The pent–heptagonal carbon nanosheets (CNSs) admit significant industrial applications. We shall compute the distance-based Hosoya polynomial of CNSs including [Formula: see text] and [Formula: see text]. The first-, second- and third-order derivatives of Hosoya polynomial, respectively, lead to the Wiener, hyper-Wiener and Tratch–Stankevitch–Zefirov (TSZ) indices which are the distance-based topological indices. These indices have many physical applications and can be correlated to physio-chemical properties of chemical structures. Results exhibit that despite the fact that both sheets have the same chemical formula, they have different Hosoya polynomials and related distance-based topological indices, resulting in distinct quantitative structure–property relation (QSPR) and quantitative structure–activity relation (QSAR) due to configuration differences.

Chemical structures are mathematically modeled using chemical graphs. The graph invariants including algebraic polynomials and topological indices are related to the topological structure of molecules. Hosoya polynomial is a distance based algebraic polynomial and is a closed form of several distance based topological indices. This article is devoted to compute the Hosoya polynomial of two different atomic configurations (C4C8(R) and C4C8(S)) of C4C8 Carbon Nanosheets. Carbon nanosheets are the most stable, flexible structure of uniform thickness and admit a vast range of applications. The Hosoya polynomial is used to calculate distance based topological indices including Wiener, hyper Wiener and Tratch–Stankevitch–Zafirov Indices. These indices play their part in determining quantitative structure property relationship (QSPR) and quantitative structure activity relationship (QSAR) of chemical structures. The three dimensional presentation of Hosoya polynomial and related distance based indices leads to the result that though the chemical formula for both the sheets is same, yet they possess different Hosoya Polynomials presenting distinct QSPR and QSAR corresponding to their atomic configuration.