The polygonal cylinder and its Hosoya polynomial

We introduce a polygonal cylinder \( C_{m,n} \), using the Cartesian product of paths \( P_m \) and \( P_n \) and using topological identification of vertices and edges of two opposite sides of \( P_m \times P_n \), and give its Hosoya polynomial, which, depending on odd and even \( m \), is covered in seven separate cases.