1. Introduction
Topological indices are very useful tools for chemists which are provided by Graph Theory. In a molecular graph, vertices denotes the atoms and edges are represented as chemical bonds in the terms of graph theory. To predict bioactivity of the chemical compounds, the topological indices such as ABC index, Wiener index, Randi index, Szeged index and Zagreb indices are very useful.
A graph is a tuple, which consists of the n-connected vertex set and the edge set ||. denotes the degree of a vertex ‘m’ in a graph . A graph can be represented by the polynomials, numeric numbers, a sequence of numbers, or a matrix. Throughout this article, all graphs examined are simple, finite, and connected.
As a chemical descriptor, the topological index has an integer attached to the graph which features the graph, and there is no change under graph automorphism. Previously, interest in the computing chemistry domain has grown in terms of topological descriptors and is mainly associated with the use of unusual quantities, the relationship between the structure property, and the relationship of the structure quantity. The topological indices that are based on distance, degree, and polynomials are some of the main classes of these indices. In a number of these segments, degree-based displayers are widely important and chemical graphs play an integral part in theory and theoretical chemistry.
In this article, we consider some important topological indices and some important derived graphs. We examine their chemical behavior by the help of topological indices. These topological indices are of use to chemists.
Chen et al. [
1] gleaned a hexagonal mesh which consists of triangles. Triangle graphs are called oxide graphs in terms of chemistry. We can construct a
hexagonal mesh by joining these triangles, as shown in
Figure 1. There does not exist any hexagonal mesh whose dimension equals 1. By the joining of six triangles, we make a hexagonal mesh of dimension 2,
(see
Figure 1 (1)). By putting the triangles around the all sides of
, we obtain hexagonal mesh of dimension 3,
(see
Figure 1 (2)). Furthermore, we assemble the nth hexagonal mesh by putting
n triangles around the boundary of each hexagon.
Drawing Algorithm of Third Type of Hex-Derived Networks
Step-1: For , we should draw a hexagonal mesh of dimension m.
Step-2: Draw a
graph in each subgraph of
and join all the vertices to the outer vertices of each
. The new graph is called an
(see
Figure 2) network.
Step-3: By
network, we can simply design
(see
Figure 3) and
(see
Figure 4).
In this paper, ‘’ is taken as a simple connected graph and the degree of any vertex is stands for .
The oldest, most desired and supremely studied degree-based topological index was introduced by Milan Randić and is known as
Randić index [
2] denoted by
and described as
The
Forgotten index, also called F-index, was discovered by Furtula and Ivan Gutman [
3] and described as
In 1982, Balaban [
4,
5] found another important index known as
Balaban index. For a graph
of ‘
n’ vertices and ‘
m’ edges, and is described as
The reclassified the Zagreb indices which are proposed by Ranjini et al. [
6], is of three types. For a graph
, it is described as
The atom-bond connectivity (ABC) index is a useful predictive index in the study of the heat of formation in alkanes [
7] and is introduced by Estrada et al. [
8].
Ghorbani et al. [
9] introduced the
index and is described as
Graovac et al. [
10] introduced the
index and is described as
2. Main Results
Simonraj et al. [
11] created the new network which is named as third type of hex-derived networks. Chang-Cheng Wei et al. [
12] found some topological indices of certain new derived networks. In this paper, we compute the exact results for all the above descriptors. For these results on different degree-based topological descriptors for a variety of graphs, we recommend [
13,
14,
15,
16,
17,
18,
19,
20]. For the basic notations and definitions, see [
21,
22].
2.1. Results for
In this part, the Forgotten index, Balaban index, reclassified the Zagreb indices, index, and index are under consideration for the third type of hex-derived network.
Theorem 1. Consider the third type of hex-derived network ; its Forgotten index is equal to Proof. Let
be the hex-derived network of Type 3,
shown in
Figure 2, where
. The hex derived network
has
vertices and the edge set of
is divided into nine partitions based on the degrees of end vertices as shown in
Table 1.
Forgotten index can be calculated by using
Table 1. Thus, from
, it follows,
After some calculations, we have the final result
□
In the subsequent theorem, we compute the Balaban index of the third type of hex-derived network, .
Theorem 2. For the third type of hex-derived network , the Balaban index is equal to Proof. Let
be the third type of hex-derived network
. The Balaban index can be calculated by using (3) and with the help of
Table 1, we have.
After some calculations, we have the result
□
Now, we compute , and indices of the third type of hex-derived network .
Theorem 3. Let be the third type of hex-derived network, then
= ,
= ,
= .
Proof. Reclassified Zagreb index can be calculated by using
Table 1, the ReZ
by using Equation (
4) as follows.
After some calculations, we have
The ReZG
can be calculated by using
as follows.
After some calculations, we have
The ReZG
index can be calculated from (6) as follows.
After some calculations, we have
□
Now, we find and indices of third type of hex-derived network .
Theorem 4. Let be the third type of hex-derived network, then
= ;
= .
Proof. The
index can be calculated by using
and by
Table 2, as follows.
After some calculations, we have
The
index can be determined from (8) as follows.
After some calculations, we have
□
2.2. Results for Third Type of Triangular Hex-Derived Network
Now, we discuss the third type of rectangular hex-derived network and compute exact results for Forgotten index and Balaban index, and reclassified the Zagreb indices, index, and index for .
Theorem 5. Consider the third type of triangular hex-derived network of ; its Forgotten index is equal to Proof. Let
be the third type of triangular hex-derived network,
shown in
Figure 3, where
. The third type of triangular hex-derived network
has
vertices and the edge set of
is divided into six partitions based on the degree of end vertices as shown in
Table 3.
By using edge partition from
Table 3, we get. Thus, from
it follows that
By doing some calculations, we get
□
In the following theorem, we compute the Balaban index of the third type of triangular hex-derived network, .
Theorem 6. For the third type of triangular hex-derived network , the Balaban index is equal to Proof. Let
be the third type of triangular hex-derived network
. By using edge partition from
Table 3, the result follows. The Balaban index can be calculated by using (3) as follows.
After some calculation, we have
□
Now, we compute , and indices of third type of triangular hex-derived network .
Theorem 7. Let be the third type of triangular hex-derived network, then
= ,
= ,
= .
Proof. By using edge partition given in
Table 3, the ReZG
can be calculated by using
as follows.
After some calculation, we have
The ReZG
can be calculated by using
as follows.
After some calculation, we have
The ReZG
index can be calculated from (6) as follows.
After some calculation, we have
□
Now, we compute and indices of third type of triangular hex-derived network .
Theorem 8. Let be the third type of triangular hex-derived network, then
= ;
= .
Proof. By using the edge partition given in
Table 4, the
index can be calculated by using
as follows.
After some calculation, we have
The
index can be calculated from (8) as follows.
After some calculation, we have
□
2.3. Results for Third Type of Rectangular Hex-Derived Network,
In this section, we calculate certain degree-based topological indices of the third type of rectangular hex-derived network, of dimension . We compute Forgotten index and Balaban index, and reclassified the Zagreb indices, forth version of index, and fifth version of index in the coming theorems of .
Theorem 9. Consider the third type of rectangular hex-derived network , its Forgotten index is equal to Proof. Let
be the third type of rectangular hex-derived network,
shown in
Figure 4, where
. The third type of rectangular hex-derived network
has
vertices and the edge set of
is divided into nine partitions based on the degree of end vertices as shown in
Table 5.
Thus, from
, it follows that.
Let
be the third type of rectangular hex-derived network,
. By using edge partition from
Table 5, the result follows.
After some calculation, we have
□
In the following theorem, we compute the Balaban index of the third type of rectangular hex-derived network, .
Theorem 10. For the third type of rectangular hex-derived network , the Balaban index is equal to Proof. Let
be the rectangular hex-derived network
. By using edge partition from
Table 5, the result follows. The Balaban index can be calculated by using (3) as follows.
After some calculation, we have
□
Now, we compute , and indices of the third type of rectangular hex-derived network .
Theorem 11. Let be the third type of rectangular hex-derived network, then
= ,
= ,
= .
Proof. By using the edge partition given in
Table 5, the ReZG
can be calculated by using
as follows.
After some calculation, we have
The ReZG
can be calculated by using
as follows.
After some calculation, we have
The ReZG
index can be calculated from (6) as follows.
After some calculation, we have
□
Now, we compute and indices of the third type of rectangular hex-derived network .
Theorem 12. Let be the third type of rectangular hex-derived network, then
= ;
= .
Proof. By using the edge partition given in
Table 6, the
can be calculated by using
as follows.
After some calculation, we have
The
index can be calculated from (8) as follows.
After some calculations, we have
□
The graphical representations of topological indices of these networks are depicted in
Figure 5 and
Figure 6 for certain values of
m. By varying the different values of
m, the graphs are increasing. These graphs show the correctness of the results.
3. Conclusions
The study of topological descriptors are very useful to acquire the basic topologies of networks. In this paper, we find the exact results for Forgotten index, Balaban index, reclassified the Zagreb indices, ABC index and GA index of the Hex-derived networks of type 3. Due to their fascinating and challenging features, hex-derived networks have studied literature in relation to different graph-ideological parameters. However, their developmental circulatory features have been read for the foremost in this paper.
We are also very keen in designing some new networks and then study their topological indices which will be quite helpful to understand their primary priorities.
Author Contributions
Software, M.A.B.; validation, M.K.S. writing—original draft preparation, H.A.; writing—review and editing, W.G.; supervision, M.K.S.; funding acquisition, W.G.
Funding
This work has been partially supported by National Science Foundation of China (11761083).
Conflicts of Interest
The authors declare no conflict of interest.
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