From graph Laplacian to star polymer dynamics: part II

Abstract This paper extends the graph-Laplacian formulation of bead–spring polymer dynamics developed in Part I from ring polymers to star and branched polymer architectures. The three-arm star polymer is considered first as the simplest branched molecule. Starting from the overdamped Langevin equations for the four beads, the equations of motion are written in terms of the graph Laplacian of the star graph K 1,3 . For identical bead frictions and identical Hookean spring constants, the Laplacian spectrum consists of one translational eigenvalue, two degenerate arm-asymmetry eigenvalues, and one symmetric radial eigenvalue. The internal dynamics therefore contain two distinct relaxation times, and the polymer stress is shown to be equivalent to a two-mode upper-convected Maxwell model, with the modal multiplicities determining the corresponding stress weights. The longest relaxation time is compared with that of an open three-segment Rouse chain having the same number of beads and springs, demonstrating directly how branching modifies the spectral gap and accelerates the slowest relaxation. The analysis is then generalized to an f -arm star polymer, for which the spectrum of K 1, f yields one radial mode and an ( f − 1)-dimensional degenerate arm-asymmetry subspace. Unequal spring stiffnesses and unequal Stokes frictions are subsequently incorporated through a weighted graph Laplacian and a generalized eigenvalue problem in the friction metric. In this case, the modal degeneracy is generally removed and the constitutive response becomes a genuine multimode upper-convected Maxwell model. These results establish a direct connection between branching topology, graph spectrum, relaxation dynamics, and macroscopic constitutive behavior, and provide a systematic continuation of the topology–spectrum–constitutive framework.