Modified Zagreb connection indices of the T-sum graphs

Abstract: The quantitative structures activity relationships (QSAR) and quantitative structures property relationships (QSPR) between the chemical compounds are studied with the help of topological indices (TI’s) which are the fixed real numbers directly linked with the molecular graphs. Gutman and Trinajstic (1972) defined the first degree based TI to measure the total π-electrone energy of a molecular graph. Recently, Ali and Trinajstic (2018) restudied the connection based TI’s such as first Zagreb connection index, second Zagreb connection index and modified first Zagreb connection index to find entropy and accentric factor of the octane isomers. In this paper, we study the modified second Zagreb connection index and modified third Zagreb connection index on the T-sum (molecular) graphs obtained by the operations of subdivision and product on two graphs. At the end, as the applications of the obtained results for the modified Zagreb connection indices of the T-sum graphs of the particular classes of alkanes are also included. Mainly, a comparision among the Zagreb indices, Zagreb connection indices and modified Zagreb connection indices of the T-sum graphs of the particular classes of alkanes is performed with the help of numerical tables, 3D plots and line graphs using the statistical tools.


Introduction
vertex and edge are equal to chemical terms atom and bond respectively. Briefly, a TI is a numerical value for correlation between chemical structure and various physical properties of biological activity or chemical reactivity. So, these TI's predict the chemical and physical aspects that encoded or shown in the molecular graphs such as heat of formation, heat of evaporation, polarizability, solubility, surface tension, connectivity, critical temperature, boiling, melting and freezing point. The medical behaviors of the different drugs and their compounds, crystalline materials and their nano materials which are very important for the pharmaceutical fields, chemical industries and its widely extended networks have studied with the help of various TI's (Gonzalez-Diaz et al., 2007;Hall and Kier, 1976;Matamala and Estrada, 2005).
In addition, the quantitative structures activity relationships (QSAR) and quantitative structures property relationships (QSPR) are useful in the study of molecules with the help of these TI's. For more details, we refer to Consonni (2009, 2010) and Yan et al. (2015). It is also investigated that these relationships play an important role in the subject of cheminformatics (Todeschini and Consonni, 2009).
Wiener defined first TI when he was working on the boiling point of paraffin (Wiener, 1947). Classical Zagreb indices was defined in Gutman and Trinajstic (1972) and Gutman et al. (1975). These are very well known and frequently used in the study of chemical graph theory. These are called first Zagreb index (M 1 (G)) and second Zagreb index (M 2 (G)). Furtula and Gutman (2015) defined another degree based TI and they called it as third Zagreb index (M 3 (G)). It was appeared after a long gap, due to that instance, (M 3 (G)) -index is also known as forgotten index (F(G)). At present time, so many TI's have been explored with their properties and they have revolutionized the fruitful results in the study of science especially in the lattest field of cheminformatics that is the combination of three subjects Mathematics, Chemistry and Information Technology (Borovicanin et al., 2017;Das and Gutman, 2004;Liu et al., 2019aShirinivas et al., 2010;Todeschini and Consonni, 2002), Therefore, degree based TI's are more studied and well applicabled (Du et al., 2019;Liu et al., 2019bLiu et al., , 2020Tang et al., 2019).
Recently, Ali and Trinajstic studied the Zagreb connection indices and also checked their capability on thirteen physicochemical properties of octane isomers. They also reported that Zagreb connection indices has better correlation value than classical Zagreb indices (Ali and Trinajstic, 2018). Till now, many work have been started on this descriptor such as Du et al. (2019), defined extremal alkanes to use modified first Zagreb connection index. Ducoffe et al. (2018aDucoffe et al. ( , 2018b presented their work on the 16th cologne-twente workshop on graphs and combinatorial optimization, France, on the topic of extremal graphs with respect to the modified first Zagreb connection index. Shao et al. (2018) studied extremal graphs for alkanes and cycloalkanes under some conditions. Tang et al. (2019) also used Zagreb connection indices and modified first Zagreb connection index to find the new results of the T-sum graphs.
In the computational graph theory, the operations on different graphs have to play an important role in the generalized optimization of graphs (Cvetkocic et al., 1980;Shao et al., 2012). By the use of various operations, a variety of graphs can be developed from the simplest graphs that show as their novel constructed pillars. Supported by this, many researchers have studied the well-frame families of graphs under the subdivision-related operations on graphs. For this ambition, Yan et al. (2007) studied the subdivisionrelated operations on graphs and computed the Wiener index of the resultant graphs. Eliasi and Taeri (2009) used these operations to develop new sums of graphs. Deng et al. (2016), Akhter and Imran (2017), Shirdel et al. (2013), and Liu et al. (2019cLiu et al. ( , 2019d computed the first Zagreb index (M 1 (G)), second Zagreb index (M 2 (G)), third Zagreb index (M 3 (G)), hyper-Zagreb index (HM(G)) and computed the exact formulas of first general Zagreb index (M α (G), where α is a real number) of the T-sum graphs and also computed the generalized T k -sum graphs. Recently, Tang et al. (2019) derived the exact formulas of the Zagreb connection indices ( ) * of the T-sum graphs with the help of these subdivision-related operations.
In this paper, we extend the study of Tang  of the T-sum graphs G 1 + T G 2 which are achieved by the cartesian product of T(G 1 ) and G 2 , where T ϵ {T 1 ,T 2 }, G 1 and G 2 are any connected graphs and T(G 1 ) is a graph gained after operating the operation T on G 1 . Mainly, we included the comparison among the classical Zagreb indices, Zagreb connection indices and modified Zagreb connection indices for the aforsaid family of graphs obtains from alkanes. The current article is framed as follows: Section II presents the preliminary definitions of the classical/novel Zagreb indices, Section III holds the general results and section IV covers the applications and conclusion.

Notations and preliminaries
Let G = (V(G), E(G)) be a simple and connected graph. The vertex and edge set of G is denoted by V(G) and E(G) respectively. The degree of a vertex b in any graph G is the number of edges incident to a particular vertex. Todeschini and Consonni (2002) defined generalized form of a degree. In particularly, • Number of incident edges at distance one from b is called simple degree, • Number of incident edges at distance two from b is called connection number as well as leap degree. For more detail see Ali and Trinajstic (2018) and Naji et al. (2017).
In molecular graph, atom and covalent bond between atoms are represented by vertex and edge respectively. Throughout the study of this paper, we assume that G 1 and G 2 are two connected graphs such that G 1 and G 2 are two connected graphs such that |V(G 1 ) |= n 1 , |V(G 2 )|= n 2 , |E(G 1 )|= e 1 and |E(G 2 )|= e 2 .

Definition 2.1
For a graph G, the first Zagreb index (M 1 (G)), second Zagreb index (M 2 (G)) and third Zagreb index (M 3 (G)) are defined as: These degree-based indices are defined by Gutman and Trinajstic (1972), Gutman et al., (1975), and Furtula and Gutman (2015). These are continuously used to predict better results in molecular structures such as heat capacity, entropy, acentric factor, absolute value of correlation coefficient and ZE-isomerism (Gutman and Polansky, 1986). These indices also played an important role in the study of QSPR and QSAR (Devillers and Balaban, 1999;Diudea, 2001).

Definition 2.2
For a graph G, the first Zagreb connection index (ZC 1 (G)) and second Zagreb connection index (ZC 2 (G)) are defined as: Suppose that V 0 (G) be the set of the isolated vertices of the graph G such that |V 0 (G)|= n 0 . In the reference of Doslic et al. (2011), the following relation was derived: where f is defined on the set of degrees of vertex set that is as a real value function of graph G. Ali and Trinajstic (2018) discussed the reality of the following identity on the same pattern: where g has same properties as of f. Moreover, they defined the modified first Zagreb connection index ( ) . Now, we describe the T-sum graphs i.e. subdivision and semi-total point operations on graphs as follow: • T 1 (G) is a graph that is obtained by including a new vertex between each edge of the graph G. • T 2 (G) is a graph that is obtained from T 1 (G) by adding the edges between old vertices which are adjacent in G.
The more details of these operations can be found in Harary (1969) and Sampathkumar and Chikkodimath (1973). For further understanding, see Figure 1. The concept of T-sum graphs is defined as in the following definition.

Definition 2.4
For i ϵ {1,2} and T ϵ {T i }, assume that G i are connected graphs and the graph T(G 1 ) having vertex set V(T(G 1 )) and edge set E(T(G 1 )) is happened after operating the operations T on G 1 . Then, the graphs and a 1 a 2 ∈E(S(G 1 )), where (a 1 ,b 1 ), (a 2 ,b 2 ) ∈V(G 1 + T G 2 ). This scheme has been presented according to cartesian product because cartesian product is the best fitted and the most stylish apparatus to construct a large network from a simple graph and is also important for designing as well as exploration of networks (Xu, 2001). For more clearance, see Figures 2 and 3. Coefficient of variation (Allan, 2010): if X is a random variable then a relative measure of dispersion is called coefficient of variation (C.V) in which standard deviation is absolute measure of dispersion. Mathematically, it will be written as: where, are called standard deviation (S) and arithmetic mean ( ) X for ungroup data. C.V is used to draw the results of at least two commodities or variables.
• All of those commodities which commodity has greater value of X is considered more variability among the corresponding or remaining commodities. • All of those commodities which commodity has less value of C.V is observed more consistent. Now, we present some important results which are used in the main results.

Lemma 2.1 (Yamaguchi, 2008)
Let G be a connected graph with n vertices and e edges. Then where equality holds if and only if G is a {C 3 , C 4 }-free graph.

Lemma 2.2 (Tang et al., 2019)
Let G be a connected and {C 3 , C 4 }-free graph with n vertices and e edges. Then, .

Lemma 2.3
Let G be a connected and {C 3 , C 4 }-free graph with n vertices and e edges. Then,

Lemma 2.4
Let G be a connected and {C 3 , C 4 }-free graph with n vertices and e edges. Then,

Lemma 2.5
Let G be a connected graph and G ≅ P n , then For more study, we refer to Naji et al. (2017).

Results and discussion
This section consists on the main results.

Applications and conclusion
Let G 1 ≅ P m and G 2 ≅ P n be two particular alkanes called by paths. Then, the second and third modified

Comparisons with 3D plots
Maple 15 software is used to construct a simple comparison of the classical Zagreb (Lemma 2.6), novel Zagreb connection (Lemma 2.7) and extended modified Zagreb connection (given above) indices related to operations T 1 and T 2 into 3D plots (Figures 4-9).

Comparisons with line graphs
Now, we close our discussion with the following conclusion: • In Figures 9a and 9b, we obtain that among all the indices (classical Zagreb indices, Zagreb connection indices and modified Zagreb connection indices), the third modified Zagreb connection index ZC 3 * of θ ≅ + P P m T n 1 1 and θ ≅ + P P m T n 2 2 are better than the other indices with most upper increasing layer.
• The numerical values of Tables 1 and 2 also present that the third modified Zagreb connection index ZC 3 * of θ ≅ + P P m T n 1 1 and θ ≅ + P P m T n 2 2 are better than the other indices having larger values. (m,n) M 1 (θ 2 ) M 2 (θ 2 ) ZC 1 (θ 2 ) ZC 2 (θ 2 ) ZC 1 * (θ 2 ) ZC 2 * (θ 2 ) ZC 3 * (θ 2 ) • Figures 10 and 11 statistically prove that M 1 is more consistent and ZC 3 * has better variability than other indicated indices respectively. Moreover, ZC 3 * is averagely consisitent by Figure 10 for both the graphs θ ≅ + P P m T n 1 1 and θ ≅ + P P m T n 2 2 . Additionally by Figures 10 and 11, all the indicated indices of θ ≅ + P P m T n 2 2 are more variability and more consisitent than θ ≅ + P P m T n 1 1 , respectively.

Data availability:
The data used to support the findings of this study are cited at relevant places within the text as references.

Conflicts of interest:
The authors declare that they have no conflicts of interest.
Authors' contributions: All authors contributed equally to this paper.    Table 4