Abstract
Topological indices or coindices are one of the graph-theoretic tools which are widely used to study the different structural and chemical properties of the under study networks or graphs in the subject of computer science and chemistry, respectively. For these investigations, the operations of graphs always played an important role for the study of the complex networks under the various topological indices or coindices. In this paper, we determine bounds for the second Zagreb coindex of a well-known family of graphs called -sum (-sum, -sum, -sum, and -sum) graphs in the form of Zagreb indices and coindices of their factor graphs, where these graphs are obtained by using four subdivision-related operations and Cartesian product of graphs. At the end, we illustrate the obtained results by providing the exact and bonded values of some specific -sum graphs.
1. Introduction
A topological index (TI) is a function from the set of graphs to the set of real numbers that assigns the different numerical values to the different graphs unless the graphs are isomorphic. Moreover, TIs are essential tools to discuss various physical and chemical properties of the graphs such as volume, density, connectivity, boiling point, freezing point, and heat of formation and evaporation [1, 2]. TIs are also used to study the quantitative structure property relationships , quantitative structure activity relationships , and clinical practices of various medications in the subject of cheminformatics and pharmaceutical industries, respectively (see [3–5]). Mainly TIs have three types such as degree, distance, and polynomial based but the degree-based TIs are more studied than others (see the most recent review [6]).
Firstly, an American Chemist Harry Wiener (1947) used a distance-based TI to calculate the boiling point of paraffin (see [7]). First and second Zagreb indices are introduced by Gutman and Trinajsti in 1972; these indices are used to calculate total -electron energy of alternant hydrocarbons [8]. Kinkar and Gutman calculated different relations between the second Zagreb index of a graph and its complement (see [9]). Yan et al. computed sharp bounds for the second Zagreb index of different unicyclic graphs [10]. Carlos et al. calculated the second Zagreb index of the graphs with minimum and maximum vertex degrees. They also investigated trees with the maximum value of the second Zagreb index among all trees with maximum vertex degree [11].
Recently, Zagreb coindices are introduced by Ashrafi et al., and they studied them for the derived graphs obtained by the operations of joining, union, disjunction, Cartesian product, and corona product (see [12, 13]). Kinkar et al. calculated the first Zagreb index and multiplicative Zagreb coindices of tree (see [14]). Gutman obtained coindices of graphs and their complements (see [15]). Nilanjan et al. calculated -coindex of some graph operations (see [16]). Javaid et al. calculated the first Zagreb connection index and coindex of some derived graphs [17]. Ramane et al. calculated coindices for the transmission and reciprocal transmission-based graphs (see [18]). Mansour and Song computed and -analogs of Zagreb indices and coindices of graphs [19]. For further studies of Zagreb indices, see [20].
There are various operations on graphs such as union, intersection, complement, product, and subdivision. These operations on graphs are useful to obtain the new graphs from the old ones. Yan et al. listed five new graphs , , , , and with the help of five operations , , , , and on a graph , respectively, and studied the behavior of Wiener index of these graphs (see [4]). Eliasi and Taeri computed the Wiener indices of the -sum graphs obtained by the Cartesian product of and , where [21]. Later on, many researchers worked on these -sum graphs such as Deng et al. [22] computed first and second Zagreb indices, Akhtar and Imran calculated the forgotten index [23], Liu et al. computed first general Zagreb index [24], Ahmad et al. calculated sharp bounds of general sum-connectivity index [11], and Alanazi et al. calculated Gutman indices [25].
In this paper, we compute the bounds for the second Zagreb coindex of -sum graphs in the form of Zagreb indices and coindices of their factor graphs. At the end, the obtained results are additionally illustrated with the assistance of examples of the exact and bonded values for some specific -sum graphs. The rest of the paper is settled as follows: Section 2 contains the basic definitions and notions, Section 3 covers the main results, and Section 4 presents conclusion with specific examples related to the derived results.
2. Preliminaries
A graph denoted by is formed by set of vertices and edges , where edge set is subset of the Cartesian product of set of vertices, i.e., . In a simple connected graph , total number of vertices is called its order (denoted by ) and total number of edges is called its size (presented by ). The degree of a vertex is number of its neighborhood vertices that is denoted by . The complement of is denoted by and defined as , and any two vertices (say and ) imply that iff . Gutman and Trinajsti in 1972 [8] introduced the first and second Zagreb indices (denoted by and ) as follows:
The second Zagreb coindex is defined in [13] as follows:
It is important to note that the above defined coindex uses degrees of but run over .
Let be a graph, then(i) is a graph obtained by inserting one vertex in every edge of (ii) is a graph obtained from by joining the adjacent vertices of (iii) is a graph formed from by joining the pairs of new vertices which are on the adjacent edges (the edges with one common vertex) of (iv) is obtained by performing both operations of and on , respectively
Let and be two simple connected graphs, then their -sum graphs are denoted by having vertex set and iff(i) and (ii) and , where
(a)
(b)
(c)
(d)
(e)
(a)
(b)
(c)
3. Main Results
In this section, main results of the second Zagreb coindex for the F-sum graphs are discussed.
Theorem 1. Let and be two simple connected graphs, then second Zagreb coindex of is given as follows:where
Proof. Using equation (2), we haveNote thatNote thatNote thatConsequently,We obtained the required result by putting the values of in equation (5).
Theorem 2. Let and be two simple connected graphs, then second Zagreb coindex of is given as follows:where
Proof. Using equation (2), we haveUsing equation (6), we directly haveNote thatsoNote thatsoNote thatsoConsequently,We obtained the required proof by putting the values of in equation (14).
Theorem 3. Let and be two simple connected graphs, then second Zagreb coindex of is given as follows:where
Proof. Using equation (2), we haveNote thatsoNote thatsoNote thatsoNote thatsoNote thatsoNote thatsoNote thatsoConsequently,Using equation (7), we directly haveNote thatNote thatNote thatConsequently,We obtained the required proof by putting the values of in equation (25).
Theorem 4. Let and be two graphs, then second Zagreb coindex of is given as follows:where
Proof. Using equation (2), we haveUsing equation (40), we directly haveUsing equation (15), we directly haveConsequently,We obtained required results by putting the values of in equation (48).
4. Conclusion
In this paper, we have computed second Zagreb coindex of -sum graphs such as , , , and . The obtained results are illustrated with the help of specific class graphs of -sum graphs. Let and , then the lower and upper bounds of first Zagreb coindex for their F-sum graph are given in Table 1.
Now, we close our discussion that the problem is still open to compute the other generalized coindices (first general Zagreb and general Randic coindices) for the -sum graphs.
Data Availability
The data used to support this study are included within the article. However, the reader may request the corresponding author for more details of the data.
Conflicts of Interest
The authors declare that they have no conflicts of interest.