Abstract
Topological index (TI) is a function that assigns a numeric value to a (molecular) graph that predicts its various physical and structural properties. In this paper, we study the sum graphs (S-sum, R-sum, Q-sum and T-sum) using the subdivision related operations and strong product of graphs which create hexagonal chains isomorphic to many chemical compounds. Mainly, the exact values of first general Zagreb index (FGZI) for four sum graphs are obtained. At the end, FGZI of the two particular families of sum graphs are also computed as applications of the main results. Moreover, the dominating role of the FGZI among these sum graphs is also shown using the numerical values and their graphical presentations.
1. Introduction
Let be a (molecular) graph with and as sets of vertex and edge respectively. The degree of a vertex is the number of edges which are incident to . A topological index (TI) is a function that assigns a numeric value to the under study (molecular) graph, see [1–3]. TIs are used to predict the various chemical and structural properties such as octane isomers including entropy, acentric factor, density, total surface area, molar volume, boiling point, capacity of heat at temperature and pressure, enthalpy of formation, connectivity of compounds and octanol water partition, see [4]. These are also used to study the quantitative structure property and activity relationships which are very important in the subject of cheminformatics, see [5].
Wiener see [6] defined a distance-based TI to compute the boiling point of paraffin. Gutman and Trinajstic see [7] investigated total -electron energy of the molecule graphs using a degree-based TI called by first Zagreb index nowadays. Later on, various Zagreb indices (hyper, forgotten, multiplicative and augmented) are defined to predict the different physicochemical properties chirality, heterosystems, complexity, branching and ZE-isomerism of the (molecular) graphs see [8]. Li and Zheng defined the first general Zagreb index (FGZI) and studied its different properties see [9].
The operations of graphs (addition, deletion, complement, product, union and intersection) also play a very important role for the construction of new graphs, see [10]. In particular, for a graph Yan et al. [11] introduced the five new graphs , , , and are defined with the help of the operations of subdivision , semitotal point , Line graph , semitotal edge and total vertex and edge respectively. Also, Wiener index of these graphs is computed, where . Eliasi and Taeri [12] constructed the -sum graphs by the Cartesian product of the graphs and , where is obtained after applying the on for . Moreover, they computed the Wiener indices of the -sums graphs , , and . Deng et al. [13] computed the first and second Zagreb indices of four operations on graphs using the concept of Cartesian product. Akhter and Imran [14] computed the forgotten topological index of four operations on graphs under Cartesian product. Liu et al. [15] computed the FGZI of -sum graphs under the operation of Cartesian product. Sarala et al. [16] computed the -index of -sum graphs under the operation of strong product. For further study, we refer to [17–29].
In this paper, we extend this study and compute FGZI of the -sums graphs () under the operation of strong product of and in term of FGZI of its factor graphs and , where , and are any two connected graphs. The obtained results are general extension of the results of Deng et al. [13], Akhter and Imran [14], Liu et al. [15] and Sarala et al. [16] works. Forthcoming sections are arranged as: Section 2 includes the basic formulae and results, Section 3 covers the main results and Section 4 includes the conclusion.
2. Preliminaries
An atom is presented by a vertex and bonding between atom is showed by edge in the molecular graphs. The first and second Zagreb indices and of are defined as follows [5, 7, 22]:
For any real number FGZ index and general Randi index are defined as
For more details of aforesaid TIs, see [30, 31]. The following graphs are defined with the help of the subdivision related operations (, , and ), see [16].(i)By inserting a new vertex in each , S (G) is obtained.(ii)By joining the end (original) vertices of the edges being incidence on each new vertex in S (G), R (G) is obtained.(iii)By joining the new vertices which have common adjacent (original) vertices in S (G), Q (G) is obtained.(iv)By operating both R(G) and Q(G) on S (G), T (G) is obtained. For more explanation, see Figure 1.
(a)
(b)
(c)
(d)
(e)
In 1960 Sabidussi introduced the strong product for any two graphs has vertex set Cartesian product such that and will be adjacent in iff: [ and is adjacent to ] or [ and is adjacent to ] or [ is adjacent to and is adjacent to ]. Now, the -sum graph under the operation of strong product is defined in [16].
Let and be two graphs. For {, , , }, is a graph constructed by the operation on with set of vertex and set of edge . Then, -sum graph under the strong product of graphs and is a graph with vertex setsuch that two vertex and of are adjacent iff, either [ and ] or [ and ] or [ and ].
It is observed that has copies of which are labeled by the vertices of . Also, and are shown as blue and red vertices in respectively, see Figure 2.
3. Main Results
Now, we compute the analytical closed expression for FGZ index of the , , and .
Theorem 1. Let and be two graphs. For and , FGZ index of -sum graph is
Proof. By the definition for any vertex , the degree of isFor each vertex and edge , we haveWe know that . Also, for every and with , then,Hence
Theorem 2. Let and be two graphs. For and , FGZ index of -sum graph is
Proof. By definitionConsider,Since For each vertex and edge where then,For each and where , then,For each and , where then,For each and , where then,Hence
Theorem 3. Let and be two graphs. For and , FGZ index of -sum graph is
Proof. By definitionConsider,For each vertex and edge where , . Note that = for ], is the vertex inserted into the edge of G for all , , we haveconsiderIn , and occurs times. ThusLetIn , and occurs two times. ThereforeNow and edge where , , we haveFor each and , where , we haveFor each and , where , , we haveHence
Theorem 4. Let and be two graphs. For and , FGZ index of -sum graph is
Theorem 5. Let and be two graphs. Then, the FGZI of -sum graphs where , we have(i)(ii)(iii)(iv)
4. Discussion and Conclusion
We conclude our work with applications of (Theorem 1–Theorem 4) in the following examples.
Example 1. For two graphs and of order and , we have
Example 2. For two graphs and of order , we have(i)In this paper, we computed the exact values of FGZ index of the four -sum graphs (, , and ) in terms of their factor graphs, where these graphs are obtained under the operation of the strong product.(ii)It is the extension of Deng et al. [13], Akhter and Imran [14], Liu et al. [15] and Sarala et al. [16] works.(iii)The FGZ index of the -sum graphs obtained from the paths and cycles are also computed as applications of the obtained results.(iv)Table 1 and Figure 3 present that FGZ index of is dominant having ordering .(v)Table 2 and Figure 4 also presents that FGZ index of is dominant having ordering .
Data Availability
All data are included within this paper. However, the reader may contact the corresponding author for more details of the data.
Disclosure
The authors have no conflicts of interest.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The author is deeply thankful to the reviewers for their valuable suggestions to improve the quality of this manuscript. This work was supported by the Humanities and Social Science Project of Anhui Provincial Education Department. Research on social crisis management in Anhui cities under the background of big data (Subject No: SK2016A0233).