Abstract
Topological indices or coindices are mathematical parameters which are widely used to investigate different properties of graphs. The operations on graphs play vital roles in the formation of new molecular graphs from the old ones. Let be a graph we perform four operations which are , , and obtained subdivisions type graphs such that , , , and , respectively. Let and be two simple graphs; then, -sum graph is defined by performing the Cartesian product on and ; mathematically, it is denoted by , where . In this article, we have calculated sum-connectivity coindex for -sum graphs. At the end, we have illustrated the results for particular -sum graphs with the help of a table consisting of numerical values.
1. Introduction
The field of science in which chemical graph theory and mathematical chemistry are studied is known as cheminformatics. In this field, computable properties of molecular graphs are investigated using different mathematical parameters, perhaps the best known as topological index (TI), which is defined as the mathematical formula that is applied to any graph which has a molecular structure. TIs play a significant role especially in quantitative structure activity relationship and quantitative structure property relationship investigations, such as optimisation and physicochemical interpretation of molecules [1]. The role of bioinformatics and chemistry in drugs discovery is explained in [2], and Rucker calculated the boiling points of different cycloalkanes in [3]. Topological indices are categorized mainly in two types: one is known as degree-based and second is known as distance-based. According to a recent survey, it is found that degree-based TIs attracted a lot of attention in recent years [4].
Wiener was pioneer that used distance-based TI and calculated accurate boiling point of paraffin [5]. Later on, Gutman and Trinajsti C (1972) worked on alternant hydrocarbons and calculated their total -electron energy using two degree-based indices known as Zagreb indices [6]. The Randi C (1975) proposed a new index known as Randi index and calculated theoretical characterization of molecular branching with the help of said index [7].
Zhou and Trinajstic [8] introduced sum-connectivity index and calculated this index for different graphs. Xing et al. [9] computed bounds for trees graphs using pendant vertices, and Wang et al. [10] determined lower bound for triangle-free graphs. Ma and Deng calculated the sharp lower bounds for cacti [11] under the sum-connectivity indices. Du and Zhou computed minimum sum-connectivity index for bicyclic graphs [12]. Das et al. [13] derived relations among Randić index and sum-connectivity index. Later, Das et al. [14] proved that for trees graphs, sum-connectivity is smaller than zeroth-order Randić index. Zhou and Trinajstić derived different relations among the sum-connectivity indices and product [15].
Farahani calculated sum-connectivity index and Randi of nanotubes [16] and calculated sum-connectivity index, Randi connectivity index ABC index, and geometric-arithmetic index of a class of dendrimer [17], Jahanbani computed the sharp lower bound on the sum-connectivity index of two trees with the minimum and the second minimum sum-connectivity. Ramane et al. [18] derived the relationship among sum-connectivity index, Randic index, harmonic index, and -electron energy for benzenoid hydrocarbons. Rodriguez et al. [19] computed sum-connectivity index and harmonic index of graphs. Su and Xu computed the general sum-connectivity coindex of different graphs such as cycle graph, path graph, complete graph, complete bipartite graph, and hypercube graph [20].
We use different operations on a graph for the formation of new families of graphs such as joining, subtraction, union, intersection, and products. Ahmad et al. [21] computed exact values and improved bounds for graph operations. Akhter and Imran computed sharp bounds four operations on graphs [22] and computed bounds for four types of graphs operations involving -graph using the general sum-connectivity index [23]. Yan et al. [24] gave the idea about new graphs with help of operations as , where on and computed Wiener index for said graphs. Deng et al. [25] defined new -sum graphs by extending the work of Yan which are obtained with the help of Cartesian product of two different graphs and , where and calculated Zagreb indices. Ibraheem et al. [26] computed -coindex. Javaid et al. [27] investigated the bounds for first and second Zagreb coindex [28].
In this study, we compute sum-connectivity coindex of sum graphs in terms of Zagreb indices and coindices of their factor graphs. We have illustrated results through table for the specific sum graphs obtained using path (alkane) graphs. The rest of article as follows: Section 2 describes elementary definitions and notations. Section 3 contains main results of work, and Section 4 contains the application and conclusion.
2. Preliminaries
A graph consists of set of vertices and edges mathematically denoted as . The total number of vertices is called order of graph and total number of edges is called size of graph. For any vertex , then is called degree of and defined as number of edges attached to . Let be a graph; then, its complement is denoted by and defined as iff .
The first Zagreb index and second Zagreb index were introduced by Gutman and Trinajsti [6] which are defined as
Ashrafi et al. [29] introduced Zagreb coindices , which are defined as
Zhou and Trinajstic [8] introduced sum-connectivity index , which is defined as
Now, the sum-connectivity coindex is defined as
Suppose that is a connected graph, then(i)The graph is formed by replacing each edge of with (ii)The graph is obtained from by joining the vertices which are adjacent in (iii)The graph is a graph formed using by attaching the new pairs of vertices which are on the adjacent edges of (iv)If both and operations are performed on , then is obtained
Considered two graphs and , then we defined their -sum graphs which is denoted by and defined as having vertex set and edge set as , and of are joined iff and and , where .
Figures 1 and 2 show the explanation of -sum graphs.
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(b)
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3. Main Results
This section contains main results of harmonic coindex for -sum graphs. Here, we defined some useful supposition that will be used in theorems.
Theorem 1. The sum-connectivity coindex for -sum graph is given as
Proof. Using equation (4), we haveWe get required result by substituting all in equation (2).
Theorem 2. The sum-connectivity coindex for -sum graph is given as
Proof. We considerThe value of follows by equation (9):We get the required result by substituting all in equation (10).
Theorem 3. The sum-connectivity coindex for -sum graph is given as
Proof. We consideredThe values of and follow by equations (9) and (43).We get required result by substituting all in equation (6).
Theorem 4. The sum-connectivity coindex for -sum graph is given as
Proof. It follows from Theorem 2 and Theorem 3.
4. Conclusion
(i)In this study, we have computed sum-connectivity coindex in the form of first and second Zagreb indices and coincides for -sum graphs such as , , , and .(ii)Table 1 consisting of numerical values present that sum-connectivity coindex of is dominant than , , and .Data Availability
The data used to support the findings of this study are included within the article and are available from the corresponding author upon request.
Additional Points
We close our discussion that the problem is still open to compute the sum-connectivity coindex for other products, i.e., lexicographic product and strong product.
Conflicts of Interest
The authors declare that they have no conflicts of interest.