This paper investigates a novel class of rounded knot networks formed by incorporating Möbius type helical linkages into phenylene–quadrilateral lattices. For the graph family \(G_{2n}\) , we establish general formulas for its structural parameters and develop a spectral framework that leverages graph automorphisms to block–diagonalize the Laplacian matrix into two tridiagonal Toeplitz components. This decomposition enables closed–form derivations of important network invariants, including the Kirchhoff index, mean first passage time, and the number of spanning trees, using Vieta’s relations and the Matrix-Tree Theorem. Theoretical analysis reveals cubic and exponential growth behaviors for these invariants, reflecting efficient diffusion dynamics and enhanced structural robustness of the network. To the best of our knowledge, this is the first comprehensive spectral and random–walk analysis of a helical Möbius–linked phenylene–quadrilateral network, providing new insights and analytical tools for modeling transport and reliability in chemically inspired helical structures.
