On the resistance distance of helicene k-polygonal chain

In resistor networks, the resistance distance rG(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 Δ-Y transformations.