1. 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
, where
is an activity or property,
is a molecular network, and
is a function that depends upon the molecular network
, 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
&
and
&
present the polynomial chain and nanotube (
), respectively, alkane (
) is the corona product of
and
, cyclobutane (
) is the corona product of
and
, and lexicographic products of
&
and
&
are fence and closed fence, respectively, where
,
and
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 and is a network with vertex-set: and edge-set: ; where and
Either or . For more detail, see Figure 1.
Definition 2. Corona product of two networks and is obtained by taking one copy of and copies of (i.e., ) then by joining each vertex of the ith
copy of to the ith
vertex of one copy of , where , and . For more detail, see Figure 2. Definition 3. Lexicographic product of two networks and is a graph with vertex-set: and edge-set: ; where and
Either or . 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
-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.
2. Preliminaries
For the vertex set and edge set , we present a simple and undirected (molecular) network by , such that and 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 by . It is also simple with same vertex set as of G, but edge set is defined as , thus , where is a complete network of order n and size . Moreover, if , then and , where and are the degrees of the vertex b in G and , respectively. In addition, we assume that 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
and
, we assume that
,
,
and
. Finally, it is important to note that Zagreb connection coindices of
G are not Zagreb connection indices of
, 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 and second Zagreb index 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 and second Zagreb connection index are defined as Definition 6. For a (molecular) network G, the first Zagreb coindex and second Zagreb coindex 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 and second Zagreb connection coindex 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, , where equality holds if and only if G is a free network. Lemma 2 (see [
38])
. Let G be a connected network with n vertices and e edges. Afterwards, . Lemma 3 (see [
36]))
. Let G be a connected network with n vertices and e edges. Subsequently, where equality holds iff G is a free network. 5. Applications, Comparisons and Conclusions
In this section, we compute Zagreb connection coindices
for the particular molecular networks, such as carbon nanotube, linear polynomial chain, alkane, cyclobutane, fence, and closed fence (see Figures 7–9, 11, 13, 15, and 17) as the consequence of the main results obtained in
Section 4. We also construct the
Table 1,
Table 2,
Table 3,
Table 4,
Table 5 and
Table 6 with the help of the numerical values of Zagreb connection coindices
and Zagreb connection indices
for the aforesaid molecular networks. The graphical presentations of the Zagreb connection coindices
and Zagreb connection indices
for these molecular networks are also presented in Figures 8, 10, 12, 14, 16, and 18. Assume that
&
be two null networks (with order 2 & 3),
,
,
&
be four particular alkanes called by paths (with order 2, 3, 4, & 6) and
,
&
be cycles (with order 4, 5, & 6).
5.1. Cartesian Product
(1) Polynomial chains: Let
and
be two particular path- alkanes, then the polynomial chains
are obtained by the Cartesian product of
and
. For
and
, see
Figure 7.
Using Theorem 1, Zagreb connection coindices and of polynomial chains are obtained, as follows:
- (a)
- (b)
The Zagreb connection indices
and
of polynomial chains are as follows [
43]:
of polynomial chains: (1) If &, ; (2) If &,
of polynomial chains: (1) If &, ; (2) If &, ; (3) If &, ; (4) If &, .
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) Carbon Nanotubes : Let
and
be a particular alkane and cycloalkane called by path and cycle, then carbon nanotubes
are obtained by the cartesian product of
and
. For
and
, see
Figure 9.
Using Theorem 1, Zagreb connection coindices and of carbon nanotubes are obtained as follows:
- (a)
- (b)
The Zagreb connection indices
and
of carbon nanotubes are as follows [
43]:
- (1)
- (2)
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.
5.2. Corona Product
(3) Alkane Let
and
be a particular alkane called by paths and a null graph, then the alkanes
are obtained by the corona product of
and
. The corona product only has a chemical sense when for arbitrary
,
, and
provide equivalence chemical networks of alkenes and alkanes, respectively. Besides this sense, for
, see no chemical context of corona product. For
and
, see
Figure 11.
Using Theorem 2, Zagreb connection coindices and of alkanes are obtained as follows:
- (a)
- (b)
The Zagreb connection indices
and
of alkanes are as follows [
43]:
- (1)
- (2)
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.
(4) Cyclobutane (): Let
and
be a cycle and a null graph, then Cyclobutanes
are obtained by the corona product of
and
. The corona product has a chemical sense only when for arbitrary
,
and
provide equivalence chemical networks of cycloalkenes and cycloalkanes, respectively. Besides this sense, for
see no chemical context (cyclic compounds) of corona product. For
and
, see
Figure 13.
Using Theorem 2, Zagreb connection coindices and of cyclobutanes are obtained, as follows:
- (a)
- (b)
The Zagreb connection indices
and
of cyclobutanes are as follows [
43]:
- (1)
- (2)
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.
5.3. Lexicographic Product
(5) Fence: Let
and
be two particular path-alkanes, then the fence
are obtained by the lexicographic product of
and
. For
and
, see
Figure 15.
Using Theorem 3, Zagreb connection coindices and of fence are obtained, as follows:
- (a)
- (b)
The Zagreb connection indices
and
of fence are as follows [
43]:
- (1)
- (2)
Table 5 and
Figure 16 present the numerical and graphical behaviours of the upper bound values of Zagreb connection indices and coindices for fence with respect to different values of
m and
n.
(6) Closed fence: Let
and
be a cycle and a particular path-alkane, then closed fence
is obtained by the lexicographic product of
and
. For
and
, see
Figure 17.
Using Theorem 3, Zagreb connection coindices and of closed fence are obtained, as follows:
- (a)
- (b)
The Zagreb connection indices
and
of the closed fence are as follows [
43]:
- (1)
- (2)
Table 6 and
Figure 18 present the numerical and graphical behaviours of the upper bound values of Zagreb connection indices and coindices for closed fence with respect to different values of
m and
n.
Now, from
Table 1,
Table 2,
Table 3,
Table 4,
Table 5 and
Table 6 and
Figure 8,
Figure 10,
Figure 12,
Figure 14,
Figure 16, and
Figure 18,
Figure 19,
Figure 20,
Figure 21 and
Figure 22, we close our discussion with the following conclusions:
The behaviours of all the connection-based Zagreb indices and coindices for the molecular networks (polynomial chain, carbon nanotube, alkane, cycloalkane, fence, and closed fence) are symmetrise with some less or more values and the following orderings:
(i) (for polynomial chain), (ii) (for carbon nanotubes, fence and closed fence) and (iii) (for alkane and cycloalkane).
For increasing values of m and n in all of the molecular networks (polynomial chain, carbon nanotube, alkane, cycloalkane, fence, and closed fence), the second Zagreb connection index, and the first Zagreb connection coindex are responding rapidly, and steadily, respectively.
In the certain intervals of the values of m and n, all the connection-based indices and coindices attain the maximum and minimum values. These values are also lifting up in the intervals on increasing values of m and n in such a way that the response of maximum values is more rapid than the minimum values. In addition, we analyse that second the Zagreb connection index has attained more upper layer than other TIs in all pf the molecular networks.
In particular,
Figure 19,
Figure 20,
Figure 21 and
Figure 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.