An exact formula for the contraction factor of a subdivided Gaussian topological polymer

We consider the radius of gyration of a Gaussian topological polymer $G$ formed by subdividing a graph $G^{\mathrm{\prime }}$ of arbitrary topology (for instance, branched or multicyclic). We give a new exact formula for the expected radius of gyration and contraction factor of $G$ in terms of the number of subdivisions of each edge of $G^{\mathrm{\prime }}$ and a new weighted Kirchhoff index for $G^{\mathrm{\prime }}$. The formula explains and extends previous results for the contraction factor and Kirchhoff index of subdivided graphs.