Abstract
Gutman index of a connected graph is a degree-distance-based topological index. In extremal theory of graphs, there is great interest in computing such indices because of their importance in correlating the properties of several chemical compounds. In this paper, we compute the exact formulae of the Gutman indices for the four sum graphs (S-sum, R-sum, Q-sum, and T-sum) in the terms of various indices of their factor graphs, where sum graphs are obtained under the subdivision operations and Cartesian products of graphs. We also provide specific examples of our results and draw a comparison with previously known bounds for the four sum graphs.
1. Introduction
Theory of topological indices (TIs) started when Wiener discovered a close correlation between boiling points of certain alkanes and sums of the distances among pairs of vertices. Later, this calculated number was named Wiener Index [1]. After 25 years, Gutman and Trinajstić discovered degree-based indices (first and second Zagreb indices) which they used to compute the total -electron energy of conjugate molecules [2]. Following these discoveries, many scientists began to introduce various TIs as invariant numbers for the prediction of the certain properties of molecular structures such as boiling point, freezing point, volume, density, vaporization, and weight. Two deeper approaches, namely, quantitative structures property relationships (QSPR) and quantitative structure activity relationships (QSAR) have also been used under the subject of cheminformatics (combination of chemistry, mathematics, statistics, and information sciences) and in conjunction with TIs, to find correlation values between the physical structures and chemical properties of molecules, see [3–5].
TIs have been classified into three main classes depending upon degrees of nodes (vertices), distances among the vertices, and enumerative polynomials of the molecular graphs. Distance-based TIs are generally considered more important than the others. Some of the distance-based TIs are Wiener index [1], average distance index [6], Harary index [7, 8], degree distance index, and the Gutman index [9]. For more details, see [10–14].
In graph theory, various operations such as union, intersection, addition, and Cartesian product are used to obtain the new graphs. Yan et al. [15] defined four subdivision-related operations S, R, Q, and T. They applied these operations on a connected graph G to obtain the four new graphs (subdivided graph), (triangle parallel graph), (line superposition graph), and (total graph), respectively. Afterwards, Das and Gutman [10] introduced the F-sum graphs using the operation of Cartesian product on the graphs and , where . Various hexagonal chains were later derived from these F-sum graphs, which have been found isomorphic to many chemical structures. They also determined the Wiener indices of the following S-sum , R-sum , Q-sum , and T-sum graphs.
Recently, Liu et al. [16] computed the first general Zagreb indices of the F-sum graphs. Akhter and Imran [17] found out the sharp bounds of the general sum-connectivity index for F-sum graphs. Ahmad et al. [18] discovered the exact formulae of the general sum-connectivity index for F-sum graphs, by improving the bounds. An et al. [19] determined the upper bounds of the degree distance indices for all the F-sum graphs. Pattabiraman and Bhat [20] derived the upper bounds of the Gutman index for all the F-sum graphs. For more studies, we refer to [21–32].
In this paper, we obtain the exact values of the Gutman index for the F-sum , , , and graphs. Moreover, the results are illustrated with special classes of the F-sum graphs and a comparison is drawn between the obtained exact values and the previously known bounded values.
The sections of paper are organized as follows. Section 2 comprises of preliminaries (some important definitions and statements of related lemmas). Section 3 contains the main result consisting of statements and proofs of theorems about Gutman indices of F-sum graphs. Lastly, Section 4 covers the applications of the main result to the computation of Gutman indices of particular classes of the F-sum graphs and a comparison among exact and known bounded values.
2. Preliminaries
We give a detailed consideration of two simple graphs and . The degree of a vertex is equal to the number of vertices connected to it. For each , degree of the vertex is denoted by . The distance between two vertices is defined as the length of the shortest path between both the vertices and . Further details can be found in [33, 34].
Definition 1. (see [1]). The Wiener index of a connected graph is defined as
Definition 2. (see [11]). The degree distance index of a connected graph is defined as
Definition 3. (see [35]). The Gutman index of a connected graph is defined asYan et al. [15] defined four special graphs derivable from a given graph by applying the four respective operations , , , and on the graph as follows:(i)Subdivided graph is formed from if distance of one between two adjacent pair of vertices is increased by two after inserting a new vertex between them. The vertices of are named as black vertices, while new vertices are called white vertices.(ii)Triangle parallel graph is formed from if the new vertex corresponded to each edge of is joined with the end vertices of the each respective edge. The vertices of are named as black vertices, while new vertices are called white vertices.(iii)Line superposition graph is formed when two white vertices obtained from are further joined if incident edges of these white vertices have one common end vertex in .(iv)Total graph is formed from by applying the further operation on it.Figure 1 illustrates the graphs obtained by the operations , , , and based on the path graph .
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Definition 4. Let . Then, is called a F-sum graph with vertex set and are adjacent such that either and or and .
Figure 2 shows instances of S-sum , R-sum , Q-sum , and T-sum graphs. We now state some important lemmas which are frequently used in the main results.
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Lemma 1. (see [36]). Let and be two simple and connected graphs.(a)For , if both vertices and are black, then (b)If one vertex is white and second is black with , then
Lemma 2. (see [36]). Let and be two simple and connected graphs. If both vertices and are white vertices and or , then
Lemma 3. (see [36]). Let and be two simple and connected graphs. If both vertices and are white vertices and or , then
3. Main Results
This section is devoted to providing main theorems on the Gutman index of F-sum graphs , , , and . Consider the set of black vertices with so that consists of white vertices and with . Then, , where .
Theorem 1. Let and be connected and simple graphs. If is the S-sum graph of and , then
Proof. Case 1: when both vertices are black, for to n and to m. For S-sum , Substituting , Substituting , Case 2: when one vertex is white and the other is black, For to , to , and to and , The summation of the distances between vertices with different colours is twice the , i.e., Case 3: when both vertices are white, This summation consists of two parts and , whereNow, Gutman index of is given by
Theorem 2. Let and are connected and simple graphs. If is the R-sum graph of and , then
Proof. 1) When both vertices are black, for to n and to m. For R-sum , Substituting Substituting , Case 2: when one vertex is white and the other is black, For to , to , and to and , The summation of the distances between vertices with different colours is twice the , i.e., Case 3: when both vertices are white, C can be calculated similar to that in the previous theorem:Now, Gutman index of is given by
Theorem 3. Let and be two simple and connected graphs. If is the Q-sum graph of and , then
Proof. Case 1: when both vertices are black, A can be determined similar to that in Case 1 of Theorem 1: Case 2: when one vertex is white and the other is black, For to , to and to , The summation of the distances between vertices with different colours is twice the , i.e., Case 3: When both vertices are white, This summation consists of three parts , whereNow, Gutman index of is given byusing values of A, B, and C from Case 1, Case 2, and Case 3, respectively,
Theorem 4. Let and be two simple and connected graphs. If is the T-sum graph of and , then
Proof. Case 1: when both vertices are black, A can be calculated similar to that in Case 1 of Theorem 2: Case 2: when one vertex is white and the other is black, For to , to , and to , The summation of the distances between vertices with different colours is twice the B1, i.e., Case 3: when both vertices are white, C can be calculated similar to that in Case 3 of Theorem 3: Now, Gutman index of is given byusing values of A, B, and C from Case 1, Case 2, and Case 3, respectively,
4. Discussion and Conclusion
In this section, we apply the main results of Section 3 by taking equal to, firstly, , and secondly, , while is set equal to . The Wiener index degree, distance index, and Gutman index of are , , and , respectively. Now, we construct Tables 1 and 2.
Then, the following results are found by using Theorems 1–4 and Tables 3 and 4:
If is cycle of n vertices, then the Wiener index, degree distance index, and Gutman index of cycle are as follows:
It is very essential to know that, in cycle, and
Then, the following results are found by using Theorems 1–4 and Tables 3 and 4:
After deriving formulae, a comparison among the exact values, computed values, and bounded values is also drawn in Table 5.
Now, we close our discussion with the comments that the upper bounds for the Gutman indices on the F-sum (S-sum, R-sum, Q-sum, and T-sum) graphs are obtained in [20]. In this paper, we considered for the improvement of the already existing bounds and determined the exact values of the Gutman indices for the F-sum graphs. However, the problem is still open to find the exact values of other distance-based TIs for these sum graphs.
Data Availability
All data used to support the findings of the study are included within the article and can be obtained from the corresponding author upon request.
Conflicts of Interest
The authors have no conflicts of interest.
Acknowledgments
This research was partially supported by the research grant S-1440-0145, Deanship of Scientific Research, University of Tabuk, Tabuk-71491, Saudi Arabia.