Certain polynomials and related topological indices for the series of benzenoid graphs (original) (raw)

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. Graph theory has numerous applications in modern chemistry. In chemical graph theory, the vertices and edges respectively represent the atoms and bonds of a molecular structure. To predict the chemical structure using numerical quantity (i.e., topological indices) graph theory plays a vital role. Topological indices have many applications in theoretical chemistry, especially in QSPR/QSAR research. Numerous researchers have conducted studies on topological indices for different graph families; these indices have important chemical significance in the fields of chemical graph theory, molecular topology, and mathematical chemistry. Diudea was the first chemist to consider the subject of computing the topological indices of nanostructures 1-3 . A small particle of an object of intermediate size (between the microscopic and molecular structures of the object) is called a nanoparticle of that object. Nanoparticles are products derived through engineering at the molecular scale. Let G (V, E) be a connected graph with a vertex set V and an edge set E. For any two vertices v 1 and v 2 in G, the distance between v 1 and v 2 is denoted by d (v 1 , v 2 )-the shortest path between v 1 , and v 2 . If e is the edge formed by joining v 1 and v 2 , and f is an edge formed by joining v 3 and v 4 , then e = v 1 v 2 and + 1 and is denoted by 'e co f' . Here, the corelation is symmetric and reflexive but not transitive. Let C (e) = {f ∈ E (G); f co e}: if the 'co' relation is transitive, then the set C (e) is called the orthogonal cut and denoted by co of G. The set of opposite edges that lie along the same face or the same ring, eventually forming a strip of adjacent faces or rings, is called an opposite edge strip and denoted by 'ops' . This concept is also termed a quasi-orthogonal cut, denoted by 'qoc' . Here, the co distance edges are defined within the entire graph G, while 'ops' are defined in the same face or ring. By m (G, c), we mean the number of strips of length c. In this paper, we constructed four polynomials: Omega, Sadhana, Theta and PI. Counting polynomials are those polynomials whose exponent is the extent of a property partition and whose coefficients are the multiplicity of the corresponding partition. We also calculated the topological indices related to these polynomials and formulae. Each counting polynomial represents interesting topological properties of the molecular graph. These polynomials are constructed on the basis of quasi-orthogonal cut edge strips for the series of concealed non-Kekulean benzenoid graphs. The counting polynomials and matching polynomials are useful for topologically describing bipartite structures as well as for counting some single-number descriptors (i.e., the topological indices). The Omega and Theta polynomials count equidistant edges of the graph, while the Sadhana and PI polynomials count nonequidistant edges. Various results related to counting polynomials and topological indices can be found in . The Omega polynomial of a graph G (V, E) is denoted byω (G, x); more information can be found in . The Omega polynomial is defined as ω = ∑ G x m G c x s ( , ) ( , )