Computing Analysis of Connection-Based Indices and Coindices for Product of Molecular Networks

Abstract: Gutman and Trinajstić (1972) defined the connection-number based Zagreb indices, where connection number is degree of a vertex at distance two, in order to find the electron energy of alternant hydrocarbons. These indices remain symmetric for the isomorphic (molecular) networks. For the prediction of physicochemical and symmetrical properties of octane isomers, these indices are restudied in 2018. In this paper, first and second Zagreb connection coindices are defined and obtained in the form of upper bounds for the resultant networks in the terms of different indices of their factor networks, where resultant networks are obtained from two networks by the product-related operations, such as cartesian, corona, and lexicographic. For the molecular networks linear polynomial chain, carbon nanotube, alkane, cycloalkane, fence, and closed fence, first and second Zagreb connection coindices are computed in the consequence of the obtained results. An analysis of Zagreb connection indices and coindices on the aforesaid molecular networks is also included with the help of their numerical values and graphical presentations that shows the symmetric behaviour of these indices and coindices with in certain intervals of order and size of the under study (molecular) networks.


Introduction
Topological indices (TIs) are functions that associate a numeric value with a finite, simple, and undirected network. The various types of TIs are widely used for the studies of the structural and chemical properties of the networks. These are also used in chemo-informatics modelings consisting of quantitative structures activity and property relationships that create a symmetrical link between a biological property and a molecular network. This symmetric relation can be shown mathematically as P = χ(N), where P is an activity or property, N is a molecular network, and χ is a function that depends upon the molecular network N, see [1,2]. Moreover, a number of drugs particles and the medical behaviors of the different compounds have established with the help of various TI's in the pharmaceutical industries, see [3]. In particular, the TIs called by connection based Zagreb indices are used to compute the correlation values among various octane isomers, such as acentric factor, connectivity, heat of evaporation, molecular weight, density, critical temperature, and stability, see [4,5].
Operations on networks play an important role to develop the new molecular networks from the old ones that are known as the resultant networks. Graovac et al. [6] was the first who used some operations on networks and computed exact formulae of Wiener index for the resultant networks.
In particular, Cartesian products of P m & P 2 and C m & P 2 present the polynomial chain and nanotube (TUC 4 (m, n)), respectively, alkane (C 3 H 8 ) is the corona product of P 3 and N 3 , cyclobutane (C 4 H 8 ) is the corona product of C 4 and N 2 , and lexicographic products of P m & P 2 and C m & P 2 are fence and closed fence, respectively, where P m , C m and N m are path, cycle and null networks of order m respectively. For further study, see [7][8][9][10][11][12][13]. Now, we define these operations, as follows: Definition 1. Cartesian product of two networks G 1 and G 2 is a network G 1 × G 2 with vertex-set: V(G 1 × G 2 ) = V(G 1 ) × V(G 2 ) and edge-set: . For more detail, see Figure 1.

Definition 2.
Corona product (G 1 G 2 ) of two networks G 1 and G 2 is obtained by taking one copy of G 1 and n 1 copies of G 2 (i.e., {G i 2 : 1 ≤ i ≤ n 1 }) then by joining each vertex of the ith copy of G 2 to the ith vertex of one copy of G 1 , where 1 ≤ i ≤ n 1 , |V(G 1 • G 2 )| = n 1 n 2 + n 1 and |E(G 1 • G 2 )| = e 1 + n 1 e 2 + n 1 n 2 . For more detail, see Figure 2.
Definition 3. Lexicographic product of two networks G 1 and G 2 is a graph G 1 · G 2 with vertex-set : . For more detail, see Figure 3.
Thus, the theory of networks gives the significant techniques in the field of modern chemistry that is exploited to develop the several types of molecular networks and also predicts their chemical properties. Gutman and Trinajstić [14] defined the first degree-based (number of vertices at distance one) TI called by the first Zagreb index to compute the total π-electron energy of the molecules in molecular networks. There are several TIs in literature but degree-based are studied more than others, see [15]. Recently, Ashrafi et al. [16] defined the concept of coindices associated with the classical Zagreb indices for the resultant networks of different operations. Relations between Zagreb coindices and some distance-based TIs are established in [17]. The multiplicative, first, second, third, and hyper Zagreb coindices with certain properties are defined in [18][19][20][21][22][23]. Munir et al. [24] found closed relations for M-polynomial of polyhex networks and also computed closed relation for degree-based TIs of networks. Moreover, the various degree-based TIs of different networks, such as icosahedral honey comb, carbon nanotubes, oxide, rhombus type silicate, hexagonal, octahedral, neural, and metal-organic, are computed in [25][26][27][28][29].
In 2018, the concept of connection-based (number of vertices at distance two) TIs is restudied [30]. The origin of these indices can be found in the work of Gutman and Trinajstić [14]. It is found that the correlation values for the various physicochemical and symmetrical properties of the octane isomers measured by Zagreb connection indices are better than the classical Zagreb indices. Ali and Javaid [31] computed the formulae for Zagreb connection indices of disjunction and symmetric difference operations on networks. For further studies of these indices on acyclic (alkane), unicycle, product, subdivided, and semi-total point networks, we refer to [32][33][34][35][36][37].
In this paper, we compute the coindices associated with the first and second Zagreb connection indices of the resultant networks as upper bounds in the terms of their factor networks, where resultant networks are obtained by Cartesian, corona and lexicographic products of two networks. As the consequences of these results, first and second Zagreb connection coindices of the linear polynomial chain, carbon nanotube, alkane, cyclobutane, fence, and closed fence networks are also obtained. Moreover, at the end, an analysis of connection-based Zagreb indices and coindices on the aforesaid molecular networks is included with the help of their numerical values and graphical presentations.
Moreover, in this note, Section 2 represents the preliminaries and some important lemmas, Section 3 covers the few molecular networks, Section 4 contains the main results of product based networks, and Section 5 includes the applications, comparisons, and conclusions.

Preliminaries
For the vertex set V(G) and edge set E(G) ⊆ V(G) × V(G), we present a simple and undirected (molecular) network by G = (V(G), E(G)), such that |V(G)| and |E(G)| are order and size of G, respectively. A network denoted by N is called null if it has at least exactly one vertex and there exists no edge. A null network becomes trivial if it has one vertex. The complement of a network G is denoted byḠ. It is also simple with same vertex set as of G, but edge set is defined as where K n is a complete network of order n and size |E(K n )| = ( n 2 ). Moreover, if |E(G)| = e, then |E(Ḡ)| = ( n 2 ) − e = µ and dḠ(b) = n − 1 − d G (b), where d G (b) and dḠ(b) are the degrees of the vertex b in G andḠ, respectively. In addition, we assume that τ G (b) denotes the connection number (number of vertices at distance 2) of the vertex b in G (distance between two vertices is number of edges of the shortest path between them). Now, throughout the paper, for two networks 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 ofḠ, because the connection number works according to G. For further basic terminologies, see [38].

Definition 4.
For a (molecular) network G, the first Zagreb index (M 1 (G)) and second Zagreb index (M 2 (G)) are defined as Gutman, Trinajstić, and Ruscic [14,39] defined these indices 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, see [4,14,29,40]. The connection-based TIs are discussed, as follows: Definition 5. For a (molecular) network G, the modified first Zagreb connection index (ZC * 1 (G)) and second Zagreb connection index (ZC 2 (G)) are defined as Definition 6. For a (molecular) network G, the first Zagreb coindex (M 1 (G)) and second Zagreb coindex (M 2 (G)) are defined as These coindices that are associated with the degree-based classical Zagreb indices are defined by Ashrafi et al. see [16]. The coindices associated with the Zagreb connection indices are defined in Definition 7.

Definition 7.
For a (molecular) network G, the first Zagreb connection coindex (ZC 1 (G)) and second Zagreb connection coindex (ZC 2 (G)) are defined as The degree/connection based coindices defined in Definitions 6 and 7 study the various physicochemical and isomer properties of molecules on the bases of the adjacency and non-adjacency pairs of vertices in the molecular networks. For more detail, see [16,30,36,41]. Now, we present some important results that are used in the main results.
Lemma 1 (see [42]). Let G be a connected network with n vertices and e edges. Subsequently, Lemma 2 (see [38]). Let G be a connected network with n vertices and e edges. Afterwards, ∑ b∈V(G) Lemma 3 (see [36])). Let G be a connected network with n vertices and e edges. Subsequently, ∑ b∈V(G)

A Few Molecular Networks
In this section, we define a few molecular networks, as follows: • Alkanes (hydrocarbon compounds) are organic compounds consisting of carbon atoms joined by single bounds. The simple and Lewis networks of alkanes are given in Figure 4. Moreover, are examples of alkanes that are given in Figure 5. This alkane series continues and follows general formula as C n H 2n+2 .

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Cyclic compounds are molecules consisting of closed chain (ring) of at least three carbon atoms.
If the closed chain has only carbon atoms, then it is an organic cyclic molecule that is called by homocyclic compound. If the closed chain has both carbon and non-carbon atoms, then it is an inorganic cyclic molecule that is called the heterocyclic compound. Moreover, Cycloalkanes (C n H 2n ) are the isomers of alkenes consisting of exactly one cyclic compound joined by a single bond. Figure 6a,b presents the cyclic compounds (homocyclic and heterocyclic, respectively).

Main Results
The first Zagreb connection coindex (ZC 1 ) and second Zagreb connection coindex (ZC 2 ) of the product based networks obtained under the operations of Cartesian product, corona product and lexicographic product are studied in third section. Theorem 1. Let G 1 and G 2 be two networks. Then,ZC 1 andZC 2 of the Cartesian product Consequently, .
We know that, We know that, ∑ Consequently, Theorem 2. Let G 1 and G 2 be two networks. Subsequently,ZC 1 andZC 2 of the corona product G 1 G 2 are where equality holds iff G 1 G 2 is a {C 3 , C 4 }− free network.

Cartesian Product
(1) Polynomial chains: Let P m and P n be two particular path-alkanes, then the polynomial chains (P m × P n ) are obtained by the Cartesian product of P m and P n . For m = 6 and n = 2, see Figure 7. Using Theorem 1, Zagreb connection coindices (ZC 1 andZC 2 ) of polynomial chains are obtained, as follows: The Zagreb connection indices (ZC * 1 and ZC 2 ) of polynomial chains are as follows [43]: Table 1 and Figure 8 present the numerical and graphical behaviours of the upper bound values of Zagreb connection indices and Zagreb connection coindices for polynomial chains with respect to different values of m and n.
(2) ZC 2 (P m × C n ) ≤ 128mn − 238n. Table 2 and Figure 10 present the numerical and graphical behaviours of the Zagreb connection indices coindices for carbon nanotubes with respect to different values of m and n.  (TUC 4 (m, n)) of θ 2 = P m × C n .
The Zagreb connection indices (ZC * 1 and ZC 2 ) of alkanes are as follows [43]: ZC 2 (P m N n ) = 2mn 3 − 2n 3 + 8mn 2 − 16n 2 + 10mn − 26n. Table 3 and Figure 12 present the numerical and graphical behaviours of the Zagreb connection indices and coindices for alkanes with respect to different values of m and n. Table 3. Alkanes of θ 3 = P m N n .   Table 3 with respect to indices and coindices.
(4) Cyclobutane (C 4 H 8 ): Let C m and N n be a cycle and a null graph, then Cyclobutanes (C m N n ) are obtained by the corona product of C m and N n . The corona product has a chemical sense only when for arbitrary m > 0, n = 1 and n = 2 provide equivalence chemical networks of cycloalkenes and cycloalkanes, respectively. Besides this sense, for n > 2 see no chemical context (cyclic compounds) of corona product. For m = 4 and n = 2, see Figure 13.
Using Theorem 2, Zagreb connection coindices (ZC 1 andZC 2 ) of cyclobutanes are obtained, as follows: The Zagreb connection indices (ZC * 1 and ZC 2 ) of cyclobutanes are as follows [43]: ZC 2 (C m N n ) ≤ 2mn 3 + 8mn 2 + 10mn + 4m. Table 4 and Figure 14 present the numerical and graphical behaviours of the upper bound values of Zagreb connection indices and coindices for cyclobutanes with respect to different values of m and n.  Table 4 with respect to indices and coindices.

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In particular, Figures 19-22 present that first Zagreb connection index, second Zagreb connection index, first Zagreb connection coindex, and second Zagreb connection coindex are dominant and auxiliary or incapable for the molecular networks from polynomial chain to closed fence, respectively. Moreover, we analyse that last molecular network i.e., closed fence has attain more upper layer than all other molecular networks for connection-based indices and coindices.
The investigation of these molecular descriptors for the resultant networks obtained from other operations of networks (switching, addition, rooted product, and Zig-zag product, etc.) is still open.