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Polynomials with an exponential equivalent to the extent of a characteristic partition and coefficients proportional to the multiplicity/occurrence of the accompanying partition are termed as counting polynomials. These polynomials are a well-known method for describing a chemical graph's molecular invariants as polynomials. It is possible to deduce several key topological invariants from polynomials by either directly taking their value at a certain point or by calculating derivatives or integrals of the polynomial. A topological invariants is a real number correlated with a network that predicts the physico-chemical properties. Chemical modeling, drug design, and structural activity relations use invariants. This paper aims to find the counting polynomials such as Sadhana, omega, PI and theta polynomial of certain graphene nanostructures. In furthermore, topological invariants of specific graphene nanostructures that are related to these counting polynomials are investigated. [ABSTRACT FROM AUTHOR]

Tessellations of antikekulenes are of interest from the standpoints of antiaromaticity, ring currents, unusual π$$ \pi $$‐electron characteristics, magnetism and their highly reactive nature. Although precursors of antikekulenes have been synthesized, up to now synthesis of antikekulenes has been proven to be formidably difficult. Yet they are predicted to have intriguing properties and their antiaromatic nature together with a combination of four and six‐membered rings makes them intriguing candidates for theoretical indices. In the present study, we have computed the topological indices of antikekulene by employing graph‐theoretical cut methods that simplify the parent tessellations of antikekulenes into simpler graphs. We have derived exact mathematical expressions of several topological descriptors for hexagonal and rectangular antikekulene structures. Furthermore we have obtained the graph‐spectral properties, algebraic structure counts, resonance stabilization and delocalization energies as well as nuclear magnetic resonance and electron spin resonance spectroscopic patterns of the two tessellations of antikekulenes. [ABSTRACT FROM AUTHOR]

Polya was the first to establish the concept of a counting polynomial in chemistry. Despite the fact that the spectra of the characteristic polynomial of graphs were exhaustively researched by quantitative means in order to ascertain the molecular orbitals of unsaturated hydrocarbons, the field garnered little attention from chemists for several decades. Polynomials can be used to produce a variety of significant topological indices, either explicitly or after calculating derivatives or numerical methods. Certain firmly related polynomials, namely Theta, Omega, PI, and Sadhana polynomials, subordinate the equidistant edges and non-equidistant edges of structural graphs, which have a rich application in computing the corresponding topological indices. A topological index is a numerical value associated with a network that predicts chemical, physical, and biological activities. This article investigates these polynomials for three classes of polycyclic aromatic hydrocarbons as hexa-peri-hexabenzocoronene, hexa-cata-hexabenzocoronene, and dodeca-benzo-circumcoronene. In addition, closed formulas for these polynomials' corresponding indices are proposed. [ABSTRACT FROM AUTHOR]

Chemical and biological properties of polycyclic aromatic compounds rely intimately upon their structures such as the fundamental molecular topology. In this way, quantitative structure-activity and property relationship (QSAR/QSPR) strategies have been formulated for predicting properties of polycyclic benzenoid hydrocarbons and related graphs from their chemical structures. In this paper, we give the exact analytical expression of three types of polycyclic aromatic compounds where the base molecule of these structure contains hexabenzocorenene (HBC). Furthermore, we research the results by MATLAB and obtain the relationship of the indices which they portray the physcio-chemical properties. [ABSTRACT FROM AUTHOR]

Conjugated aromatic macrocycles are engaging due to their remarkable photophysical and optoelectronic properties. Specifically, the cyclic radially-oriented π-arrangement of cycloparaphenylenes (CPPs) offers to ascend to photophysical properties dissimilar to small particle or carbon nanomaterial. A couple of sizes of CPPs, CPP derivatives, and related carbon nanorings have been integrated and their entrancing properties have gotten uncovered. Structure-based topological indices of these frameworks empower expectation of chemical properties and the bioactivities of these compounds through quantitative structure-activity and structure-property connections strategies. In this article, we derive the mathematical expression of CPP and its side wall contains pyrene and hexabenzocorenene. [ABSTRACT FROM AUTHOR]

Innovative discovery in bottom‐line research and industrial utilization of graphene material has made its mass and low‐cost creation an essential advance. This one‐atom thick luminous of carbon accumulates a set of different physico‐chemical properties, going from its extraordinary mechanical conduct to its uncommon electrical and warm conductivities, which are making graphene a genuine choice to trade numerous customary materials for different applications. Hexabenzocoronene (HBC) is one of the model frameworks for graphene. Because of extremely solid π−π stacking cooperations prompting a columnar plan inside the HBC crystal the subsequent material is as insoluble as graphene. In this paper, we derive the explicit analytic expressions of distance and degree‐based topological indices of super‐polycyclic aromatic compounds with HBC as a base molecule, such as superphenalene and supertriphenylene. [ABSTRACT FROM AUTHOR]

Let G (V, E) be a Graph. A set W ⊆ V of vertices resolves a graph G if every vertex of G is uniquely determined by its vector of distances to the vertices in W. The metric dimension of G is the minimum cardinality of a resolving set. By imposing different conditions on W we get conditional resolving sets. A resolving set W is said to be an independent resolving set if W contains isolated vertices. Independent resolving number denoted by ir(G) is referred to its cardinality. In this paper we investigate independent resolving number for Titanium dioxide Nanotube. [ABSTRACT FROM AUTHOR]