Investigating Signless Laplacian Spectra and Network Topology in Helical Phenylene-Quadrilateral Structures

This study investigates the spectral and topological properties of rounded knot networks K2n, a helical extension of phenylene quadrilateral structures, through signless Laplacian spectral analysis. Motivated by the need to understand how helical topology influences network dynamics and robustness, we derive exact analytical expressions for three key invariants: the Kirchhoff index RK2n, mean first-passage time TK2n, and spanning tree count τK2n. By decomposing the signless Laplacian matrix into symmetric tridiagonal blocks Q+Γ and Q−Γ, we obtain closed-form eigenvalue solutions governed by helical periodicity following 2±32n scaling. The results reveal that the Kirchhoff index grows approximately as On2, indicating enhanced global connectivity with network expansion, while the mean first-passage time scales linearly with the network order, reflecting efficient diffusion behavior. The spanning tree enumeration shows exponential growth proportional to 2+32n, demonstrating high structural redundancy and robustness. These findings establish precise mathematical links between helical topology, transport efficiency, and network reliability, offering potential implications for analyzing biopolymer helices and designing nanomaterials with tunable conductivity and resilience.