Abstract
Chemical descriptors are numeric numbers that contain a basic chemical structure and describe the structure of a graph. A graph’s topological indices are linked to its chemical characteristics. Biological activity of chemical compounds can be predicted using topological indices. Numerous chemical indices have been developed in theoretical chemistry, including the Zagreb index, the Randić index, the Wiener index, and many others. In this paper, we compute the exact results for the Randić, Zagreb, Harmonic, augmented Zagreb, atom-bond connectivity, and geometric-arithmetic indices for the Benzenoid networks theoretically.
1. Introduction and Preliminary Results
Topological indices, which are particularly useful tools for chemists, are provided by graph theory. In terms of graph theory, vertices represent atoms and edges indicate chemical bonding in a molecular graph [1]. Topological indices such as the index, Wiener index, Randić index, Szeged index, and Zagreb index are highly useful for predicting the bioactivity of chemical compounds.
A graph can be represented by polynomials, numeric numbers, a sequence of integers, or a matrix. All graphs are simple, finite, and connected. All graphs discussed in this article are simple, finite, and connected.
A topological index is a numerical quantity for the chemical graph and it is expressed through chemical graph theory. Interest in topological descriptors has already increased in the computer chemistry sector, and is mostly related with the use of unexpected quantities, the relationship between structure properties, and the relationship between structure quantities.
Topological indices based on distance, degree, and polynomials are some of the most popular forms [2]. Chemical graphs play an important part in theory and theoretical chemistry, and degree-based indices are often utilized in a number of these segments. In this article, we explore at some important topological indices and how they areused to assess benzenoid graphs’ chemical activity. Chemists can benefit from these topological indices.the
2. Construction for Benzenoid Planar Octahedron Networks
Step 1: consider a sheet oxide network [3] of dimension Step 2: then, place in each of oxide network Step 3: connecting alternating adjacent vertices of to each opposite vertex, the resultant graph is called benzenoid planar octahedron network Step 4: by using the previous algorithm, we can construct the benzenoid dominating planar octahedron network and the benzenoid hex planar octahedron networkWe defined to be a network with as a set of vertices and as a set of edges in this article, where is the degree of vertex .
The Estrada index is a graph-spectrum-based structural descriptor that was introduced in 2000 by Estrada and is defined as follows [4]:
In full resemblance with the Estrada index, Fath-Tabar et al. in [5] introduced the Laplacian Estrada index, which is formalised as follows:
Randić index [6] is an oldest degree-based topological index, denoted by , and was proposed by Milan Randić and is defined as follows:
The sum of over all the edges is general Randić index [6] and is defined as follows:
The Zagreb index, represented by and defined by Gutman and Das [7], is an important topological index:
Zhong [8] established the most important harmonic index, which is defined as follows:
The prominent topological index is augmented Zagreb index which was proposed by Furtula et al. in [9], and it is defined as follows:
The atom-bond connectivity (ABC) index, proposed by Estrada et al. in [10], is a prominent degree-based topological indicator that is defined as follows:
Another prominent topological index is the Geometric-arithmetic (GA) index, which was proposed by Furtula in reference [11] and described as follows:
3. Primary Results of Benzenoid Networks
In this article, the general Randić, first Zagreb, , , , and indices are studied and closed equations for these indices for the benzenoid planar octahedron networks are given. The and indices, also their derivatives, are now the subject of substantial research, see [12, 13] topological indices and their invariants in different graph families for more information.
3.1. Results for the Benzenoid Planar Octahedron Network
We construct some degree-based topological indices of the benzenoid planar octahedron network, denoted by , in this section. We calculate the general Randić for , first Zagreb, , , , and indices for benzenoid planar octahedron network in this section.
In the following theorem, we calculate the general Randić index for the benzenoid planar octahedron network.
Theorem 1. Let be the benzenoid planar octahedron network, then its general Randić index is equal to the following equation:
Proof. Let be the benzenoid planar octahedron network as shown in Figure 1, with and edge set divided into five divisions based on the degree of end vertices. The first edge partition has edges, having . The second edge division has edges, having and . The third edge division has edges, having and . The fourth edge division has edges, having and . The fifth edge division has edges, having . For The general Randić index formula from equation (4) is used as follows: We can achieve the following result by using Table 1 edge division: For The general Randić index formula from equation (4) is used as follows: We can achieve the following result by using the Table 1 edge division: For The general Randić index formula from equation (4) is used as follows: For The general Randić index formula from equation (4) is used as follows: The first Zagreb index of the benzenoid planar octahedron network is computed in the following theorem.
Theorem 2. For the benzenoid planar octahedron network , the first Zagreb index is equal to the following equation:
Proof. Let denote the bezenoid planar octahedron network. The following is the result of using the edge division from Table 1. As a result of equation (5), we haveWe get following result by doing calculation:
Theorem 3. Let be the benzenoid planar octahedron network ; then, we have
Proof. We get the required result by finding the edge division in Table 1, and then, applying the definition. It follows from equation (6) thatBy doing the calculation, we obtained the following result:Equation (7) can be used to compute the augmented Zagreb index as follows:By doing the calculation, we obtained the following result:Equation (8) can be used to compute the atom-bond connectivity index as follows:By doing the calculation, we obtained the following result:Equation (9) can be used to compute the geometric-arithmetic index as follows:By doing the calculation, we obtained the following result:
3.2. Results for the Benzenoid Dominating Planar Octahedron Network
We construct some degree-based topological indices of the benzenoid planar octahedron network, denoted by . We compute the general Randić for , first Zagreb, , , , and indices for benzenoid dominating planar octahedron network in this section.
We compute the general Randić index for benzenoid dominating planar octahedron network in the following theorem.
Theorem 4. Let be the benzenoid dominating planar octahedron network, and then, its general Randić index is equal to the following equation:
Proof. Let be the benzenoid planar octahedron network as shown in Figure 2, with and edge set divided into five divisions based on the degree of end vertices. The first edge division has edges, having . The second edge division has edges, having and . The third edge division has edges, having and . The fourth edge division has edges, having and . The fifth edge division has edges, having . For The general Randić index formula from equation (4) is used as follows: We can achieve the following results by using the edge division in Table 2. For The general Randić index formula from equation (4) is used as follows: We can achieve the following results by using the edge division in Table 2. For The general Randić index formula from equation (4) is used as follows: For The general Randić index formula from equation (4) is used as follows: The first Zagreb index of the benzenoid dominating planar octahedron network is computed in the following theorem.
Theorem 5. For the benzenoid dominating planar octahedron network , the first Zagreb index is equal to the following equation:
Proof. Let be the bezenoid dominating planar octahedron network. The following is the result of using the edge division from Table 2. As a result of equation (5), we haveBy doing the calculation, we obtained the following result:
Theorem 6. Let be the benzenoid dominating planar octahedron network ; then, we have
Proof. We get the required result by finding the edge division in Table 1, and then, applying the definition. It follows from equation (6) thatBy doing the calculation, we obtained the following result:Equation (7) can be used to compute the augmented Zagreb index as follows:By doing the calculation, we obtained the following result:Equation (8) can be used to compute the atom-bond connectivity index as follows:By doing the calculation, we obtained the following result:Equation (9) can be used to compute the geometric-arithmetic index as follows:By doing the calculation, we obtained the following result:
3.3. Results for Benzenoid Hex Planar Octahedron Network
We construct some degree-based topological indices of the benzenoid planar octahedron network, denoted by , in this section. We compute the general Randić for , , , , and indices for benzenoid hex planar octahedron network in this section.
We compute the general Randić index for benzenoid hex planar octahedron network in the following theorem.
Theorem 7. Let be the benzenoid hex planar octahedron network, then its general Randić index is equal to the following:
Proof. Let be the benzenoid hex planar octahedron network as shown in Figure 3, with and edge set divided into seven divisions based on the degree of end vertices. The first edge division has 12 edges, having and . The second edge division has edges, having . The third edge division has edges, having and . The fourth edge division has edges, having and . The fifth edge division has edges, having . The sixth edge division has edges, having and . The seventh edge division has edges, having . For The general Randić index formula from equation (4) is used as follows: We can obtain the following results by using the edge division in Table 3. For The general Randić index formula from equation (4) is used as follows: We can achieve the following result by using the edge division in Table 3. For The general Randić index formula from equation (4) is used as follows: For The general Randić index formula from equation (4) is used as follows: The first Zagreb index of the benzenoid hex planar octahedron network is computed in the following theorem.
Theorem 8. For the benzenoid planar octahedron network , the first Zagreb index is equal to the following equation:
Proof. Let be the bezenoid hex planar octahedron network. The following is the result of using the edge division from Table 3. As a result of equation (5), we haveBy doing the calculation, we obtained the following result:
Theorem 9. Let be the benzenoid hex planar octahedron network ; then, we have
Proof. We obtained the required result by finding the edge division in Table 3, and then, applying the definition. It follows from equation (6) thatBy doing the calculation, we obtained the following result:Equation (7) can be used to compute the augmented Zagreb index as follows:By doing the calculation, we obtained the following result:Equation (8) can be used to compute the atom-bond connectivity index as follows:By doing the calculation, we obtained the following result:Equation (9) can be used to compute the geometric-arithmetic index as follows:By doing the calculation, we obtained the following result:To compare topological indices numerically for , , and , we calculated all of the indices for different values of . Tables 4–6 clearly show that when the value of r increases, all indices increase in ascending order.
4. Conclusion
In this paper, we computed the required results of Randić, Zagreb, Harmonic, augmented Zagreb, atom-bond connectivity, and geometric-arithmetic indices for , , and . We also discovered all of the networks, numerical computations. These key insights lay the groundwork for understanding the underlying topologies of the following networks, which are useful from a variety of chemical and pharmaceutical perspectives. In the future, we want to create some networks and then analyse their topological indices to learn more about their underlying topologies.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.