2e1μ2−n12M¯1G2+4e1M¯1G2.
Also, we take(34)∑b∈VG2∑a1a2∉EG1τG1⊖G2a1,b+τG1⊖G2a2,b.Similarly=2n1n22μ1−2n2μ1−n22M¯1G1−4n1e2μ1+4e2M¯1G1.
Again, we take (null case)(35)N=∑a1a2∉EG1a1≠a2∧a1∼a2∑b1b2∉EG2b1≠b2∧b1↑b2τG1⊖G2a1,b1+τG1⊖G2a2,b2=0.
We further take (also null case)(36)N=∑b1b2∉EG1b1≠b2∧b1∼b2∑a1a2∉EG2a1≠a2∧a1↑a2τG1⊖G2a1,b1+τG1⊖G2a2,b2=0.
Consequently,(37)Z¯C1G1⊖G2=4e2−n22M¯1G1+4e1−n12M¯1G2+2μ1n1n22−n2−2n1e2+2μ2n12n2−n1−2n2e1.Theorem 4.
LetG1 and G2 be two connected and C3,C4-free graphs. Then, ZC1 and Z¯C1 of the disjunction G1⊕G2 are(38)ZC1G1⊕G2=n2n22−4e2M1G1+n1n12−4e1M1G2+M1G1M1G2+n1n2n12n22−4n22e1−4n12e2−2n1n2+16e1e2+1+4n12e2+n22e1−2e1e2,(39)Z¯C1G1⊕G2=2e2−n22−2n2μ2M¯1G1+2e1−n12−2n1μ1M¯1G2+2μ1n1n22−n2−2n1e2+2μ2n12n2−n1−2n2e1+4μ1μ2n1n2−1+2∑a1a2∉EG1a1≠a2∧a1∼a2∑b1b2∉EG2b1≠b2∧b1∼b2dG1a1dG2b1+dG1a2dG2b2.Proof.
SinceτG1⊕G2a,b=n2n1−dG1a+dG2bdG1a−n1−1, where a∈VG1, b∈VG2, and a,b∈G1⊕G2,(40)ZC1G1⊕G2=∑a,b∈VG1⊕G2τG1⊕G2a,b2=∑a∈VG1∑b∈VG2n2n1−dG1a+dG2bdG1a−n1−12.
Similarly,(41)ZC1G1⊕G2=n2n22−4e2M1G1+n1n12−4e1M1G2+M1G1M1G2+n1n2n12n22−4n22e1−4n12e2−2n1n2+16e1e2+1+4n12e2+n22e1−2e1e2.(42)Z¯C1G1⊕G2=∑a1,b1a2,b2∉EG1⊕G2τG1⊕G2a1,b1+τG1⊕G2a2,b2=∑a∈VG1∑b1b2∉EG2τG1⊕G2a,b1+τG1⊕G2a,b2+∑b∈VG2∑a1a2∉EG1τG1⊕G2a1,b+τG1⊕G2a2,b∑a1a2∉EG1a1≠a2∧a1∼a2∑b1b2∉EG2b1≠b2∧b1∼b2τG1⊕G2a1,b1+τG1⊕G2a2,b2∑a1a2∉EG1a1≠a2∧a1∼a2∑b1b2∉EG2b1≠b2∧b1↑b2τG1⊕G2a1,b1+τG1⊕G2a2,b2+∑b1b2∉EG1b1≠b2∧b1∼b2∑a1a2∉EG2a1≠a2∧a1↑a2τG1⊖G2a1,b1+τG1⊖G2a2,b2.
We take(43)∑a∈VG1∑b1b2∉EG2τG1⊕G2a,b1+τG1⊕G2a,b2=∑a∈VG1∑b1b2∉EG2n1n2−1−n2dG1a−n1dG2b1+dG1adG2b1+n1n2−1−n2dG1a−n1dG2b2+dG1adG2b2.Similarly=2n12n2μ2−2n1μ2−4n2e1μ2−n12M¯1G2+2e1M¯1G2.
Also, we take(44)∑a1a2∉EG1a1≠a2∧a1∼a2∑b1b2∉EG2b1≠b2∧b1∼b2τG1⊕G2a1,b1+τG1⊕G2a2,b2=2∑a1a2∉EG1a1≠a2∧a1∼a2∑b1b2∉EG2b1≠b2∧b1∼b2n1n2−1−n2dG1a1−n1dG2b1+dG1a1dG2b1+n1n2−1−n2dG1a2−n1dG2b2+dG1a2dG2b2.Similarly=4n1n2μ1μ2−4μ1μ2−2n2μ2M¯1G1−2n1μ1M¯1G2+2∑a1a2∉EG1a1≠a2∧a1∼a2∑b1b2∉EG2b1≠b2∧b1∼b2dG1a1dG2b1+dG1a2dG2b2.
Again, we take(45)∑b∈VG2∑a1a2∉EG1τG1⊕G2a1,b+τG1⊕G2a2,b,Similarly=2n1n22μ1−2n2μ1−n22M¯1G1−4n1e2μ1+2e2M¯1G1.
We further take (null case)(46)N=∑a1a2∉EG1a1≠a2∧a1∼a2∑b1b2∉EG2b1≠b2∧b1↑b2τG1⊕G2a1,b1+τG1⊕G2a2,b2=0.
Furthermore, we take (also null case)(47)N=∑b1b2∉EG1b1≠b2∧b1∼b2∑a1a2∉EG2a1≠a2∧a1↑a2τG1⊕G2a1,b1+τG1⊕G2a2,b2=0.
Consequently,(48)Z¯C1G1⊕G2=2e2−n22−2n2μ2M¯1G1+2e1−n12−2n1μ1M¯1G2+2μ1n1n22−n2−2n1e2+2μ2n12n2−n1−2n2e1+4μ1μ2n1n2−1+2∑a1a2∉EG1a1≠a2∧a1∼a2∑b1b2∉EG2b1≠b2∧b1∼b2dG1a1dG2b1+dG1a2dG2b2.


## 4. Analysis and Conclusions

In this section, we compute the analysis for the first Zagreb connection indexZC1 and coindex Z¯C1 of product on graphs such as tensor product, strong product, symmetric difference, and disjunction with the help of Tables 1–5 which are constructed by using numerical values of the aforesaid Zagreb index and coindex, respectively. The graphical depictions of the exact formulae and upper bounds for ZC1 and Z¯C1 are also depicted in Figures 5–9. Moreover, in this section, for particular cases of main results, we compute all the results in the shape of upper bounds as C3,C4 are not free graphs and G1andG2 are two undirected graphs.Table 1
Analysis for index and coindex of exact formulae and upper bounds ofθ1=Pm⊘Pn, λ1=Pm⊘Pn, and ψ1=Pm⊘Pn, respectively.
m=nZC1θ1ZC1λ1Z¯C1ψ1180–1602–4–42340136124212404305512800566940132490714961976132821802756182929923664240103932470030611500058643801261967156462137520857655214897210124650151055211800756Table 2
Analysis for index and coindex of exact formulae and upper bounds ofθ2=Pm⊘Pn, λ2=Pm⊘Pn, and ψ2=Pm⊘Pn, respectively.
m=nZC1θ2ZC1λ2Z¯C1ψ2198–1322256–408832189123445846444605115419797666192845521152729068915161884088157162164954742569927901070643970434961188585866742821210856836205148131305811569160941415464156104712015180742061798226Table 3
Analysis for index and coindex of exact formulae and upper bounds ofθ3=Pm⊘Pn, λ3=Pm⊘Pn, and ψ3=Pm⊘Pn, respectively.
m=nZC1θ3ZC1λ3Z¯C1ψ311233024340311615344491239026454348745888615332121822407440041809474081087362518890492398523345153441048406842902476811909652535337980121612304653455880132721756783379464144409092925010982415689478810785148148Table 4
Analysis for index and coindex of exact formulae and upper bounds ofθ4=Pm⊘Pn, λ4=Pm⊘Pn, and ψ4=Pm⊘Pn, respectively.
m=nZC1θ4ZC1λ4Z¯C1ψ41014102040−23401894044925883785292812371456611680213639107361203285856889366046841645092134726333287681044092882324692611842760103817252012151294012780107338132579280154291533601442107521832821275815667615321477287896Table 5
Particular numeric values of obtained results for index and coindex.
Product graphsZC1θZC1λZ¯C1ψP3⊘P34013612P4⊗P4584644460P3⊖P311615344P3⊕P34018940Figure 5
Analysis for index and coindex of tensor product with respect to Table1.Figure 6
Analysis for index and coindex of strong product with respect to Table2.Figure 7
Analysis for index and coindex of symmetric difference with respect to Table3.Figure 8
Analysis for index and coindex of disjunction with respect to Table4.Figure 9
Analysis of obtained results for product on graphs with respect to Table5.


### 4.1. Tensor Product

LetPm and Pn be two particular alkanes called by paths, then the tensor product Pm⊘Pn is obtained by the product of Pm and Pn. For m=3 and n=3, see Figure 10.Figure 10
(a)G1≅P3 (b) G2≅P3, and (c) tensor product (P3⊘P3).
(a)(b)(c)Using Theorem1, the exact formulae for the first Zagreb connection index ZC1θ1 and first Zagreb connection coindex Z¯C1ψ1 of tensor product are obtained as follows:(a)
ZC1Pm⊘Pn≤64mn−138m−138n+292(b)
Z¯C1Pm⊘Pn≤4mn−4m−6n+6The upper bounds for the first Zagreb connection indexZC1λ1 of tensor product are obtained as follows [39]:(49)ZC1Pm⊘Pn≤64mn−56m−124n+100.Table1 and Figure 5 depict the numerical and graphical behaviours of the analysis between exact formulae and upper bounds for the first Zagreb connection index and coindex of tensor product by using values m=n.


### 4.2. Strong Product

LetPm and Pn be two particular alkanes called by paths, then the strong product Pm⊗Pn is obtained by the product of Pm and Pn. For m=4 and n=4, see Figure 11.Figure 11
(a)G1≅P4 (b) G2≅P4, and (c) strong product (P4⊗P4).
(a)(b)(c)Using Theorem2, the exact formulae for the first Zagreb connection index ZC1θ2 and first Zagreb connection coindex Z¯C1ψ2 of strong product are obtained as follows:(a)
ZC1Pm⊗Pn≤102mn−174m−174n+344(b)
Z¯C1Pm⊗Pn≤40mn−26m−28n+36The upper bounds for first Zagreb connection indexZC1λ2 of strong product are obtained as follows [39]:(50)ZC1Pm⊗Pn≤4mn3+16mn2−5mn−4m−12n3−40n2+24n+4.Table2 and Figure 6 depict the numerical and graphical behaviours of the analysis between exact formulae and upper bounds for the first Zagreb connection index and coindex of strong product by using values m=n.


### 4.3. Symmetric Difference

LetPm and Pn be two particular alkanes called by paths, then the symmetric difference Pm⊖Pn is obtained by the product of Pm and Pn. For m=3 and n=3, see Figure 12.Figure 12
(a)G1≅P3 (b) G2≅P3, and (c) symmetric difference (P3⊖P3).
(a)(b)(c)Using Theorem3, the exact formulae for the first Zagreb connection index ZC1θ3 and first Zagreb connection coindex Z¯C1ψ3 of symmetric difference are obtained as follows:(51)ZC1Pm⊖Pn≤m3n3−4m3n2+8m3n−6m3−4m2n3+22m2n2−52m2n+44m2+8mn3−52mn2+137mn−128m−6n3+44n2−128n+128,(52)Z¯C1Pm⊖Pn≤4m2n2−9m2n+5m2−9mn2+20mn−12m+5n2−12n+8.The upper bounds for first Zagreb connection indexZC1λ3 of symmetric difference are obtained as follows [40]:(53)ZC1Pm⊖Pn≤59mn−88m−88n+150.Table3 and Figure 7 depict the numerical and graphical behaviours of the analysis between exact formulae and upper bounds for first Zagreb connection index and coindex of symmetric difference by using values m=n.


### 4.4. Disjunction

LetPm and Pn be two particular alkanes called by paths, then the disjunction Pm⊕Pn is obtained by the product of Pm and Pn. For m=3 and n=3, see Figure 13.Figure 13
(a)G1≅P3 (b) G2≅P3, and (c) disjunction (P3⊕P3).
(a)(b)(c)Using Theorem4, the exact formulae for the first Zagreb connection index ZC1θ4 and first Zagreb connection coindex Z¯C1ψ4 of disjunction are obtained as follows:(54)ZC1Pm⊕Pn≤m3n3−4m3n2+8m3n−6m3−4m2n3+14m2n2−28m2n+20m2+8mn3−28mn2+41mn−40m−6n3+36n2−40n+28,(55)Z¯C1Pm⊘Pn≤8m2n2−19m2n+7m2−19mn2+36mn−2m+7n2−4n−14.The upper bounds for first Zagreb connection indexZC1λ4 of disjunction are obtained as follows [39]:(56)ZC1Pm⊕Pn≤125mn−238m−238n+492.Table4 and Figure 8 depict the numerical and graphical behaviours of the analysis between exact formulae and upper bounds for the first Zagreb connection index and coindex of disjunction by using values m=n.Now, from Tables1–5 and Figures 5–9 and 14–16, we close our discussion with the following conclusions:The behaviours of all the connection number-based Zagreb index and coindex for operations on graphs such as tensor product, strong product, symmetric difference, and disjunction are increased in the following order, respectively, asZ¯C1ψ≥ZC1λ≥ZC1θ.For increasing values ofm and n, the upper bound for the first Zagreb connection index of products on graphs are working rapidly than all the exact formula for the first Zagreb connection index, respectively.In certain intervals of the values ofm and n, all the first Zagreb connection coindices attain the maximum values on increasing values of m and n. In Figures 5–8, we analyse that the first Zagreb connection coindex attains more upper layer than other TIs in all the operations.Table5 and Figure 9 interpret the particular analysis of the obtained results for index and coindex on operations such as tensor product, strong product, symmetric difference, and disjunction. This particular analysis also concludes that the first Zagreb connection coindex attains more upper layer than other TIs in all the operations.In particular, Figures14–16 interpret the exact formula for the first Zagreb connection index, upper bound for the first Zagreb connection index, and exact formula for the first Zagreb connection coindex which are dominant on operations from tensor product to disjunction, respectively. In addition, we analyse that the first Zagreb connection coindex of operation disjunction has attained more upper layer than all the other operations for connection number-based index and coindex.Figure 14
Analysis forZC1θ of exact formula.Figure 15
Analysis forZC1λ of upper bound.Figure 16
Analysis forZ¯C1ψ of exact formula.The investigation of these indices and coindices for the resultant graphs obtained from other operations of graphs (subtraction, switching, zig-zag product, addition, rooted product, modular product etc.) is still open.


## 4.1. Tensor Product

LetPm and Pn be two particular alkanes called by paths, then the tensor product Pm⊘Pn is obtained by the product of Pm and Pn. For m=3 and n=3, see Figure 10.Figure 10
(a)G1≅P3 (b) G2≅P3, and (c) tensor product (P3⊘P3).
(a)(b)(c)Using Theorem1, the exact formulae for the first Zagreb connection index ZC1θ1 and first Zagreb connection coindex Z¯C1ψ1 of tensor product are obtained as follows:(a)
ZC1Pm⊘Pn≤64mn−138m−138n+292(b)
Z¯C1Pm⊘Pn≤4mn−4m−6n+6The upper bounds for the first Zagreb connection indexZC1λ1 of tensor product are obtained as follows [39]:(49)ZC1Pm⊘Pn≤64mn−56m−124n+100.Table1 and Figure 5 depict the numerical and graphical behaviours of the analysis between exact formulae and upper bounds for the first Zagreb connection index and coindex of tensor product by using values m=n.


## 4.2. Strong Product

LetPm and Pn be two particular alkanes called by paths, then the strong product Pm⊗Pn is obtained by the product of Pm and Pn. For m=4 and n=4, see Figure 11.Figure 11
(a)G1≅P4 (b) G2≅P4, and (c) strong product (P4⊗P4).
(a)(b)(c)Using Theorem2, the exact formulae for the first Zagreb connection index ZC1θ2 and first Zagreb connection coindex Z¯C1ψ2 of strong product are obtained as follows:(a)
ZC1Pm⊗Pn≤102mn−174m−174n+344(b)
Z¯C1Pm⊗Pn≤40mn−26m−28n+36The upper bounds for first Zagreb connection indexZC1λ2 of strong product are obtained as follows [39]:(50)ZC1Pm⊗Pn≤4mn3+16mn2−5mn−4m−12n3−40n2+24n+4.Table2 and Figure 6 depict the numerical and graphical behaviours of the analysis between exact formulae and upper bounds for the first Zagreb connection index and coindex of strong product by using values m=n.


## 4.3. Symmetric Difference

LetPm and Pn be two particular alkanes called by paths, then the symmetric difference Pm⊖Pn is obtained by the product of Pm and Pn. For m=3 and n=3, see Figure 12.Figure 12
(a)G1≅P3 (b) G2≅P3, and (c) symmetric difference (P3⊖P3).
(a)(b)(c)Using Theorem3, the exact formulae for the first Zagreb connection index ZC1θ3 and first Zagreb connection coindex Z¯C1ψ3 of symmetric difference are obtained as follows:(51)ZC1Pm⊖Pn≤m3n3−4m3n2+8m3n−6m3−4m2n3+22m2n2−52m2n+44m2+8mn3−52mn2+137mn−128m−6n3+44n2−128n+128,(52)Z¯C1Pm⊖Pn≤4m2n2−9m2n+5m2−9mn2+20mn−12m+5n2−12n+8.The upper bounds for first Zagreb connection indexZC1λ3 of symmetric difference are obtained as follows [40]:(53)ZC1Pm⊖Pn≤59mn−88m−88n+150.Table3 and Figure 7 depict the numerical and graphical behaviours of the analysis between exact formulae and upper bounds for first Zagreb connection index and coindex of symmetric difference by using values m=n.


## 4.4. Disjunction

LetPm and Pn be two particular alkanes called by paths, then the disjunction Pm⊕Pn is obtained by the product of Pm and Pn. For m=3 and n=3, see Figure 13.Figure 13
(a)G1≅P3 (b) G2≅P3, and (c) disjunction (P3⊕P3).
(a)(b)(c)Using Theorem4, the exact formulae for the first Zagreb connection index ZC1θ4 and first Zagreb connection coindex Z¯C1ψ4 of disjunction are obtained as follows:(54)ZC1Pm⊕Pn≤m3n3−4m3n2+8m3n−6m3−4m2n3+14m2n2−28m2n+20m2+8mn3−28mn2+41mn−40m−6n3+36n2−40n+28,(55)Z¯C1Pm⊘Pn≤8m2n2−19m2n+7m2−19mn2+36mn−2m+7n2−4n−14.The upper bounds for first Zagreb connection indexZC1λ4 of disjunction are obtained as follows [39]:(56)ZC1Pm⊕Pn≤125mn−238m−238n+492.Table4 and Figure 8 depict the numerical and graphical behaviours of the analysis between exact formulae and upper bounds for the first Zagreb connection index and coindex of disjunction by using values m=n.Now, from Tables1–5 and Figures 5–9 and 14–16, we close our discussion with the following conclusions:The behaviours of all the connection number-based Zagreb index and coindex for operations on graphs such as tensor product, strong product, symmetric difference, and disjunction are increased in the following order, respectively, asZ¯C1ψ≥ZC1λ≥ZC1θ.For increasing values ofm and n, the upper bound for the first Zagreb connection index of products on graphs are working rapidly than all the exact formula for the first Zagreb connection index, respectively.In certain intervals of the values ofm and n, all the first Zagreb connection coindices attain the maximum values on increasing values of m and n. In Figures 5–8, we analyse that the first Zagreb connection coindex attains more upper layer than other TIs in all the operations.Table5 and Figure 9 interpret the particular analysis of the obtained results for index and coindex on operations such as tensor product, strong product, symmetric difference, and disjunction. This particular analysis also concludes that the first Zagreb connection coindex attains more upper layer than other TIs in all the operations.In particular, Figures14–16 interpret the exact formula for the first Zagreb connection index, upper bound for the first Zagreb connection index, and exact formula for the first Zagreb connection coindex which are dominant on operations from tensor product to disjunction, respectively. In addition, we analyse that the first Zagreb connection coindex of operation disjunction has attained more upper layer than all the other operations for connection number-based index and coindex.Figure 14
Analysis forZC1θ of exact formula.Figure 15
Analysis forZC1λ of upper bound.Figure 16
Analysis forZ¯C1ψ of exact formula.The investigation of these indices and coindices for the resultant graphs obtained from other operations of graphs (subtraction, switching, zig-zag product, addition, rooted product, modular product etc.) is still open.



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