Certain polynomials and related topological indices for the series of benzenoid graphs

A topological index of a molecular structure is a numerical quantity that differentiates between a base molecular structure and its branching pattern and helps in understanding the physical, chemical and biological properties of molecular structures. In this article, we quantify four counting polynomials and their related topological indices for the series of a concealed non-Kekulean benzenoid graph. Moreover, we device a new method to calculate the PI and Sd indices with the help of Theta and Omega polynomials.


series of Concealed Non-Kekulean Benzenoid Graph
The Kekulean and non-Kekulean structures in benzenoids have important properties from a chemical point of view. It is known that most benzenoids with different numbers of starred and unstarred vertices have no Kekulean structure; these possess color excess and are referred to non-Kekulean benzenoids. In contrast, benzenoids with equal numbers of starred and unstarred vertices necessarily possess Kekulean structures. According to Gutman 19 , equal numbers of starred and unstarred vertices is a necessary and sufficient condition for a benzenoid structure to be Kekulean. However, it is not true that non-Kekulean with equal numbers of starred and unstarred vertices were detected and later identified as concealed non-Kekulean benzenoids. The total number of edges in the series of concealed non-Kekulean benzenoid graphs shown in Figure 1.1 is 17n + 14, where n is the number of connected edges in the middle of the graph. It has been demonstrated that exactly eight systems of this category exist. If we eliminate the edge cut, which consists of the connected edges, then the graph is decomposed into two parts. Such a structure is called the Kekulean structure of the benzenoid graph. In the present work, we use the concealed non-Kekulean benzenoid graph shown in Fig. 1 [19][20][21][22][23] .
The series of concealed non-Kekulean benzenoid graphs in Fig. 1 has six quasi-orthogonal cuts (i.e., S i i = 1, 2 … 6) of different lengths. The lengths of the cuts (qoc) and the number of cuts in the series of concealed non-Kekulean benzenoid graphs are n, n + 1, n + 2, 2, 3, and 6 and 1, 2, 2, 4, 4, and 2(n − 1), respectively. Proof : To calculate the Omega and Theta polynomials of the concealed non-Kekulean benzenoid graphs shown in Figure 1.1, we need to find both the quasi-orthogonal cuts (qoc) and the number of quasi orthogonal cuts of each type. Let S i , where i = 1, 2 … 6 be six 'qoc' in a concealed non-Kekulean benzenoid graph. The lengths and cardinalities of these quasi-orthogonal cuts (i.e., S i , where i = 1, 2 … 6) are n, n + 1, n + 2, 2, 3, and 6 and 1, 2, 2, 4, 4, and 2(n − 1), respectively. Because the Omega polynomial is defined as, ω (G, where c is the length of the cut and m (G, c) represents the number of quasi-orthogonal cuts of length c. Hence, the Omega polynomial calculated from the qocs of a concealed non-Kekulean benzenoid graph is: Also, by the definition of the Theta polynomial, Hence, the Theta polynomial calculated from the cuts of a concealed non-Kekulean benzenoid graph becomes  Proof : In the series of concealed non-Kekulean benzenoid graphs, the total number of edges is |E (G)| = 17n + 14, and there are six strips (of qocs) of different lengths, namely, S i, i = 1, 2 … 6. The lengths of these strips are n, n + 1, n + 2, 2, 3, and 6, respectively. The cardinality of the length of S i, i = 1, 2 … 6 is1, 2, 2, 4, 4, and 2(n − 1). From the definition of the Sadhana polynomial, sd (G, . Therefore, the Sadhana polynomial constructed from the cuts of a concealed non-Kekulean benzenoid graph is: Additionally, the π Polynomial is defined as π (G, . By using the lengths of the strips and the number of strips in a concealed non-Kekulean benzenoid graph, the π polynomial becomes;

topological Indices for the series of Concealed Non-Kekulean Benzenoid Graph
The numerical value of the first derivatives of these counting polynomials at x = 1 yields the interesting properties of the molecular graph. These values are called the topological indices of the graphs. At x = 1, the value of the first derivative of the Omega polynomial gives the total number of edges of the graph, and at x = 1, the Theta polynomials give the same result. The relations for the topological indices related to these polynomials are as follows: x c 1 2 The following Table 2. shows the Omega, Theta, PI, and Sadhana indices calculated from their related polynomials.
In 24 , John et al. proposed the following formulae to calculate the PI index in terms of the Omega and Theta indices by considering relations (1) and (2).

Results
The Omega and Theta polynomials count the equidistant edges of the graph, while the Sadhana and PI polynomials count the nonequidistant edges of the graph. These polynomials help researchers discuss and predict the molecular structure without necessarily having to refer to quantum mechanics. Hence, we sum up this paper with the following results: