On the resistance distance of helicene $k$-polygonal chain

Abstract In resistor networks, the resistance distance $r_G(u,v)$ represents the equivalent resistance between two nodes $u$ and $v$. This paper investigates a special polygonal chain structure---the helicene $k$-polygonal chain. Polygonal chains are of particular interest due to their rich geometric and topological properties, which play important roles in both theoretical studies and practical applications. Specifically, the helicene $k$-polygonal chain derived from ladder graphs belongs to the planar graph family. Its construction involves adding $k-4$ vertices to the upper side of each square in the ladder graph while leaving the lower side unmodified. Owing to the structural characteristics of helicene $k$-polygonal chains, their resistance distances hold important applications in communication network modeling, power grid analysis, chemistry, and materials science. We compute the resistance distances between arbitrary vertex pairs in helicene $k$-polygonal chains, where these resistances are determined through series connection configurations. Our analytical approach is based on electrical network theory, incorporating series-parallel resistance principles and $\Delta$-Y transformations.