Abstract
In mathematical chemistry, molecular structure of any chemical substance can be expressed by a numeric number or polynomial or sequence of number which represents the whole graph is called topological index. An important branch of graph theory is the chemical graph theory. As a consequence of their worldwide uses, chemical networks have inspired researchers since their development. Determination of the expressions for topological indices of different derived graphs of graphs is a new and interesting problem in graph theory. In this article, some graphs which are derived from honeycomb structure are studied and obtained their exact results for sum degree-based indices. Additionally, a comparison is shown graphically among all the indices.
1. Introduction and Preliminary Results
Atoms are connected through covalent bonds in molecular frames. Atoms are called vertices and covalent bonds as edges in graph theory. Information sciences, Mathematics, and Chemistry are combined in Cheminformatics. This is a new research field that attracts the attention of researchers. Dominating David Derived networks are under consideration in the present research article which are further extracted from honeycomb structures. Honeycomb structures taken from bee honeycombs had observed a lot of applications in different fields including chemical engineering, transportation, mechanical engineering, architecture, nanofabrication, and recently bioinformatics. The greatest challenge in this research direction is to comprehend the unique characteristics of honeycomb structures, relying on their structures, scales, and the materials used to allow the minimization of the materials to get minimum weight and maximum strength. Nowadays, there is a substantial use of honeycomb networks in computer graphics, processing of images, chemistry as the portrayal of benzenoid hydrocarbons, and cellular phone base stations by using hexagon arrangements with repetitive building of honeycomb structures [1].
A topological index , sometimes also known as a graph-theoretic index, is a numerical invariant of a chemical graph [2]. There are many types of , but most popular and authentic are distance-based, degree-based, and neighbourhood degree-based indices. These indices contain a lot of information within themselves.
The method of drawing Dominating David Derived networks (dimension ) is as follows: Step 1: consider a honeycomb network dimension as shown in Figure 1(a). Step 2: split each edge into two by embedding another vertex as shown in Figure 1(b). Step 3: in each hexagon cell, connect the new vertices by an edge if they are at a distance of 4 units within the cell as shown in Figure 1(c). Step 4: place vertices at new edge crossings as shown in Figure 1(d). Step 5: remove initial vertices and edges of honeycomb network as shown in Figure 1(e). Step 6: split each horizontal edge into two edges by inserting a new vertex. The resulting graph is called Dominating David Derived system of measurement as shown in Figure 1(f) [1, 3].
(a)
(b)
(c)
(d)
(e)
(f)
The first type of Dominating David Derived network can be obtained by connecting vertices of degree two by an edge, which are not in the boundary (see Figure 2).
The second type of Dominating David Derived network can be obtained by subdividing once the new edge introduced in (see Figure 3).
The third type of Dominating David Derived network can be obtained from by introducing parallel path of length 2 between the vertices of degree two which are not in the boundary. See Figure 4 for third type of Dominating David Derived network of dimension 2, . Moreover, isomorphic graph of Dominating David Derived network of dimension 2 can be seen in Figure 5.
In this article, is considered a network with a vertex set and an edge set of , and is the degree of vertex . Let denote the sum of the degrees of all vertices adjacent to a vertex . Graovac et al. [4] defined fifth M-Zagreb indices as polynomials for a molecular graph and these are characterized as follows.
Let be a graph. Then,
Kulli [5, 6] motivated by above indices described some new topological indices and named them as the fifth M-Zagreb indices of first and second type and fifth hyper-M-Zagreb indices of first and second type of a graph . They are defined as
They also define a new version of Zagreb index which they call as third Zagreb index or fifth -Zagreb [7].
2. Results
We have studied the new topological indices described by Kulli named as fifth M-Zagreb indices, fifth M-Zagreb polynomials, and -Zagreb index and give closed formulae of these indices for Dominating David Derived networks. In the following publications, the authors found out and described many other interesting characterizations for topological indices for different networks. For example, Ali et al. [8] studied degree-based topological indices and computed some new topological indices [9] for various networks. Bača et al. [10] and Baig et al. [11] found some topological indices of carbon nanotube network and polyoxide networks, respectively. Baig et al. [12] and Imran [13] studied about topological polynomials of certain nanostructures and mesh-derived networks, respectively. Eliasi et al. [14] and Liu et al. [15] computed multiplicative indices of first Zagreb and carbon graphite -levels, respectively. Furthermore, Gao et al. [16] computed multiple ABC and GA index; Imran et al. [17] studied topological properties of Dominating David Derived network; Imran et al. [18] investigated topological properties of diamond-like networks. To know more about topological indices of various graph families, see [19, 20]. For the basic notations and definitions, see [21, 22].
2.1. Results for First Type of Dominating David Derived Networks
In this section, we calculate degree-based topological indices of the dimension for first type of Dominating David Derived networks. In the coming theorems, we compute M-Zagreb indices and polynomials.
Theorem 1. Let be the first type of Dominating David Derived network, then the first and second fifth M-Zagreb indices are equal to
Proof. The outcome can be obtained by using the edge partition in Table 1. By using equation (1),
By doing some calculations, we getThus, from equation (2),By doing some calculations, we get
Theorem 2. Consider the first type of DDD network for . Then the first and second general fifth M-Zagreb indices are equal to
Proof. Let be the first type of DDD network. Table 1 shows such an edge partition of . Thus, from equation (3), it follows thatBy using edge partitions in Table 1, we getBy doing some calculations, we haveFrom equation (4), we haveBy using edge partitions in Table 1, we getBy doing some calculations, we have
Theorem 3. Consider the first type of DDD network for . Then, the first and second hyper fifth M-Zagreb indices are equal to
Proof. Let be the first type of DDD network. Table 1 shows such an edge partition of . Thus, from equation (5), it follows thatBy using edge partitions in Table 1, we getBy doing some calculations, we haveFrom equation (6), we haveBy using edge partitions in Table 1, we getBy doing some calculations, we have
Theorem 4. Consider the first type of DDD network for . Then, the third M-Zagreb index is equal to
Proof. Let be the first type of DDD network. Table 1 shows such an edge partition of . Thus, from equation (7), it follows thatBy using edge partitions in Table 1, we getBy doing some calculations, we have
2.2. Results for Second Type of Dominating David Derived Network
Now, we are calculating fifth M-Zagreb topological indices of the , where for second type of Dominating David Derived network.
Theorem 5. Let be the second type of Dominating David Derived network, then the first and second fifth M-Zagreb indices are equal to
Proof. The outcome can be obtained by using the edge partition in Table 2. By using equation (1),
By doing some calculations, we getThus, from equation (2),By doing some calculations, we get
Theorem 6. Consider the second type of DDD network for . Then the first and second general fifth M-Zagreb indices are equal to
Proof. Let be the second type of DDD network. Table 2 shows such an edge partition of . Thus, from equation (3), it follows thatBy using edge partitions in Table 2, we getBy doing some calculations, we haveFrom equation (4), we haveBy using edge partitions in Table 2, we getBy doing some calculations, we have
Theorem 7. Consider the second type of DDD network for . Then the first and second hyper fifth M-Zagreb indices are equal to
Proof. Let be the second type of DDD network. Table 2 shows such an edge partition of . Thus, from equation (5), it follows thatBy using edge partitions in Table 2, we getBy doing some calculations, we haveFrom equation (6), we haveBy using edge partitions in Table 2, we getBy doing some calculations, we have
Theorem 8. Consider the second type of DDD network for . Then the third M-Zagreb index is equal to
Proof. Let be the second type of DDD network. Table 2 shows such an edge partition of . Thus, from equation (7), it follows thatBy using edge partitions in Table 2, we getBy doing some calculations, we have
2.3. Results for Third Type of Dominating David Derived Network
In this section, we calculate degree-based topological indices of the dimension for third type of Dominating David Derived networks. In the coming theorems, we compute M-Zagreb indices and polynomials.
Theorem 9. Let be the third type of Dominating David Derived network, then the first and second fifth M-Zagreb indices are equal to
Proof. The outcome can be obtained by using the edge partition in Table 3. By using equation (1),
By doing some calculations, we getThus, from equation (2),By doing some calculations, we get
Theorem 10. Consider the third type of DDD network for . Then the first and second general fifth M-Zagreb indices are equal to
Proof. Let be the first type of DDD network. Table 3 shows such an edge partition of . Thus, from equation (3), it follows thatBy using edge partitions in Table 3, we getBy doing some calculations, we haveFrom equation (4), we haveBy using edge partitions in Table 3, we getBy doing some calculations, we have
Theorem 11. Consider the third type of DDD network for . Then the first and second hyper fifth M-Zagreb indices are equal to
Proof. Let be the third type of DDD network. Table 3 shows such an edge partition of . Thus, from equation (5), it follows thatBy using edge partitions in Table 3, we getBy doing some calculations, we haveFrom equation (6), we haveBy using edge partitions in Table 3, we getBy doing some calculations, we have
Theorem 12. Consider the third type of DDD network for . Then the third M-Zagreb index is equal to
Proof. Let be the third type of DDD network. Table 3 shows such an edge partition of . Thus from equation (7) it follows thatBy using edge partitions in Table 3, we getBy doing some calculations, we have
3. Comparison of Indices
This section contains the comparison of indices and graphical details of different types of DDD networks as given in Figures 6–14.
4. Conclusion
In this manuscript, we computed sum of degree-based indices for some derived graphs of honeycomb structure. We also computed certain sum of degree-based polynomials such as fifth M-Zagreb, fifth hyper M-Zagreb, generalized fifth M-Zagreb indices for all types of Dominating David Derived networks and we also provide comparison of indies in form of graphs, all the graphs are increasing, increase with the value of . These facts may be useful for people working in computer science and chemistry who encounter honeycomb networks. These results can also play a vital part in the determination of the significance of honeycomb derived networks. As other topological indices, determining the representations of derived graphs like these, is an open question for many other topological indices.
Honeycomb structure is present in different products of sports, aerospace, woodworking and loudspeaker technology due to its physical and chemical properties. Today, mathematicians are working on derived structures of honeycomb to enhance its physical and chemical properties. In future, for more stable structures we will be able to derive some new graphs of honeycomb structure and will find their physical and chemical properties via topological indices.
Data Availability
The data that used for this research paper are included in this article.
Conflicts of Interest
The authors declare no conflicts of interest.