1. Introduction
Chemical graph theory is a field of graph theory with molecular graphs obtained by representing chemical structures with graphs. Molecular descriptors or topological indices are numerically representative of a chemical structure and represented by a graph. They are used to predict the physical, biological, and chemical properties of new structures obtained from the molecule or molecular compound [
1,
2]. The first-known application was studied to find the physical properties of a chemical structure in 1947 [
3].
Molecular descriptors depend on the vertex degree of the graph, the distance between the two vertices, the eigenvalues of the graph, etc. Finding the topological polynomials instead of calculating the molecular descriptors one by one makes it easier to provide information about the molecular graph. M-polynomials are polynomials of molecular descriptors based on the vertex degree of a G graph and were defined by Deutsch and Klavžar [
4]. Using this definition, NM-polynomials are defined. These polynomials depend on the sum of adjacent vertex degrees [
5,
6].
The quality of a molecular descriptor is measured by its ability to successfully predict molecules. Kirmani et al. argued that the neighborhood second-modified Zagreb index is best to predict the index for the molar refractivity and polarizability properties of COVID-19 drugs [
6]. Havare argued that the neighborhood harmonic index is the best-predicting index for the molar volume of cancer drugs [
7]. NM-polynomials and these molecular descriptors for various chemical graphs were studied (see for detail [
8,
9,
10,
11]).
Cycle-related graphs are molecular graphs of many chemical structures in chemistry, such as cycloalkanes. It is also a graph representation of many networks [
12]. Sowmya studied total-eccentricity polynomials [
13]. Asif et al. computed the Mostar index of cycle-related chemical structures [
14]. Natarajan et al. computed leap-Zagreb indices of cycle-related special graphs [
15]. Basanagoud et al. studied the M-polynomials of cycle-related graphs [
16]. Havare computed the Mostar index and edge-Mostar index of cycle-related graphs [
17]. Javaraju et al. studied the reciprocal-leap indices of wheel graphs [
18].
2. Materials and Methods
If the
graph has
and
, then the numbers of these sets are defined as
and
, respectively. The degree of the vertex
of the
graph is denoted by
. The neighborhood of a vertex
in a graph
G,
), is the set of all vertices adjacent to
[
19]. See Reference [
19] for basic definitions and notations on graph theory. Let
and
.
The neighborhood M-polynomial is defined by Verma et al. [
20] as
Table 1 shows descriptors based on the sum of neighborhood degrees.
The following operators are used in
Table 1:
If and , then neighborhood degree sum-based molecular descriptors are obtained.
3. Results
In this section, NM-polynomials of the wheel, gear, helm, friendship, and flower graphs are obtained. Molecular descriptors based on the neighborhood degree sum using these polynomials are computed.
The wheel graph for
is obtained from the cycle with
orders and a central vertex
. The orders of the wheel graph are
and the size of it is
[
19].
Theorem 1. Letbe the wheel graph withorders. The neighborhood M-polynomial ofis Proof of Theorem 1. From the definition of the wheel graph, it is divided as
From Equation (1),
or
□
Corollary 1. The molecular descriptors based on the neighborhood degree of thegraph are
| - 5.
- 6.
- 7.
- 8.
|
Proof of Corollary 1. Using the formulas in
Table 1, the following equations can be written.
Using the above equations and
Table 1, the following equations are obtained.
The Helm graph is obtained by joining a pendant edge attached to each vertex of
of the wheel graph [
24]. The orders of the helm graph are
and the size of the helm graph is
. □
Theorem 2. Letbe the helm graph withorders.
Then,
for.
Proof of Theorem 2. From the definition of the helm graph, it is divided as
From Equation (1),
or
□
Corollary 2. The molecular descriptors based on the neighborhood degree of thegraph are
| - 5.
- 6.
- 7.
- 8.
|
Proof of Corollary 2. Using the formulas in
Table 1, the following equations can be written.
□
The gear graph is obtained by a vertex added between each pair of adjacent vertices of
of the wheel graph [
24]. The orders of the gear graph are
and the size of the helm graph is
.
Theorem 3. Letbe the helm graph withorders.
Then,
for.
Proof of Theorem 3. From the definition of the gear graph, it is divided as
From Equation (1),
or
□
Corollary 3. The molecular descriptors based on the neighborhood degree of thegraph are
| - 5.
- 6.
- 7.
- 8.
|
Proof of Corollary 3. Using the formulas in
Table 1, the following equations can be written.
□
The flower graph is obtained from a helm by combining the pendant vertices of the helm graph with the central vertex of the helm graph with one edge [
24]. The orders of the flower graph are
and the size of the flower graph is
.
Theorem 4. Letbe the flower graph withorders.
Then,
for.
Proof of Theorem 4. From the definition of
, it is divided as
From Equation (1),
or
□
Corollary 4. The molecular descriptors based on the neighborhood degree of thegraph are
| - 5.
- 6.
- 7.
- 8.
|
Proof of Corollary 4. Using the formulas in
Table 1, the following equations can be written.
□
The
friendship graph is obtained by deleting the alternate edges of the
of
[
24]. The orders of the friendship graph are
and the size of the friendship graph is
.
Theorem 5. Letbe the friendship graph withorders.
Then,
for.
Proof of Theorem 5. From the definition of
, it is divided as
From Equation (1),
or
□
Corollary 5. The molecular descriptors based on the neighborhood degree of thegraph are
| - 5.
- 6.
- 7.
- 8.
|
Proof of Corollary 5. Using the formulas in
Table 1, the following equations can be written.
□
Figure 1,
Figure 2 and
Figure 3 show the 3D plots of the NM-polynomial of the wheel, helm, gear, flower, and friendship graphs.
4. Discussion and Conclusions
This paper shows that the gear, helm, and friendship graphs have the same orders and size, but NM-polynomials are different from each other. In
Figure 4,
Figure 5 and
Figure 6, green * =
, blue − =
, blue * =
, red * =
, black * =
, red + =
, red − =
, and blue + =
.
If it is compared, topological indices depend on the sum of the neighboring degrees of the wheel graph; the
index has a very large value and creates a very fast curve, while
has the lowest value. Although the
index is close to the
index, it has larger values and creates a faster curve (see
Figure 4a).
Figure 4b shows the molecular descriptors based on the neighborhood sum degree of the
graph.
Figure 4b shows that
is the fastest curve-forming index. The index that produces values closest to zero is the mM index. If we order the indices from that with the largest value to that with the smallest value, the list would be as follows:
,
,
,
,
,
,
, and
.
For indices dependent on the sum of the adjacent vertex degrees of the gear graph, is the most curvilinear-growing, followed by . The most linear-growing is . If they are sorted in order from the index with the lowest value to the index with the highest value, the list would be as follows: , , , , , , , and .
If we list the indices for the flower graph, as the number increases, the index with the smallest value would be , , , , , , , and .
From
Figure 6, if the topological index values for
are ordered from smallest to the largest, the list would be as follows:
,
,
,
,
,
,
, and
.
For the NM-polynomials of the cycle-related graphs and the topological indices based on them, the index with the fastest increasing and the slowest increasing values is . These results can be used to predict the properties of new cycle-related molecules in QSPR/QSPR studies.