  Topological indices are graph invariants numeric quantities, which are
utilized by researchers to analyze a variety of physiochemical aspects of
molecules. The goal of developing topological indices is to give each chemical
structure a numerical value while maintaining the highest level of
differentiation. Using these indices, the classification of various structures,
and their physiochemical and biological properties can be predicted. In this
paper, the leap and leap hyper Zagreb indices, as well as their polynomials for
a zigzag benzenoid system $Z_{p}$ and a rhombic benzenoid system $R_{p}$ are
determined. In addition, new $k$-distance degree-based topological indices such
as leap-Somber index, hyper leap forgotten index, leap $Y$ index, and leap $Y$
coindex are also computed for the molecular graphs of $Z_p$ and $R_p$.
Furthermore, their numerical computation and discussion are performed to
determine the significance of their physiochemical properties.
