Recently, polymers of complex chemical\nconnectivity expressed with graphs have been synthesized in experiments [1-3]. It\nis indeed marvelous that even such polymers expressed with K_{3,3} bipartite graph have been produced [2]. We call\npolymers with nontrivial structures in chemical connectivity *topological polymers. We also call\npolymers with nontrivial topology of spatial graphs as embeddings in three\ndimensions topological polymers [4].*

The Rouse dynamics of a polymer\nplays an important role in the dynamical aspects of the polymer in dilute\nsolution, and also in melts if the molecular weight is small [5]. In this talk,\nwe formulate the Rouse dynamics of the topological polymer with a given graph. We\nderive several physical consequences of the model. In particular, we compare\nthe experimental data of Size Exclusion Chromatography (SEC) of some\ntopological polymers with their theoretical estimates of the mean-square radius\nof gyration obtained by the Gaussian method [6]. We argue physical backgrounds\nof SEC data and discuss how they are consistent.

\n\nAfter reviewing the method for\nconstructing Gaussian random configurations of a topological polymer, i.e.\nGaussian random graph embeddings [6], we derive the normal coordinates and modes\nfor the topological polymer. Here we remark that while the Moore-Penrose\ngeneralized inverse matrix has been addressed for general Gaussian molecules about\nthree decades ago [7], it seems that important consequences were found later\nindependently [8]. Moreover, it has not been known until quite recently how to\ngenerate Gaussian random configurations of a given topological polymer [6]. In\nfact, we can calculate any physical quantity at least numerically by taking the\nensemble averages over generated configurations. It is quite nontrivial to\ngenerate such random walks that satisfy the constraints of all independent loops\nin the graph.

\n\nThe results of this talk are\nobtained in collaboration with Jason Cantarella, Clayton Shonkwiler, and Erica\nUehara.

\n\n*1) \n Topological Polymer Chemistry:\nProgress of cyclic polymers in synthesis, properties and*

* functions*, Y. Tezuka ed., World\nScientific, Singapore, 2013.

2) \nT. Suzuki, T. Yamamoto and Y. Tezuka. *J. Am. Chem.\nSoc.*, 2014, **136**, 10148–10155.

3) \nY. Tezuka. *Acc. Chem. Res.*, 2017, **50, 2661–2672.**

4) \nE. Uehara and T. Deguchi, *J. Chem.\nPhys.* 2016, **145**, 164905.

5) \nM. Doi and S. F. Edwards, *The Theory\nof Polymer Dynamics*, Oxford University Press, Oxford, 1986.

6) \nJ. Cantarella, T. Deguchi, C. Shonkwiler and E. Uehara, in preparation.

\n\n7) \nB. E. Eichinger, *Macromolecules*,\n1980, **13**, 1-11.

8) E. Estrada and N. Hatano, *Chem.\nPhys. Let. *2010, **486**, 166–170.