Fuzzy Sets and Their Applications in MathematicsView this Special Issue
Study of Neighborhood Degree-Based Topological Indices via Direct and NM-Polynomial of Starphene Graph
Our objective is to compute the neighborhood degree-based topological indices via NM-polynomial for starphene. In the neighborhood degree-based topological indices, we compute the third version of the Zagreb index; neighborhood second Zagreb index; neighborhood forgotten topological index; neighborhood second modified Zagreb index; neighborhood general Randic index; neighborhood harmonic index; neighborhood inverse sum index; first, second, third, fourth, and fifth NDe indices; fourth atom bond connective index; fifth geometric arithmetic index; fifth arithmetic-geometric index; fifth hyper-first and second Zagreb index; general first neighborhood index; and Sanskruti index. These neighborhood topological indices are computed both direct and via the NM-polynomial approach.
Graph theory deals with the study of lines and points. It is the division of mathematics which is concerned with graphs (structures that include points and lines and which frequently pictorially denote mathematical facts). Graph theory is the study of the relationship between edges and vertices. Formally, a graph is a pair where and are the vertex set and edge set of a graph , respectively. A simple graph is a graph that has no loops and multiple edges.
The total number of edges incident to is known as the degree of and is denoted by. Here, denotes the degree sum of neighbors of in. By neighbors of a vertex, we mean the vertices adjacent to that vertex.
Various physical properties, chemical reactivity, and biological activities of a chemical molecule are strongly connected to its graphical structure, and this fact is the main topic of interest in chemical graph theory. The topological index plays a key role in predicting such a connection without involving a wet lab. A topological index is a function from the collection of graphs to the set of real numbers that describe the topology of the graph and are used in QSPR/QSAR analysis. It remains unchanged for isomorphic graphs. Thousands of indices are developed in the literature on chemical graph theory.
A molecular graph is a graph such that its vertices represent the atoms and the edges of the bonds. Chemical graph theory is a branch of mathematical chemistry whose focus of interest is on finding topological indices of a molecular graph that correlate well with the chemical properties of the chemical molecules. Several topological indices have been considered in theoretical chemistry and have found some applications, especially in QSPR/QSAR.
Topological indices are numerical parameters associated with a graph that characterize its topology. These indices are usually graphed invariant. The topology of chemical structures is described by these indices . To make the computation of neighborhood degree-based topological indices easier,  introduced neighborhood M-polynomial whose role for neighborhood degree-based indices is parallel to the role of the M-polynomial for degree-based indices. NM-polynomial is another correct method to prove the neighborhood degree-based topological indices.
A starphene can be considered as a structure obtained by fusing three linear polyacenes of length , and , respectively. Starphenes are widely used in many electronic devices and played a key role in the revolution of miniaturization of electronic devices. I want to explore the structure of starphene by neighborhood degree-based topological indices by direct and NM-polynomial .
For a graph , neighborhood degree-based topological invariants are defined as
By counting edges that have the same end degrees in the chemical graph, then we can rewrite equation (1) as where the relation is satisfied and is the total count of edges of the graph .
In 2021, Mondal et al. introduced the indices :
Ghorbani and Hosseinzadeh defined the third version of the Zagreb index in 2013 .
Mondal et al. introduced the neighborhood second Zagreb index in 2019 .
Verma and Mondal defined the neighborhood second modified Zagreb index in 2019 .
Mondal et al. introduced the neighborhood forgotten topological index in 2019 .
Verma and Mondal defined the neighborhood general Randic index in 2019 .
Verma and Mondal defined the neighborhood harmonic index in 2019 .
Verma and Mondal defined the neighborhood inverse sum index in 2019 .
Ghorbani and Hosseinzadeh present in 2010 the fourth atom bond connectivity index as 
The fifth geometric arithmetic index was proposed by Grovac et al. in 2011 and defined as 
Kulli introduced the fifth arithmetic geometric index in 2017 and defined it as 
Kulli  proposed the fifth hyper-first and second Zagreb index in 2017 and defined it as
Hosamani proposed the Sanskruti index in 2020 .
Verma and Mondal defined the neighborhood M-polynomial in 2019 [7, 11].
3. Induced Neighborhood Degree-Based Topological Indices via NM-Polynomial
Some operators which are used in the above table are defined as In Table 1, we have computed Neighbourhood degree dependent topological indices via NM-polynomial.
4. Starphene Graph
Starphenes are polycyclic aromatic hydrocarbons and build by three different acene arms. Starphenes are the basic building blocks for the miniaturization of different especially organic electronic devices. It also played an important role in different logical gates. A starphene shown in Figure 1 can be considered as a structure obtained by fusing three linear polyacenes of length , and , respectively. The edge partition of is shown in 2. A starphene , shown in Figure 1 can be considered as a structure obtained by fusing three linear polyacenes of length , , and , respectively. The edge partition of is shown Table 2.
To compute the edge partitions, first, we count the number of vertices and edges of the molecular graph and find the degree and neighborhood degree of the vertices and then, we count the edges that have a specific neighborhood vertex degree. First, we compute the smallest dimension and then generalize.
5. NM-Polynomial of Starphene Graph
In this section, we compute the NM-polynomial of .
Theorem 1. If starphene is denoted by , then for ,, , NM-polynomial of is
Proof. Let represent the starphene; then, by using Figure 1 and Table 2, we have the following edge partition of which is The following result was obtained by using the definition of NM-polynomial:
The plot of NM-polynomial of is shown in Figure 2.
6. Topological Indices of Starphene
In this section, we calculate a few topological indices by direct formulae for .
Theorem 2. Let be a starphene, and then, (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)
Proof. (1)First NDe index (2)Second NDe index (3)Third NDe index (4)Fourth NDe index (5)Fifth NDe index (6)Third version of Zagreb index (7)Neighborhood second Zagreb index (8)Neighborhood second modified Zagreb index (9)Neighborhood forgotten topological index (10)Neighborhood general Randic index (11)Neighborhood harmonic index (12)Neighborhood inverse sum index (13)Fourth atom bond connectivity index (14)Fifth geometric arithmetic index (15)Fifth arithmetic geometric index (16)Fifth hyper-first Zagreb index (17)Fifth hyper-second Zagreb index (18)Sanskruti index
7. Topological Indices of Starphene via NM-Polynomial
In this section, we calculate some topological indices via NM-polynomial, computed in Section 5, of .
Theorem 3. Let be a starphene and then