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

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 Trinajsti´c studied the ﬁrst Zagreb connection index ( ZC 1 ) to evaluate the value of a molecule. This concept was ﬁrst studied by Gutman and Trinajsti´c 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.


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. e reformulated, multiplicative, hyper, first, second, and third Zagreb coindices with certain properties are defined in [8][9][10][11][12][13]. Relations between Zagreb coindices and some distance-based indices are computed in [14]. For more details, see [15][16][17][18]. For this purpose, we can define some operations such as tensor product, strong product, symmetric difference, and disjunction in the following definitions.
Definition 2. Strong product or normal product (G 1 ⊗ G 2 ) of two graphs G 1 and G 2 is obtained by taking the vertex set and edge set as (2) e connection number of a vertex (a, b) of Q 1 ⊗ Q 2 is defined by if n 1 , n 2 ≥ 4. For more details, see Figure 2.
but not both hold at the same time, respectively. e connection number of a vertex (a, b) of G 1 ⊖G 2 is defined by For more details, see Figure 3.
Definition 4. Disjunction of two graphs G 1 and G 2 is a graph e connection number of a vertex (a, b) of G 1 ⊕G 2 is defined by For more details, see Figure 4.
e 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.
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. ey 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][33][34][35][36][37][38][39][40]. Recently, Gong et al. [41,42] developed blocking lot-streaming flow shop scheduling problems and dynamic interval multiobjective optimization problems. ese 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. e rest of the paper is as follows: Section 2 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.

Preliminaries
Let G � (V(G), E(G)) be a simple graph such that the order and size of G are |V(G)| � n and |E(G)| � e. e distance d(a, 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 ∈ V(G) and a positive integer p, in any graph G. e degree of a vertex b in a graph G is the number of edges incident on it, and it is denoted by d G (b). In particular: e complement of a graph G is denoted by G. It is also simple with the same vertex set as of G, but edge set is defined as Figure 1: (a) G 1 � P 4 , (b) G 2 � P 4 , and (c) tensor product (P 4 ⊘P 4 ).   (1, s) Figure 3: (a) G 1 � C 3 , (b) G 2 � P 2 , and (c) symmetric difference (C 3 ⊖P 2 ).

Mathematical Problems in Engineering 3
order n and size |E(K n )| � where d G (b) and d G (b) are the degrees of the vertex b in G and G, respectively. Now, throughout the paper, for two graphs G 1 and G 2 , we assume that |V(G 1 )| � n 1 , |V(G 2 )| � n 2 , |E(G 1 )| � e 1 , and |E(G 2 )| � e 2 . 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 graph G, the first Zagreb index (M 1 (G)) and second Zagreb index (M 2 (G)) are defined as ese 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 graph G, the first Zagreb connection index (ZC 1 (G)) is defined as Definition 7. For a graph G, the modified first Zagreb connection index (ZC * 1 (G)) and second Zagreb connection index (ZC 2 (G)) are defined as Definition 8. For a graph G, the first Zagreb coindex (M 1 (G)) and second Zagreb coindex (M 2 (G)) are defined as ese coindices associated with the degree-based classical Zagreb indices are defined by Ashrafi et al. (see [6]). e coindices associated with the connection-based Zagreb indices are defined in Definition 9.
Definition 9. For a graph G, the first Zagreb connection coindex (ZC 1 (G)) and second Zagreb connection coindex (ZC 2 (G)) are defined as e modified coindices associated with the connection number-based Zagreb indices are defined in Definition 10.
ese modified coindices associated with the connection number-based Zagreb indices are defined by Ali et al. (see [49]).
Definition 10. For a graph G, the modified first Zagreb connection coindex (MZC 1 (G)) and the modified second Zagreb connection coindex (MZC 2 (G)) are defined as e degree/connection-based coindices defined in Definitions 8-10 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]). Let G be a connected graph and G be its complement with n vertices and e edges. en, Lemma 2 (see [36]). Let G be a connected and C 3 , C 4 -free graph with n vertices and e edges. en,

Main Results
is section contains the main results for the first Zagreb connection index (ZC 1 ) and first Zagreb connection coindex (ZC 1 ) of the product on graphs obtained under the operations of tensor product, strong product, symmetric difference, and disjunction. Theorem 1. Let G 1 and G 2 be two connected and C 3 , C 4 -free graphs. en, ZC 1 and ZC 1 of the tensor product (G 1 ⊘G 2 ) are □ Mathematical Problems in Engineering Theorem 2. Let G 1 and G 2 be two connected and C 3 , C 4 -free graphs. en, ZC 1 and ZC 1 of the strong product (G 1 ⊗ G 2 ) are 6 Mathematical Problems in Engineering Using Lemma 2 (a) and (b), We take We know that Also, we take Mathematical Problems in Engineering 7 Again, we take Consequently, □ Theorem 3. Let G 1 and G 2 be two connected and C 3 , C 4 -free graphs. en, ZC 1 and ZC 1 of the symmetric difference (G 1 ⊖G 2 ) are Mathematical Problems in Engineering + 16n 1 n 2 e 1 e 2 − 4n 3 1 n 2 e 2 − 8n 2 e 2 M 1 G 1 + 8n 1 n 2 e 1 e 2 − 16e 1 e 2 + 4n 2 1 e 2 − 2n 2 1 n 2 2 + 4n 2 2 e 1 , We take Also, we take Similarly � 2n 1 n 2 2 μ 1 − 2n 2 μ 1 − n 2 2 M 1 G 1 − 4n 1 e 2 μ 1 + 4e 2 M 1 G 1 .

Analysis and Conclusions
In this section, we compute the analysis for the first Zagreb connection index (ZC 1 ) and coindex (ZC 1 ) 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. e graphical depictions of the exact formulae and upper bounds for ZC 1 and ZC 1 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 C 3 , C 4 are not free graphs and G 1 and G 2 are two undirected graphs.

Tensor Product.
Let P m and P n be two particular alkanes called by paths, then the tensor product (P m ⊘P n ) is obtained by the product of P m and P n . For m � 3 and n � 3, see Figure 10.
Using eorem 1, the exact formulae for the first Zagreb connection index (ZC 1 (θ 1 )) and first Zagreb connection coindex (ZC 1 (ψ 1 )) of tensor product are obtained as follows: e upper bounds for the first Zagreb connection index (ZC 1 (λ 1 )) of tensor product are obtained as follows [39]: Table 1 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.

Strong Product.
Let P m and P n be two particular alkanes called by paths, then the strong product (P m ⊗ P n ) is obtained by the product of P m and P n . For m � 4 and n � 4, see Figure 11.
Using eorem 2, the exact formulae for the first Zagreb connection index (ZC 1 (θ 2 )) and first Zagreb connection coindex (ZC 1 (ψ 2 )) of strong product are obtained as follows: e upper bounds for first Zagreb connection index (ZC 1 (λ 2 )) of strong product are obtained as follows [39]:  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.

Symmetric Difference.
Let P m and P n be two particular alkanes called by paths, then the symmetric difference (P m ⊖P n ) is obtained by the product of P m and P n . For m � 3 and n � 3, see Figure 12. Table 1: Analysis for index and coindex of exact formulae and upper bounds of θ 1 � P m ⊘P n , λ 1 � P m ⊘P n , and ψ 1 � P m ⊘P n , respectively.
Using eorem 3, the exact formulae for the first Zagreb connection index (ZC 1 (θ 3 )) and first Zagreb connection coindex (ZC 1 (ψ 3 )) of symmetric difference are obtained as follows: e upper bounds for first Zagreb connection index (ZC 1 (λ 3 )) of symmetric difference are obtained as follows [40]: Table 3 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.

Disjunction.
Let P m and P n be two particular alkanes called by paths, then the disjunction (P m ⊕P n ) is obtained by the product of P m and P n . For m � 3 and n � 3, see Figure 13.
For increasing values of m 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 of m 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 Mathematical Problems in Engineering attains more upper layer than other TIs in all the operations. Table 5 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. is particular analysis also concludes that the first Zagreb connection coindex attains more upper layer than other TIs in all the operations. In particular, Figures 14-16 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. e 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.

Data Availability
All data used to support the findings of this study are included within the article. However, additional data will be made available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.